In this paper, we shall discuss the existence and multiplicity of non-negative solutions for the following nonlinear boundary value problem

$ \begin{eqnarray} \label{eq.1.1} &-\Delta_{\mathbb{H}, p}u = \lambda f(\xi)|u|^{p-2}u+g(\xi)|u|^{r-2}u \quad \text{in }\Omega\, ; \end{eqnarray} $

(1.1)

$ \begin{eqnarray} &u(\xi) = 0 \quad \text{on }\partial \Omega \, , \label{eq.1.2} \end{eqnarray} $

(1.2)

where $\Omega$ is a bounded region with smooth boundary in $\mathbb{H}^N$, $1 <p <r <\frac{pQ}{Q-p}$, $\lambda > 0$ is a real parameter and $f, g: \Omega \to \mathbb{R}$ are given functions which change sign on $\Omega$, i.e. $f, g$ are indefinite weight functions. We assume that $f(\xi), ~ g(\xi)\in L^{\infty}(\Omega)$, $\{ u \, \in \mathcal{D}^{1, p}_0(\Omega):\displaystyle\int_\Omega f(\xi) |u|^p d\xi > 0 \}\neq \emptyset$ and $\{u\, \in\mathcal{D}^{1, p}_0(\Omega):\displaystyle\int_\Omega g(\xi) |u|^r d\xi > 0 \}\neq \emptyset$.

Problems (1.1)–(1.2) are studied in connection with the corresponding eigenvalue problem for the $p$-sub-Laplacian

$ \begin{equation}\left\{ \begin{array}{ll}\label{eq.1.3} -\Delta_{\mathbb{H}, p}u = \lambda f(\xi)|u|^{p-2}u & \quad \text{in } \Omega\, ; \\ \displaystyle u=0 & \quad \text{on }\partial \Omega \, .\\ \end{array} \right. \end{equation} $

(1.3)

Over the recent past decade, several authors used the Nehari manifold and fibering maps (i.e., maps of the form $t \to J_{\lambda}(tu)$, where $J_{\lambda}$ is the Euler function associated to the equation) to solve semilinear and quasilinear problems (see [3-9, 12]). By the fibering method, Drabek and Pohozaev [9], Bozhkov and Mitidieri [12] studied respectively the existence of multiple solutions to the following $p$-Laplacian equation

$ \begin{equation}\left\{ \begin{array}{ll}\label{eq.1.4} -\Delta_{p}u = \lambda f(x)|u|^{p-2}u+g(x)|u|^{r-2}u & \quad \text{in } \Omega\, ; \\ \displaystyle u=0 & \quad \text{on }\partial \Omega \, .\\ \end{array} \right. \end{equation} $

(1.4)

In [6], from the viewpoint of the Nehari manifold, the authors studied the following subcritical semilinear elliptic equation with a sign-changing weight function

$ \begin{equation}\left\{ \begin{array}{ll}\label{eq.1.5} -\Delta u = \lambda f(x) u+g(x)|u|^{r-2}u & \quad \text{in } \Omega\, ; \\ \displaystyle u=0 & \quad \text{on }\partial \Omega , \\ \end{array} \right. \end{equation} $

(1.5)

where $2 <r <\frac{2N}{N-2}$, $\lambda$ is constant, and $f(x), g(x)$ are smooth functions which may change sign in $\Omega$. Exploiting the relationship between the Nehari manifold and fibering maps, they gave an interesting explanation of the well-known bifurcation result. In fact, the nature of the Nehari manifold changes as the parameter $\lambda$ crosses the bifurcation value. In [8], the author dealt with the similar problem for the case $1 <r <2$ and discussed the existence and multiplicity of non-negative solutions of (1.5) from a variational viewpoint making use of the Nehari manifold.

The Dirichlet problems (1.1)-(1.2) on the Heisenberg group is a natural generalization of the classical problem on $\mathbb{R}^N$, see [6-11] and their references. It is well known that (1.4) and (1.5) are counterparts of (1.1)-(1.2) in $\mathbb{R}^N$. In this work, we use a variational method which is similar to the fibering method (see [9]) to prove the existence and multiplicity of positive weak solution for problems (1.1)-(1.2), particularly, by using the method of [6].

This paper, except for the introduction, is divided into four sections. In Section 2, we firstly recall some basic facts and necessary known results on the Heisenberg group, and then we consider the eigenvalue problem (1.3). In Section 3, we focus on the Nehari manifold and the connection between the Nehari manifold and the fibrering maps. In Section 4, we discuss the Nehari manifold when $\lambda <\lambda_1(f)$ and show how the behaviour of the manifold as $\lambda\rightarrow\lambda_1^-(f)$ depends on the sign of $\displaystyle\int_{\Omega } g(\xi)\phi_1^{r}\, d\xi$. In Section 5, using the properties of Nehari manifold we give simple proofs of the existence of two positive solutions.

Let $\xi=(x_1, \cdots, x_N, y_1, \cdots, y_N, t)=(x, y, t)=(z, t)\in \mathbb{R}^{2N+1}$ with $N\ge1$. The Heisenberg group $\mathbb{H}^{N}$ is the set $\mathbb{R}^{2N+1}$ equipped with the group law

$ \begin{equation*} ( x, y, t)\circ( x', y', t')=( x+x', y+y', t+t'+2\left( \langle {x}', \ y \right\rangle -\left\langle x, \ {y}' \right\rangle ))\, , \end{equation*} $

where $\left\langle \cdot , \cdot \right\rangle$ denotes the inner product in $\mathbb{R}^{N}$. This group multiplication endows $\mathbb{H}^N$ with a structure of a Lie group. A family of dilations on $\mathbb{H}^{N}$ is defined as $\delta_{\tau}(x, y, t)=(\tau x, \tau y, \tau^2t), $ $\tau > 0.$ The homogeneous dimension with respect to dilations is $Q=2N+2$. The sub-Laplacian $\Delta_\mathbb{H}$ is obtained from the vector fields $X_i=\partial_{x_i}+2y_i\partial_t$, $Y_i=\partial_{y_i}-2x_i\partial_t$, $i=1, \cdots , N$, as

$ \begin{equation} \Delta_\mathbb{H}:=\nabla_\mathbb{H}\cdot\nabla_\mathbb{H}=\sum\limits_{i=1}^N X_i\circ X_i + Y_i\circ Y_i, \end{equation} $

(2.1)

i.e.,

$ \begin{equation} \Delta_\mathbb{H}=\sum\limits_{i=1}^N\partial_{x_i}^2+\partial_{y_i}^2+4y_i\partial_{x_i}\partial_t -4x_i\partial_{y_i}\partial_t+4(x_i^2+y_i^2)\partial_t^2, \end{equation} $

(2.2)

where $\nabla_\mathbb{H}$ is the $2n$-vector $(X_1, \cdots, X_N, Y_1, \cdots, Y_N)$.

For $p>1$, the sub-$p$-Laplacian $\Delta_{\mathbb{H}, p}$ is defined as

$ \begin{equation} \Delta_{\mathbb{H}, p}:=\nabla_\mathbb{H}(|\nabla_\mathbb{H}|^{p-2}\nabla_\mathbb{H}u)\, . \end{equation} $

(2.3)

For more details concerning the Heisenberg group, see [1, 2].

The space $\mathcal{D}_0^{1, p}(\Omega)$ is defined as the closure of $C_0^{\infty}(\Omega)$ under the norm

$ \|u \| = \left(\displaystyle\int_{\Omega} | \nabla_\mathbb{H} u | ^p d\xi \right) ^{\frac1{p}}. $

For notational convenience, we denote $X:=D_0^{1, p}(\Omega)$ and define the norm in $L^p( \Omega )$ by $\| u \|_p$. The following lemma will be referred to as the Folland-Stein embedding theorem.

Lemma 2.1 (see [19]) Let $\Omega \subset \mathbb{H}^N$ be a bounded domain. Then the following inclusion is compact

$ \mathcal{D}_0^{1, p}(\Omega)\subset\subset L^q( \Omega ) \quad \mbox{for} \quad 1 <q <\frac{pQ}{Q-p}. $

According to the continuity of the Nemytskii operator (see [20, 22]) and Lemma 2.1, $f\in L^{\infty}(\Omega)$ implies that

(f) the functional

$ u \mapsto \displaystyle\int_{\Omega}f(\xi)|u|^p d\xi $

is weakly continuous on $X$.

Analogously, it follows from $g\in L^{\infty}(\Omega)$ and $1 <r <\frac{pQ}{Q-p}$ that

(g) the functional

$ u \mapsto \displaystyle\int_{\Omega}g(\xi)|u|^r d\xi $

is weakly continuous on $X$.

Now, we consider the nonlinear eigenvalue problem (1.3). This eigenvalue problem is also of independent interest (see [13-15]). Set

$ \begin{gather*} I(u)=\displaystyle\int_{\Omega}|{\nabla_{\mathbb{H}} }u|^p d\xi, \quad \mathcal{M}(f)= \{ u\in \mathcal{D}^{1, p}_0(\Omega):\displaystyle\int_{\Omega}f(\xi)|u|^p d\xi =1\}. \end{gather*} $

From the definition of the space $\mathcal{D}^{1, p}_0(\Omega)$, it is obvious that $I$ is coercive and weakly lower semi-continuous. We have the following theorem.

Theorem 2.1 If $1 < p < Q$ and $f(\xi)$ satisfies the conditions above, then

(ⅰ) there exists the first positive eigenvalue $\lambda_1(f)$ of (1.3) which is variationally expressed as

$ \begin{equation}\label{eq.2.4} \lambda_1(f) = \inf\limits_{u\in \mathcal{M}(f)}{I(u)}\, ; \end{equation} $

(2.4)

(ⅱ) $\lambda_1(f)$ is simple, i.e., the eigenfunctions associated to $\lambda_1(f)$ are merely a constant multiple of each other;

(ⅲ) $\lambda_1(f)$ is unique, i.e., if $v\geq 0$ is an eigenfunction associated with an eigenvalue $\lambda$ with $\displaystyle\int_{\Omega}f(\xi)|v|^p d\xi=1$, then $\lambda = \lambda_1(f)$.

A key point of the proof of Theorem 2.1 lies on the following lemma.

Lemma 2.2 (see [16]) Let $u\geq0$ and $v>0$ be differentiable functions on $\Omega \subset \mathbb{H}^N$, where $\Omega$ is a bounded or unbounded domain in $\mathbb{H}^N$. Then we have

$ \begin{equation}\label{eq.2.5} L(u, v)=R(u, v)\geq 0\, , \end{equation} $

(2.5)

where

$ \begin{align*} &L(u, v)=|\nabla_{\mathbb{H}} u|^p +(p-1)\frac{u^p}{v^p}|\nabla_{\mathbb{H}} v|^p -p\frac{u^{p-1}}{v^{p-1}}|\nabla_{\mathbb{H}} v|^{p-2}\nabla_{\mathbb{H}} u \cdot \nabla_{\mathbb{H}} v, \\ &R(u, v)=|\nabla_{\mathbb{H}} u|^p - |\nabla_{\mathbb{H}} v|^{p-2} \nabla_{\mathbb{H}}(\frac{u^p}{v^{p-1}})\cdot \nabla_{\mathbb{H}}v \end{align*} $

for $p>1$. Moreover, $L(u, v)=0$ a.e. on $\Omega$ if and only if $\nabla_{\mathbb{H}}(\frac{u}{v})=0$ a.e. on $\Omega$.

A direct consequence of Theorem 2.1 is

Corollary 2.1 If $1 < p < Q$, $0 <\lambda <\lambda_1(f)$ and $f(\xi)$ satisfies the conditions above, then the eigenvalue problem

$ \begin{equation}\left\{ \begin{array}{ll}\label{eq.2.9} -\Delta_{\mathbb{H}, p}u - \lambda f(\xi)|u|^{p-2}u = \mu |u|^{p-2}u & \quad \text{in } \Omega\, ; \\ \displaystyle u=0 & \quad \text{on }\partial \Omega \, \\ \end{array} \right. \end{equation} $

(2.6)

has the first positive eigenvalue $\mu_1(\lambda)$ which is variationally expressed as

$ \begin{equation}\label{eq.2.10} \mu_1(\lambda) = \inf\limits_{u\in \mathcal{M}(1)}{\displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u | ^p - \lambda f(\xi)|u|^p\right)} d\xi}\, . \end{equation} $

(2.7)

Moreover, $\mu_1(\lambda)$ is simple and unique.

The Euler-Lagrange functional associated with (1.1)-(1.2) is

$ \begin{equation} J_{\lambda } (u) = \frac{1}{p} \displaystyle\int_{\Omega} | \nabla_{\mathbb{H}} u|^p\, d\xi - \frac{\lambda }{p} \displaystyle\int_{\Omega } f(\xi) | u |^p \, d\xi - \frac{1}{r} \displaystyle\int_{\Omega} g(\xi) | u | ^r\, d\xi \end{equation} $

(3.1)

for all $u \in X$. $J_{\lambda }$ may be not bounded from below on whole $X$, since $p <r$. In order to obtain existence results in this case, we introduce the Nehari manifold

$ \mathcal{N}(\lambda)=\{u\in X:\, \langle J_{\lambda}'(u), u\rangle=0\}, $

where$\langle\, , \, \rangle$ denotes the usual duality between $X$ and $X^\ast$. Thus $u \in \mathcal{N}(\lambda)$ if and only if

$ \displaystyle\int_{\Omega} | \nabla_{\mathbb{H}} u|^p\, d\xi - \lambda\displaystyle\int_{\Omega } f(\xi) | u |^p \, d\xi - \displaystyle\int_{\Omega} g(\xi) | u | ^r\, d\xi =0\, . $

Clearly $\mathcal{N}(\lambda)$ is a much smaller set than $X$ and, as we will see below, $J_\lambda$ is much better behaved on $\mathcal{N}(\lambda)$. In particular, on $\mathcal{N}(\lambda)$ we have that

$ \begin{equation}\label{eq:3.2} J_{\lambda } (u)= ( \frac{1}{p} - \frac{1}{r}) \displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u | ^p - \lambda f|u|^p\right)} d\xi=(\frac{1}{p} - \frac{1}{r}) \displaystyle\int_{\Omega}g|u|^r d\xi \, . \end{equation} $

(3.2)

The Nehari manifold is closely linked to the behaviour of the functions of the form $\phi_u : t \to J_{\lambda}(tu)$ $(t > 0)$. Such maps are known as fibrering maps and were introduced by Drabek and Pohozaev in [9]. They were also discussed in [6] and [8]. If $u \in X $, we have

$ \begin{eqnarray} &\phi_u(t) = \frac{t^p}{p}\displaystyle\int_{\Omega} \left[ | \nabla_{\mathbb{H}} u | ^p - \lambda \, f(\xi) | u | ^p \right] d\xi - \frac{t^{r}}{r} \displaystyle\int_{\Omega}g(\xi)| u |^r d\xi\, , \end{eqnarray} $

(3.3)

$ \begin{eqnarray} &\phi_u'(t) = t^{p-1} \displaystyle\int_{\Omega}\left[ | \nabla_{\mathbb{H}} u | ^p - \lambda \, f(\xi) | u | ^{p}\right] d\xi - t^{r-1} \displaystyle\int_{\Omega}g(\xi)| u |^{r} d\xi\, , \label{e2.6}\end{eqnarray} $

(3.4)

$ \begin{eqnarray} &\phi_u''(t) = (p-1)t^{p-2}\displaystyle\int_{\Omega} \left[ |\nabla_{\mathbb{H}} u |^p - \lambda f(\xi)|u|^p \right] d\xi -(r-1)t^{r-2}\displaystyle\int_{\Omega} g(\xi)| u |^{r}d\xi\, . \label{e2.7} \end{eqnarray} $

(3.5)

It is easy to see that $u \in\mathcal{N}(\lambda) $ if and only if $\phi_u'(1) = 0$. More generally, $\phi_u'(t) = 0$ if and only if $tu \in \mathcal{N}(\lambda)$, i.e., elements in $\mathcal{N}(\lambda)$ correspond to stationary points of fibering maps. Thus it is natural to subdivide $\mathcal{N}(\lambda)$ into sets corresponding to local minima, local maxima and points of inflection, respectively. It follows from (3.4) and (3.5) that, $\phi_u'(t)=0$ implies

$ \begin{equation}\phi_u''(t) = (p-r)t^{p-2}\displaystyle\int_{\Omega} \left[ |\nabla_{\mathbb{H}} u |^p - \lambda f(\xi)|u|^p \right] d\xi = (p-r)t^{r-2}\displaystyle\int_{\Omega} g(\xi)| u |^{r}d\xi. \end{equation} $

(3.6)

Thus we define

$ \begin{eqnarray*} &\mathcal{N}^{+}(\lambda) = \{ u \in \mathcal{N}(\lambda): \displaystyle\int_{\Omega} g(\xi)| u |^{r}d\xi < 0 \}, \\ &\mathcal{N}^{-}(\lambda) = \{ u \in \mathcal{N}(\lambda): \displaystyle\int_{\Omega} g(\xi)| u |^{r}d\xi > 0 \}, \\ &\mathcal{N}^{0}(\lambda) = \{ u \in \mathcal{N}(\lambda): \displaystyle\int_{\Omega} g(\xi)| u |^{r}d\xi = 0 \}, \end{eqnarray*} $

so that $\mathcal{N}^{+}(\lambda)$, $\mathcal{N}^{-}(\lambda)$, $\mathcal{N}^{0}(\lambda)$ correspond to minima, maxima and points of inflection, respectively.

Let $u\in X$. Then

(ⅰ) if $\displaystyle\int_{\Omega} \left[ |\nabla_{\mathbb{H}} u |^p -\lambda f(\xi)|u|^p \right] d\xi$ and $\displaystyle\int_{\Omega} g(\xi)| u |^{r}d\xi$ have the same sign, $\phi_u$ has exactly one turning point at

$ t(u) =\left[ \frac{\displaystyle\int_{\Omega} \left[ |\nabla_{\mathbb{H}} u |^p -\lambda f(\xi)|u|^p \right] d\xi} {\displaystyle\int_{\Omega} g(\xi)| u |^{r}d\xi} \right]^{\frac{1}{r-p}}, $

this turning point is a minimum (maximum) so that $t(u)u\in \mathcal{N}^{+}(\lambda)$ $(\mathcal{N}^{-}(\lambda))$ if and only if $\displaystyle\int_{\Omega} g(\xi)| u |^{r}d\xi < 0\, (>0);$

(ⅱ) if $\displaystyle\int_{\Omega} \left[ |\nabla_{\mathbb{H}} u |^p -\lambda f(\xi)|u|^p \right] d\xi$ and $\displaystyle\int_{\Omega} g(\xi)| u |^{r}d\xi$ have opposite sign, $\phi_u$ has no turning points and so no multiples of $u$ lie in $\mathcal{N}(\lambda)$.

Hence we define

$ \begin{eqnarray} &\mathcal{L}^{+} = \{ u \in X: \|u \|=1, \, \displaystyle\int_{\Omega} \left[ |\nabla_{\mathbb{H}} u |^p -\lambda f(\xi)|u|^p \right] d\xi > 0 \}, \nonumber\\ &\mathcal{B}^{+} = \{ u \in X: \|u \|=1, \, \displaystyle\int_{\Omega} g(\xi)| u |^{r}d\xi > 0 \}. \label{e3.7} \end{eqnarray} $

(3.7)

Analogously we can define $\mathcal{L}^{-}$, $\mathcal{L}^{0}$, $\mathcal{B}^{-}$, $\mathcal{B}^{0}$ by replacing `$>0\, $' in (3.7) by `$ <0\, $' or `$=0\, $', respectively. Then we have

(ⅰ) if $u\in \mathcal{L}^{+}\cap \mathcal{B}^{+}$, then the fibering map $\phi_u$ has a unique critical point which is a local maximum. Moreover, $t(u)u\in \mathcal{N}^{-}(\lambda)$;

(ⅱ) if $u\in \mathcal{L}^{-}\cap \mathcal{B}^{-}$, then the fibering map $\phi_u$ has a unique critical point which is a local minimum. Moreover, $t(u)u\in \mathcal{N}^{+}(\lambda)$;

(ⅲ) if $u\in \mathcal{L}^{+}\cap \mathcal{B}^{-}$, then the fibering map $\phi_u$ is strictly increasing and no multiple of $u$ lies in $\mathcal{N}(\lambda)$;

(ⅳ) if $u\in \mathcal{L}^{-}\cap \mathcal{B}^{+}$, then the fibering map $\phi_u$ is strictly decreasing and no multiple of $u$ lies in $\mathcal{N}(\lambda)$.

Thus the following theorem holds.

Theorem 3.1 If $u\in X\backslash \{0\}$, then

(a) a multiple of $u$ lies in $\mathcal{N}^{-}(\lambda)$ if and only if $\frac{u}{{\|u\|}}$ lies in $\mathcal{L}^{+}\cap \mathcal{B}^{+}$;

(b) a multiple of $u$ lies in $\mathcal{N}^{+}(\lambda)$ if and only if $\frac{u}{{\|u\|}}$ lies in $\mathcal{L}^{-}\cap \mathcal{B}^{-}$;

(c) no multiple of $u$ lies in $\mathcal{N}(\lambda)$ if $u\in \mathcal{L}^{+}\cap \mathcal{B}^{-}$ or $u\in\mathcal{L}^{-}\cap \mathcal{B}^{+}$.

The following lemma was stated in [6] (see also [17]) which showed that minimizers on $\mathcal{N}(\lambda)$ are also critical points for $J_{\lambda}$ on $X$.

Lemma 3.1 Suppose that $u_0$ is a local maximum or minimum for $J_{\lambda}$ on $\mathcal{N}(\lambda)$. Then if $u_0 \not \in \mathcal{N}^{0}(\lambda)$, $u_0$ is a critical point of $J_{\lambda}$.

In this section, we discuss the Nehari manifold when $\lambda <\lambda_1(f)$ and show how the behaviour of the manifold as $\lambda\rightarrow \lambda_1^{-}(f)$ depends on the sign of $\displaystyle\int_{\Omega} g(\xi)\phi_1^{r}d\xi$. As a consequence, we obtain that equations (1.1)-(1.2) have at least one positive solution in this case.

Suppose $0 <\lambda <\lambda_1(f)$. It follows from (2.7) that there exists $\delta(\lambda)>0$ such that

$ \begin{equation}\label{eq:4.1} \displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u | ^p - \lambda f(\xi)|u|^p\right)} d\xi \geq \delta(\lambda)\|u\|^p, \quad \forall u\in X. \end{equation} $

(4.1)

Thus by (4.1), we have

Lemma 4.1 If $1 < \lambda < \lambda_1(f)$, then $\mathcal{L}^{-}$, $\mathcal{L}^{0}$ and $\mathcal{N}^{+}(\lambda)$ are empty and $\mathcal{N}^{0}(\lambda)=\{0\}$. Moreover, $\mathcal{N}^{-}(\lambda)=\{t(u)u: u\in \mathcal{B}^{+}\}$ and $\mathcal{N}(\lambda)=\mathcal{N}^{-}(\lambda)\cap\{0\}$.

We now investigate the behavior of $J_\lambda$ on $\mathcal{N}^{-}(\lambda)$. In view of the preceding lemma, we have

Theorem 4.1 If $0 <\lambda < \lambda_1(f) $, then $\inf\limits_{u\in\mathcal{N}^{-}(\lambda)}J_\lambda(u)>0$.

Proof By (3.2) and the structure of $\mathcal{N}^{-}(\lambda)$, we easily obtain $J_\lambda(u)>0$ whenever $u\in\mathcal{N}^{-}(\lambda)$ and so $J_\lambda(u)$ is bounded from below by $0$ on $\mathcal{N}^{-}(\lambda)$. Let $u\in\mathcal{N}^{-}(\lambda)$, then $v =\frac{u}{\|u\|}\in \mathcal{L}^{+}\cap \mathcal{B}^{+}$ and $u=t(v)v$ where $t(v) =\left[ \frac{\displaystyle\int_{\Omega} \left[ |\nabla_{\mathbb{H}} v |^p -\lambda f(\xi)|v|^p \right] d\xi}{\displaystyle\int_{\Omega} g(\xi)| v |^{r}d\xi} \right]^{\frac{1}{r-p}}$. Denote $b^*=\sup\limits_{\xi\in\Omega}g(\xi)$ and $K^{1/r}$ is a Folland-Stein embedding constant. Then $b^*>0$ and

$ \begin{equation}\label{eq:4.6} \displaystyle\int_{\Omega}g(\xi)|v|^r d\xi \leq b^*\displaystyle\int_{\Omega}|v|^r d\xi \leq b^*K\|v\|^r = b^*K\, . \end{equation} $

(4.2)

Combining (4.1) and (4.2), it yields that

$ \begin{align*} J_{\lambda }(u) &=J_{\lambda } (t(v)v)=( \frac{1}{p} - \frac{1}{r})t^p(v)\displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} v | ^p - \lambda f|v|^p\right)}d\xi \\ &=( \frac{1}{p} - \frac{1}{r})\frac{\left({\displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} v | ^p - \lambda f|v|^p \right)d\xi}}\right)^{\frac{r}{r-p}}}{\left(\displaystyle\int_{\Omega}g|v|^p d\xi \right)^{\frac{p}{r-p}}} \geq ( \frac{1}{p} - \frac{1}{r})\frac{{\delta(\lambda)}^{\frac{r}{r-p}}}{(b^*K)^{\frac{p}{r-p}}}, \end{align*} $

and $\inf\limits_{u\in\mathcal{N}^{-}(\lambda)}J_\lambda(u)>0$.

Next, we will show that there exists a minimizer on $\mathcal{N}^{-}(\lambda)$ which is a critical point of $J_{\lambda}$ and also a nontrivial solution of (1.1)-(1.2).

Theorem 4.2 If $0 < \lambda < \lambda_1(f) $, then there exists a minimizer of $J_{\lambda}$ on $\mathcal{N}^{-}(\lambda)$ which is a critical point of $J_{\lambda}$.

Proof Let $\{u_m\}\subset\mathcal{N}^{-}(\lambda)$ be a minimizing sequence, i.e.,

$ \lim\limits_{m \to \infty} J_{\lambda}(u_m) = \inf\limits_{u \in\mathcal{N}^{-}(\lambda)}J_{\lambda}(u). $

Using (3.2) and (4.1), we obtain

$ \begin{equation}\label{eq:4.7} J_{\lambda } (u_m)= ( \frac{1}{p} - \frac{1}{r}) \displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u_m | ^p - \lambda f|u_m|^p\right)} d\xi\geq(\frac{1}{p} - \frac{1}{r})\delta(\lambda) \|u_m\|^p\, , \end{equation} $

(4.3)

thus the sequence $\{u_m\}$ is bounded in $X$, and so we may assume passing to a subsequence that $u_m\rightharpoonup u_0$ in $X$. Since

$ \begin{equation*} J_{\lambda } (u_m)=(\frac{1}{p} - \frac{1}{r}) \displaystyle\int_{\Omega}g|u_m|^r d\xi\leq b^*K(\frac{1}{p} - \frac{1}{r})\|u_m\|^r\, \end{equation*} $

together with (4.3) implies that $\|u_m\|\geq \varepsilon$ for some $\varepsilon > 0$. Using (4.3) again, we deduce that

$ \begin{equation*} (\frac{1}{p} - \frac{1}{r})\delta(\lambda) \varepsilon^p\leq\lim\limits_{m \to \infty} J_{\lambda}(u_m) =\lim\limits_{m \to \infty}(\frac{1}{p} - \frac{1}{r}) \displaystyle\int_{\Omega}g|u_m|^r d\xi= (\frac{1}{p} - \frac{1}{r})\displaystyle\int_{\Omega}g|u_0|^r d\xi\, , \end{equation*} $

which implies that $u_0 \neq 0$. By (4.1), we get

$ \begin{equation}\label{eq:4.8} \displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u_0 | ^p - \lambda f |u_0|^p\right)} d\xi \geq \delta(\lambda)\|u_0\|^p>0\, . \end{equation} $

(4.4)

Hence $\frac{u_0}{\|u_0\|}\in \mathcal{L}^{+}\cap \mathcal{B}^{+}$.

We claim that $u_m\rightarrow u_0$ in $X$. Suppose the contradiction, then we have $\|u_0\| <\liminf\limits_{m\rightarrow \infty}\|u_m\|$. Together with (f) and (g) in Section 2, it yields

$ \begin{equation*} \displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u_0 | ^p - \lambda f|u_0|^p-g|u_0|^r\right)} d\xi < \liminf\limits_{m\rightarrow \infty}\displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u_m | ^p - \lambda f|u_m|^p-g|u_m|^r\right)} d\xi=0 \, , \end{equation*} $

which means $\phi_{u_0}'(1) = \displaystyle\int_{\Omega}\left( | \nabla_{\mathbb{H}} u_0|^p - \lambda \, f|u_0| ^{p}- g|u_0|^{r}\right) d\xi <0$. Since

$ \phi_{u_0}'(t) = t^{p-1}\left( \displaystyle\int_{\Omega}\left( | \nabla_{\mathbb{H}} u_0 | ^p - \lambda \, f | u_0 | ^{p}\right) d\xi - t^{r-p} \displaystyle\int_{\Omega}g | u_0 |^{r} d\xi\right) \, , $

it follows from (4.4) that $\phi_{u_0}'(t) > 0$ for $t$ sufficiently small. Then there exists $0 <\alpha <1$ such that $\phi_{u_0}'(\alpha)=0$, i.e., $\alpha u_0\in \mathcal{N}^{-}(\lambda)$. In virtue of $u_m\in \mathcal{N}^{-}(\lambda)$, we conclude that $\phi_{u_m}(t)$ attains its maximum at $t = 1$. Hence

$ J_{\lambda}(tu_m)=\phi_{u_m}(t)\leq \phi_{u_m}(1)=J_{\lambda}(u_m), \quad \forall t>0, $

and $J_{\lambda}(\alpha u_m)\leq J_{\lambda}(u_m)$. Note that $\alpha u_m\rightharpoonup \alpha u_0$ and $\|u_0\| <\liminf\limits_{m\rightarrow \infty}\|u_m\|$, we obtain

$ J_{\lambda}(\alpha u_0) <\liminf\limits_{m\rightarrow \infty} J_{\lambda}(\alpha u_m) \leq \lim\limits_{m\rightarrow \infty} J_{\lambda}(u_m)=\inf\limits_{u \in\mathcal{N}^{-}(\lambda)}J_{\lambda}(u)\, , $

i.e., it is a contradiction. Therefore $u_m\rightarrow u_0$ in $X$. This implies that

$ \begin{equation}\label{eq:4.9} \displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u_0 | ^p - \lambda f|u_0|^p-g|u_0|^r\right)} d\xi=0\, \end{equation} $

(4.5)

and

$ \begin{equation}\label{eq:4.10} J_{\lambda}(u_0)=\lim\limits_{m\rightarrow \infty} J_{\lambda}(u_m)=\inf\limits_{u \in\mathcal{N}^{-}(\lambda)}J_{\lambda}(u)\, . \end{equation} $

(4.6)

From (4.5), (4.4) and (4.6), we conclude that $u_0$ is a minimizer for $J_{\lambda}$ on $\mathcal{N}^{-}(\lambda)$. Since $\displaystyle\int_{\Omega} g(\xi) |u_0|^{r} d\xi >0$, $u_0 \notin \mathcal{N}^{0}(\lambda)$. By Lemma 3.1 $u_0$ is a critical point of $J_{\lambda}$. Since $J_{\lambda}(|u|)=J_{\lambda}(u)$, by applying Harnack inequality [18], we may assume that $u_0$ is positive.

As a direct consequence of Theorem 4.2, we obtain the following existence theorem.

Theorem 4.3 Equations (1.1)-(1.2) have at least one positive solution whenever $0 < \lambda < \lambda_1(f) $.

We conclude this section by proving some properties of the branch of solutions bifurcated from $\lambda_1(f)$ whenever the condition $\displaystyle\int_{\Omega } g(\xi)\phi_1^{r}\, d\xi>0$ is satisfied. The case where $\displaystyle\int_{\Omega } g(\xi)\phi_1^{r}\, d\xi <0$ which gives rise to multiple solutions when $\lambda > \lambda_1(f)$ will be discussed in the next section.

Theorem 4.4 Suppose $\displaystyle\int_{\Omega } g(\xi)\phi_1^{r}\, d\xi>0$. Then

(ⅰ) $\lim\limits_{\lambda \rightarrow{\lambda_1^{-}(f)}}\inf\limits_{u\in\mathcal{N}^{-}(\lambda)}J_\lambda(u)=0$;

(ⅱ) if $ \lambda_m \rightarrow{\lambda_1^{-}(f)}$ and $u_m$ is a minimizer of $J_{\lambda_m}$ on $\mathcal{N}^{-}(\lambda)$ (we may assume that $u_m>0$), then $\lim\limits_{m\rightarrow \infty}u_m=0$. Moreover, $\lim\limits_{m\rightarrow \infty}\frac{u_m}{\|u_m\|}=\phi_1$.

Proof (ⅰ) We may assume, without loss of generality, that $\|\phi_1\|=1$. Since $\displaystyle\int_{\Omega } g(\xi)\phi_1^{r}\, d\xi>0$ and $\lambda <\lambda_1(f)$, we have $\phi_1\in \mathcal{L}^{+}\cap\mathcal{B}^{+}$. Hence $t(\phi_1)\phi_1 \in \mathcal{N}^{-}(\lambda)$, where

$ t(\phi_1) =\left[ \frac{\displaystyle\int_{\Omega} ( |\nabla_{\mathbb{H}} \phi_1 |^p -\lambda f(\xi)\phi_1^p ) d\xi} {\displaystyle\int_{\Omega} g(\xi) \phi_1^{r}d\xi} \right]^{\frac{1}{r-p}} =\left[ (\lambda_1(f)-\lambda)\frac{\displaystyle\int_{\Omega} f(\xi)\phi_1^p d\xi}{\displaystyle\int_{\Omega} g(\xi) \phi_1^{r}d\xi} \right]^{\frac{1}{r-p}}. $

Therefore we have

$ \begin{align*} J_{\lambda }(t(\phi_1)\phi_1) &=( \frac{1}{p} - \frac{1}{r})\displaystyle\int_{\Omega} g(\xi)|t(\phi_1)\phi_1|^r d\xi \\ &=( \frac{1}{p} - \frac{1}{r})(\lambda_1(f)-\lambda)^{\frac{r}{r-p}} \frac{\left(\displaystyle\int_{\Omega} f(\xi)\phi_1^p d\xi\right)^{\frac{r}{r-p}}}{\left(\displaystyle\int_{\Omega} g(\xi)\phi_1^{r}d\xi\right)^{\frac{p}{r-p}}} \\ &\rightarrow 0, \quad \mbox{as} \, \lambda \rightarrow{\lambda_1^{-}(f)}. \end{align*} $

Since $0 <\inf\limits_{u\in\mathcal{N}^{-}(\lambda)}J_\lambda(u)\leq J_\lambda(t(\phi_1)\phi_1)$, it follows from the above relation that

$ \lim\limits_{\lambda \rightarrow{\lambda_1^{-}(f)}}\inf\limits_{u\in\mathcal{N}^{-}(\lambda)}J_\lambda(u)=0\, . $

(ⅱ) First, we show that every minimizing sequence $\{u_m\}$ on $\mathcal{N}^{-}(\lambda)$ is bounded. Suppose otherwise, then we may assume without loss of generality that $\|u_m\|\rightarrow\infty$. Let ${v_m}=\frac{u_m}{\|u_m\|}$, we may assume that $v_m\rightharpoonup v_0$ in $X$. By (f) and (g), we have

$ \begin{equation*} \lim\limits_{m\rightarrow\infty}\displaystyle\int_{\Omega} f(\xi)|v_m|^p d\xi=\displaystyle\int_{\Omega} f(\xi)|v_0|^p d\xi, \quad \lim\limits_{m\rightarrow\infty}\displaystyle\int_{\Omega} g(\xi)|v_m|^r d\xi=\displaystyle\int_{\Omega} g(\xi)|v_0|^r d\xi\, . \end{equation*} $

Using (ⅰ), we obtain

$ (\frac{1}{p}-\frac{1}{r})\displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u_m | ^p - \lambda_m f(\xi)|u_m|^p\right)} d\xi =(\frac{1}{p}-\frac{1}{r})\displaystyle\int_{\Omega} g(\xi)|u_m|^r d\xi=J_{\lambda_m}(u_m) \rightarrow 0\, $

as $m\to \infty$. Divided by $\|u_m\|^p$, we have

$ \lim\limits_{m\rightarrow\infty}\displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} v_m | ^p - \lambda_m f(\xi)|v_m|^p\right)} d\xi=0, \quad \lim\limits_{m\rightarrow\infty}\|u_m\|^{r-p}\displaystyle\int_{\Omega} g(\xi)|v_m|^r d\xi=0\, . $

Hence $\displaystyle\int_{\Omega} g(\xi)|v_0|^r d\xi=\lim_{m\rightarrow\infty}\displaystyle\int_{\Omega} g(\xi)|v_m|^r d\xi=0$. Now we show that $v_m\rightarrow v_0$ in $X$. Suppose not, and then

$ \begin{equation}\label{eq:4.11} \displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} v_0 | ^p - \lambda_1 f(\xi)|v_0|^p\right)} d\xi < \lim\limits_{m\rightarrow\infty}\displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} v_m | ^p - \lambda_m f(\xi)|v_m|^p\right)} d\xi=0\, , \end{equation} $

(4.7)

which is a contradiction for (2.4). Hence $v_m\rightarrow v_0$ in $X$. Thus we have

$ \displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} v_0 | ^p - \lambda_1 f(\xi)|v_0|^p\right)} d\xi = \lim\limits_{m\rightarrow\infty}\displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} v_m | ^p - \lambda_m f(\xi)|v_m|^p\right)} d\xi=0\, . $

Since $\|v_m\|=1$, we have $\|v_0\|=1$. It then follows from Theorem 2.1 that $v_0=\phi_1$ and

$ \displaystyle\int_{\Omega} g(\xi)\phi_1^r d\xi=\displaystyle\int_{\Omega} g(\xi)|v_0|^r d\xi=0\, , $

a contradiction. Therefore $\{u_n\}$ is bounded.

Thus we may assume, without loss of generality, that $u_m\rightharpoonup u_0$. Then by analogous argument above on $\{u_m\}$, it follows that $u_m \rightarrow u_0$ and $u_0=0$. Moreover, $\frac{u_m}{\|u_m\|}\rightarrow \phi_1$ and so the proof is complete.

In this section, with the properties of Nehari manifold, we shall give simple proofs of the existence of two positive solutions, one in $\mathcal{N}^-(\lambda)$ and the other in $\mathcal{N}^+(\lambda)$.

If $\lambda>\lambda_1(f)$, then

$ \displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} \phi_1 | ^p - \lambda f(\xi)\phi_1^p\right)} d\xi =(\lambda_1(f)-\lambda) \displaystyle\int_{\Omega}f(\xi)\phi_1^p d\xi<0\, . $

This yields $\phi_1 \in \mathcal{L}^{-}$. Hence, $\phi_1 \in \mathcal{L}^{-}\cap \mathcal{B}^{-}$ and $\mathcal{N}^{+}(\lambda)\neq \emptyset$ if $\displaystyle\int_{\Omega}g(\xi)\phi_1^r d\xi <0$. As we shall see, $\mathcal{N}(\lambda)$ may consist of two distinct components. Problems (1.1)-(1.2) have at least two positive solutions, if we show that $J_\lambda$ has an appropriate minimizer on each component.

The following lemma provides a useful property of the positive solutions to our problem.

Lemma 5.1 Suppose $\displaystyle\int_{\Omega}g(\xi)\phi_1^r d\xi <0$. Then there exists $\delta >0$ such that $\overline{\mathcal{L}^{-}}\cap \overline{\mathcal{B}^{+}} = \emptyset $ whenever $\lambda_1(f) \leq \lambda < \lambda_1(f)+\delta $.

Proof Suppose that the result is false. Then there exist sequences $\{\lambda_m\}$ and $\{u_m\}$ such that $\|u_m\|=1$, $\lambda_m\rightarrow \lambda_1^+(f)$ and

$ \begin{equation}\label{eq:5.1} \displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u_m | ^p - \lambda_m f(\xi)|u_m|^p\right)} d\xi\leq 0\, , \quad \displaystyle\int_{\Omega} g(\xi)|u_m|^r d\xi\geq 0. \end{equation} $

(5.1)

Since $\{u_m\}$ is bounded, we assume without loss of generality that $u_m\rightharpoonup u_0$ in $X$.

We now show that $u_m\rightarrow u_0$ in $X$. Suppose otherwise, then $\|u_0\| <\liminf\limits_{m\rightarrow \infty}\|u_m\|$ and (f) implies that

$ \displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u_0 | ^p - \lambda_1(f) f|u_0|^p\right)} d\xi < \liminf\limits_{m\rightarrow \infty}\displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u_m | ^p - \lambda_m f|u_m|^p \right)} d\xi\le0 \, , $

which is a contradiction to (2.4). It follows from (5.1), (f) and (g) that

(ⅰ) $\displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u_0 | ^p - \lambda_1 f(\xi)|u_0|^p\right)} d\xi\leq 0\, , $

(ⅱ) $\displaystyle\int_{\Omega} g(\xi)|u_0|^r d\xi\geq 0.$

Using (ⅰ) and Theorem 2.1, we obtain $u_0=k\phi_1$ for some constant $k$. Hence, from (ii) we deduce that $k=0$ which is impossible as $\|u_0\|=1$.

We next show that, if $\overline{\mathcal{L}^{-}}\cap \overline{\mathcal{B}^{+}}=\emptyset$, it is possible to obtain more information about the nature of the Nehari manifold.

Theorem 5.1 Suppose $\overline{\mathcal{L}^{-}}\cap \overline{\mathcal{B}^{+}}=\emptyset$, then

(ⅰ) $\mathcal{N}^0(\lambda)=\{0\}$;

(ⅱ) $0\notin \overline{\mathcal{N}^{-}(\lambda)}$ and $\mathcal{N}^{-}(\lambda)$ is closed;

(ⅲ) $\mathcal{N}^{-}(\lambda)$ and $\mathcal{N}^{+}(\lambda)$ are separated, i.e., $\mathcal{N}^{-}(\lambda)\cap \overline{\mathcal{N}^{+}(\lambda)}=\emptyset$;

(ⅳ) $\mathcal{N}^{+}(\lambda)$ is bounded.

Proof (ⅰ) Suppose $u_0 \in \mathcal{N}^0(\lambda)\backslash \{0\}$, then $\frac{u_0}{\|u_0\|}\in \mathcal{L}^0\cap \mathcal{B}^0 \subset \mathcal{L}^0\cap \overline{\mathcal{B}^+}= \emptyset$. Hence $\mathcal{N}^0(\lambda)=\{0\}$.

(ⅱ) Suppose $0\in \overline{\mathcal{N}^{-}(\lambda)}$, then there exists $\{u_m\} \subseteq \mathcal{N}^{-}(\lambda)$ such that $u_m\rightarrow 0$ in $X$. Hence

$ 0 <\displaystyle\int_{\Omega} {\left(|\nabla_{\mathbb{H}} u_m | ^p - \lambda f(\xi)|u_m|^p\right)} d\xi =\displaystyle\int_{\Omega} g(\xi)|u_m|^r d\xi \rightarrow 0\, . $

Let $v_m=\frac{u_m}{\|u_m\|}$, then we may assume that $v_m \rightharpoonup v_0$ in $X$. Clearly

$ 0 <\displaystyle\int_{\Omega} {\left(|\nabla_{\mathbb{H}} v_m | ^p - \lambda f(\xi)|v_m|^p\right)} d\xi =\|u_m\|^{r-p}\displaystyle\int_{\Omega} g(\xi)|v_m|^r d\xi \rightarrow 0\, . $

Thus by (f), we have

$ 0=\lim\limits_{m\rightarrow\infty}\displaystyle\int_{\Omega} {\left(|\nabla_{\mathbb{H}} v_m | ^p - \lambda f|v_m|^p\right)} d\xi =1-\lim\limits_{m\rightarrow\infty}\lambda \displaystyle\int_{\Omega} {f|v_m|^p} d\xi=1-\lambda \displaystyle\int_{\Omega} { f|v_0|^p} d\xi\, , $

and then $v_0\ne 0$. Moreover, by the weak lower semicontinuity of the norm and (f), we get

$ \displaystyle\int_{\Omega} {\left(|\nabla_{\mathbb{H}} v_0 | ^p - \lambda f|v_0|^p\right)} d\xi \leq \lim\limits_{m\rightarrow\infty}\displaystyle\int_{\Omega} {\left(|\nabla_{\mathbb{H}} v_m | ^p - \lambda f|v_m|^p\right)} d\xi=0\, , $

and then $\frac{v_0}{\|v_0\|}\in \mathcal{L}^0\cup \mathcal{L}^{-}$. Since $\displaystyle\int_{\Omega} g(\xi)|v_m|^r d\xi>0$, it follows that $\displaystyle\int_{\Omega} g(\xi)|v_0|^r d\xi\geq 0$ and $\frac{v_0}{\|v_0\|}\in \overline{\mathcal{B}^+}$. Hence $\frac{v_0}{\|v_0\|}\in \overline{\mathcal{L}^-}\cap \overline{\mathcal{B}^{+}}$, and this is a contradiction. Thus $0\notin \overline{\mathcal{N}^{-}(\lambda)}$.\\ \indent By (i), $\overline{\mathcal{N}^{-}(\lambda)}\subseteq \mathcal{N}^{-}(\lambda)\cup \mathcal{N}^{0}(\lambda)= \mathcal{N}^{-}(\lambda)\cup \{0\}$. Since $0\notin \overline{\mathcal{N}^{-}(\lambda)}$, it follows that $\overline{\mathcal{N}^{-}(\lambda)}=\mathcal{N}^{-}(\lambda)$, i.e., $\mathcal{N}^{-}(\lambda)$ is closed.

(ⅲ) By (ⅰ) and (ⅱ),

$ \overline{\mathcal{N}^{-}(\lambda)}\cap \overline{\mathcal{N}^{+}(\lambda)}\subseteq \mathcal{N}^{-}(\lambda)\cap (\mathcal{N}^{+}(\lambda)\cup \mathcal{N}^{0}(\lambda)) =(\mathcal{N}^{-}(\lambda)\cap\mathcal{N}^{+}(\lambda))\cup (\mathcal{N}^{-}(\lambda)\cap \{0\}) =\emptyset, $

and then $\mathcal{N}^{-}(\lambda)$ and $\mathcal{N}^{+}(\lambda)$ are separated.

(ⅳ) Suppose that $\mathcal{N}^{+}(\lambda)$ is unbounded, then there exists $\{u_m\}\subseteq \mathcal{N}^{+}(\lambda)$ such that $\|u_m\|\rightarrow \infty$ as $m\rightarrow \infty$. By definition,

$ \displaystyle\int_{\Omega} {\left(|\nabla_{\mathbb{H}} u_m | ^p - \lambda f(\xi)|u_m|^p\right)} d\xi =\displaystyle\int_{\Omega} g(\xi)|u_m|^r d\xi < 0\, . $

Let $v_m=\frac{u_m}{\|u_m\|}$. According to the above formula, we get

$ \begin{equation}\label{eq:5.2} \displaystyle\int_{\Omega} {\left(|\nabla_{\mathbb{H}} v_m | ^p - \lambda f(\xi)|v_m|^p\right)} d\xi =\|u_m\|^{r-p}\displaystyle\int_{\Omega} g(\xi)|v_m|^r d\xi \, . \end{equation} $

We can assume that $v_m\rightharpoonup v_0$ in $X$. Since the left-hand side (l.h.s) of (5.2) is bounded but $\|u_m\|\rightarrow \infty$, it follows that $\lim\limits_{m\rightarrow\infty} \displaystyle\int_{\Omega} g(\xi)|v_m|^r d\xi=0$. Then (g) implies $\displaystyle\int_{\Omega} g(\xi)|v_0|^r d\xi=0$.

Now we prove that $v_m \rightarrow v_0$ in $X$. Suppose otherwise, then $\|v_0\| <\liminf\limits_{m\rightarrow \infty}\|v_m\|$ and

$ \displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} v_0 | ^p - \lambda f|v_0|^p\right)} d\xi < \lim\limits_{m\rightarrow \infty}\displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} v_m | ^p - \lambda f|v_m|^p \right)} d\xi \leq 0 \, . $

Thus $\frac{v_0}{\|v_0\|}\in \mathcal{L}^{-}\cap \overline{\mathcal{B}^{+}}$ which is impossible. Hence $v_m\rightarrow v_0$ in $X$.

Since $v_m\rightarrow v_0$, we have $\|v_0\|=1$. Hence $v_0\in \mathcal{B}^{0}$ and moreover $v_0\in \overline{\mathcal{B}^{+}}$. By (f), we have

$ \displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} v_0 | ^p - \lambda f|v_0|^p\right)} d\xi = \lim\limits_{m\rightarrow \infty}\displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} v_m | ^p - \lambda f|v_m|^p \right)} d\xi \leq 0 \, , $

and then $v_0 \in \overline{\mathcal{L}^{-}}$. Thus $v_0 \in \overline{\mathcal{L}^{-}}\cap \overline{\mathcal{B}^{+}}$ which is again impossible. Hence $\mathcal{N}^{+}(\lambda)$ is bounded.

When $\mathcal{N}^{-}(\lambda)$ and $\mathcal{N}^{+}(\lambda)$ are separated and $\mathcal{N}^{0}(\lambda)=\{0\}$, any non-zero minimizer for $J_\lambda$ on $\mathcal{N}^{-}(\lambda)$ (or on $\mathcal{N}^{+}(\lambda)$) is also a local minimizer on $\mathcal{N}(\lambda)$ which is a critical point for $J_\lambda$ on $\mathcal{N}(\lambda)$ and a solution of (1.1)-(1.2).

Theorem 5.2 Suppose $\overline{\mathcal{L}^{-}}\cap \overline{\mathcal{B}^{+}}=\emptyset$. Then

(ⅰ) every minimizing sequence for $J_{\lambda}$ on $\mathcal{N}^{-}(\lambda)$ is bounded;

(ⅱ) $\inf_{u\in\mathcal{N}^{-}(\lambda)}J_\lambda(u)>0$;

(ⅲ) there exists a minimizer of $J_{\lambda}$ on $\mathcal{N}^{-}(\lambda)$.

Proof (ⅰ) Suppose that $\{u_m\}\in \mathcal{N}^{-}(\lambda)$ is a minimizing sequence of $J_{\lambda}$. Then

$ \begin{equation}\label{eq:5.3} \displaystyle\int_{\Omega} ( |\nabla_{\mathbb{H}} u_m |^p -\lambda f(\xi)|u_m|^p ) d\xi=\displaystyle\int_{\Omega} g(\xi)|u_m|^r d\xi \rightarrow c\, , \end{equation} $

(5.3)

where $c\geq 0$.

Assume that $\{u_m\}$ is unbounded, i.e., $\|u_m\|\rightarrow \infty$ as $m\rightarrow \infty$. Let ${v_m}=\frac{u_m}{\|u_m\|}$. Divided (5.3) by $\|u_m\|^p$ gives

$ \begin{equation}\label{eq:5.4} \displaystyle\int_{\Omega} {\left(|\nabla_{\mathbb{H}} v_m | ^p - \lambda f(\xi)|v_m|^p\right)} d\xi =\|u_m\|^{r-p}\displaystyle\int_{\Omega} g(\xi)|v_m|^r d\xi \, . \end{equation} $

(5.4)

Since $\|v_m\|=1$, we may assume that $v_m\rightharpoonup v_0$ in $X$. Since the l.h.s of (5.4) is bounded, it follows that $\lim\limits_{m\rightarrow\infty} \displaystyle\int_{\Omega} g(\xi)|v_m|^r d\xi=0$ and therefore $\displaystyle\int_{\Omega} g(\xi)|v_0|^r d\xi=0$.

We now show that $v_m \rightarrow v_0$ in $X$. Suppose otherwise, then

$ \displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} v_0 | ^p - \lambda f|v_0|^p\right)} d\xi < \lim\limits_{m\rightarrow \infty}\displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} v_m | ^p - \lambda f|v_m|^p \right)} d\xi = 0 \, . $

Thus $v_0\neq 0$ and $\frac{v_0}{\|v_0\|}\in \mathcal{L}^{-}\cap \mathcal{B}^{0}$ which is impossible. Hence $v_m\rightarrow v_0$ in $X$. It follows that $\|v_0\|=1$. Moreover, by (f)

$ \displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} v_0 | ^p - \lambda f|v_0|^p\right)} d\xi =\displaystyle\int_{\Omega} { g|v_0|^r } d\xi = 0 \, . $

This implies that $v_0 \in \mathcal{L}^{0}\cap \mathcal{B}^{0}$ which contradicts to the assumption $\overline{\mathcal{L}^{-}}\cap \overline{\mathcal{B}^{+}}=\emptyset$. Hence $u_m$ is bounded.

(ⅱ) Since $J_{\lambda}(u)>0$ on $\mathcal{N}^{-}(\lambda)$, we have $\inf\limits_{u\in\mathcal{N}^{-}(\lambda)}J_\lambda(u)\geq 0$. Suppose $\inf\limits_{u\in\mathcal{N}^{-}(\lambda)}J_\lambda(u) = 0$. Let $\{u_m\}\in \mathcal{N}^{-}(\lambda)$ is a minimizing sequence, then

$ \displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u_m | ^p - \lambda f|u_m|^p\right)} d\xi =\displaystyle\int_{\Omega} { g|u_m|^r } d\xi = (\frac{1}{p}-\frac{1}{r})^{-1}J_\lambda(u_m)\rightarrow 0 \, . $

By (ⅰ) we know that $\{u_m\}$ is bounded and we may suppose $u_m\rightharpoonup u_0$ in $X$. By using exactly the same argument on $\{v_m\}$ in (i), it may be shown that $u_m\rightarrow u_0$ in $X$. By Theorem 5.1 we know that $0\notin \overline{\mathcal{N}^{-}(\lambda)}$ and so $u_0\neq 0$. It then follows exactly as in the proof in (i) that $\frac{u_0}{\|u_0\|}\in \mathcal{L}^{0}\cap \mathcal{B}^{0}$ and this contradicts the assumption $\overline{\mathcal{L}^{-}}\cap \overline{\mathcal{B}^{+}}=\emptyset$.

(ⅲ) Let $\{u_m\}\in \mathcal{N}^{-}(\lambda)$ is a minimizing sequence of $J_{\lambda}$, then

$ \begin{align*} J_{\lambda }(u_m) &=( \frac{1}{p} - \frac{1}{r})\displaystyle\int_{\Omega} ( |\nabla_{\mathbb{H}} u_m |^p -\lambda f(\xi)|u_m|^p ) d\xi \\ &=( \frac{1}{p} - \frac{1}{r})\displaystyle\int_{\Omega} g(\xi)|u_m|^r d\xi \rightarrow \inf\limits_{u\in\mathcal{N}^{-}(\lambda)}J_\lambda(u)>0, \end{align*} $

and then

$ \displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u_m | ^p - \lambda f|u_m|^p\right)} d\xi =\displaystyle\int_{\Omega} { g|u_m|^r } d\xi \rightarrow (\frac{1}{p}-\frac{1}{r})^{-1}\inf\limits_{u\in\mathcal{N}^{-}(\lambda)}J_\lambda(u)> 0\, . $

By (ⅰ) we know that $\{u_m\}$ is bounded. We may assume that $u_m\rightharpoonup u_0$ in $X$. Then by (g), we have $\displaystyle\int_{\Omega} g(\xi)|u_0|^r d\xi=\lim_{m\rightarrow\infty}\displaystyle\int_{\Omega} g(\xi)|u_m|^r d\xi > 0$ and so $\frac{u_0}{\|u_0\|}\in \mathcal{B}^{+}$. Since $(\mathcal{L}^0\cup\mathcal{L}^-)\cap \mathcal{B}^{+}=\emptyset$, it follows that $\frac{u_0}{\|u_0\|}\in \mathcal{B}^{+}\subseteq \mathcal{L}^{+}$. Hence $\frac{u_0}{\|u_0\|}\in \mathcal{B}^{+}\cap \mathcal{L}^{+}$ and $\displaystyle\int_{\Omega} {\left(|\nabla_{\mathbb{H}} u_0 | ^p - \lambda f|u_0|^p\right)} d\xi >0$. Furthermore, we have $t(u_0)u_0\in \mathcal{N}^{-}(\lambda)$ where $t(u_0) =\left[ \frac{\displaystyle\int_{\Omega} ( |\nabla_{\mathbb{H}} u_0 |^p -\lambda f(\xi)|u_0|^p ) d\xi} {\displaystyle\int_{\Omega} g(\xi)| u_0 |^{r}d\xi} \right]^{\frac{1}{r-p} }$.

We will show that $u_m\rightarrow u_0$ in $X$. Suppose not, then

$ \begin{align*} \displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u_0 | ^p - \lambda_1 f(\xi)|u_0|^p\right)} d\xi &<\lim\limits_{m\rightarrow\infty}\displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u_m | ^p - \lambda f(\xi)|u_m|^p\right)} d\xi \\ &=\lim\limits_{m\rightarrow\infty}\displaystyle\int_{\Omega} g(\xi)| u_m |^{r}d\xi=\displaystyle\int_{\Omega} g(\xi)| u_0 |^{r}d\xi\, , \end{align*} $

so $t(u_0) <1$. Since $t(u_0)u_m\rightharpoonup t(u_0)u_0$ and the map $t\mapsto J_{\lambda}(tu_m)$ attains its maximum value at $t=1$, we obtain

$ J_{\lambda}(t(u_0) u_0) <\liminf\limits_{m\rightarrow \infty} J_{\lambda}(t(u_0) u_m) \leq \lim\limits_{m\rightarrow \infty} J_{\lambda}(u_m)=\inf\limits_{u \in\mathcal{N}^{-}(\lambda)}J_{\lambda}(u)\, , $

a contradiction. Hence $u_m\rightarrow u_0$.

We can easily deduce that

$ \displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u_0 | ^p - \lambda f(\xi)|u_0|^p\right)} d\xi =\displaystyle\int_{\Omega} g(\xi)|u_0|^r d\xi\, , $

and therefore $u_0\in \mathcal{N}(\lambda)$. Since $\displaystyle\int_{\Omega} g(\xi)| u_0 |^{r}d\xi>0$, $u_0\in \mathcal{N}^{-}(\lambda)$. Also

$ J_{\lambda}(u_0)=\lim\limits_{m\rightarrow \infty} J_{\lambda}(u_m)=\inf\limits_{u \in\mathcal{N}^{-}(\lambda)}J_{\lambda}(u), $

which implies $u_0$ is a minimizer for $J_{\lambda}(u)$ on $\mathcal{N}^{-}(\lambda)$.

We now turn our attention to $\mathcal{N}^{+}(\lambda)$.

Theorem 5.3 Suppose $\mathcal{L}^{-}$ is non-empty and $\overline{\mathcal{L}^{-}}\cap \overline{\mathcal{B}^{+}}=\emptyset$, then there exists a minimizer of $J_{\lambda}(u)$ on $\mathcal{N}^{+}(\lambda)$.

Proof Since $\overline{\mathcal{L}^{-}}\cap \overline{\mathcal{B}^{+}}=\emptyset$, then $\mathcal{L}^{-}\cap \mathcal{B}^{-}=\mathcal{L}^{-}\neq \emptyset$ and so $\mathcal{N}^{+}(\lambda)\neq \emptyset$. Denote $b_0=\inf\limits_{\xi\in \Omega}g(\xi)$. Then $\mathcal{N}^{+}(\lambda)\neq \emptyset$ implies that $b_0 <0$. As $\mathcal{N}^{+}(\lambda)$ is bounded, there exists $M>0$ such that $\|u\|\leq M$ for all $u\in \mathcal{N}^{+}(\lambda)$. Hence by Lemma 1.2, for $u\in \mathcal{N}^{+}(\lambda)$, we obtain

$ \begin{align*} J_{\lambda }(u) &=(\frac{1}{p} - \frac{1}{r})\displaystyle\int_{\Omega} g|u|^r d\xi\geq(\frac{1}{p} - \frac{1}{r})b_0 \displaystyle\int_{\Omega}|u|^r d\xi \\ &\geq ( \frac{1}{p} - \frac{1}{r})b_0K\|u\|^r\geq ( \frac{1}{p} - \frac{1}{r})b_0KM^r. \end{align*} $

It follows that $J_{\lambda}(u)$ is bounded from below on $\mathcal{N}^{+}(\lambda)$ and $\inf\limits_{u\in\mathcal{N}^{+}(\lambda)}J_\lambda(u)$ exists. Clearly $\inf\limits_{u\in\mathcal{N}^{+}(\lambda)}J_\lambda(u) <0$. Suppose that $\{u_m\}\subseteq \mathcal{N}^{+}(\lambda)$ is a minimizing sequence of $J_{\lambda}$, then

$ \begin{align*} J_{\lambda }(u_m) &=( \frac{1}{p} - \frac{1}{r})\displaystyle\int_{\Omega} ( |\nabla_{\mathbb{H}} u_m |^p -\lambda f(\xi)|u_m|^p ) d\xi \\ &=( \frac{1}{p} - \frac{1}{r})\displaystyle\int_{\Omega} g(\xi)|u_m|^r d\xi \rightarrow \inf\limits_{u\in\mathcal{N}^{+}(\lambda)}J_\lambda(u) <0 \end{align*} $

as $m\rightarrow\infty$. Since $\mathcal{N}^{+}(\lambda)$ is bounded, we may assume $u_m\rightharpoonup u_0$ in $X$. Then by (g) and (f), we have

$ \displaystyle\int_{\Omega} g(\xi)|u_0|^r d\xi=\lim\limits_{m\rightarrow\infty}\displaystyle\int_{\Omega} g(\xi)|u_m|^r d\xi < 0 $

and

$ \displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u_0 | ^p - \lambda f(\xi)|u_0|^p\right)} d\xi \leq \lim\limits_{m\rightarrow\infty}\displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u_m | ^p - \lambda f(\xi)|u_m|^p\right)} d\xi <0. $

Hence $\frac{u_0}{\|u_0\|}\in \mathcal{L}^{-}\cap \mathcal{B}^{-}$ and $t(u_0)u_0\in \mathcal{N}^{+}(\lambda)$.

Suppose $u_m \not \to u_0 $ in $X$. Then we get

$ \begin{align*} \displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u_0 | ^p - \lambda f(\xi)|u_0|^p\right)} d\xi &< \lim\limits_{m\rightarrow \infty}\displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u_m | ^p - \lambda f(\xi)|u_m|^p \right)} d\xi \\ &=\lim\limits_{m\rightarrow\infty}\displaystyle\int_{\Omega} g(\xi)|u_m|^r d\xi=\displaystyle\int_{\Omega} g(\xi)|u_0|^r d\xi \end{align*} $

and $t(u_0) =\left[ \frac{\displaystyle\int_{\Omega} ( |\nabla_{\mathbb{H}} u_0 |^p -\lambda f(\xi)|u_0|^p ) d\xi} {\displaystyle\int_{\Omega} g(\xi)| u_0 |^{r}d\xi} \right]^{\frac{1}{r-p} }>1$. It follows that $J_{\lambda}(t(u_0) u_0)\leq J_{\lambda}(u_0) <\lim\limits_{m\rightarrow \infty} J_{\lambda}(u_m)=\inf\limits_{u \in\mathcal{N}^{+}(\lambda)}J_{\lambda}(u)$ and this is impossible. Hence $u_m \rightarrow u_0 $ in $X$. We thus deduce that

$ \displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u_0 | ^p - \lambda f(\xi)|u_0|^p\right)} d\xi =\displaystyle\int_{\Omega} g(\xi)|u_0|^r d\xi <0\, , $

which shows $u_0\in \mathcal{N}^{+}(\lambda)$ and then $J_{\lambda}(u_0)=\lim\limits_{m\rightarrow \infty} J_{\lambda}(u_m)=\inf\limits_{u \in\mathcal{N}^{+}(\lambda)}J_{\lambda}(u)$. Hence $u_0$ is a minimizer for $J_{\lambda}(u)$ on $\mathcal{N}^{+}(\lambda)$

Corollary 5.1 Suppose $\displaystyle\int_{\Omega}g(\xi)\phi_1^r d\xi <0$ and $\delta$ is as in Lemma 5.1. Then equations (1.1)-(1.2) have at least two positive solutions whenever $\lambda_1(f) < \lambda < \lambda_1(f)+\delta$.

Proof Since $\lambda > \lambda_1(f)$, we have that $\phi_1\in \mathcal{L}^{-}$. By Theorems 5.2 and 5.3, there exist minimizers $u_{\lambda}^+$ and $u_{\lambda}^-$ of $J_{\lambda}(u)$ on $\mathcal{N}^{+}(\lambda)$ and $\mathcal{N}^{-}(\lambda)$, respectively. According to Theorem 5.1, $\mathcal{N}^{-}(\lambda)$ and $\mathcal{N}^{+}(\lambda)$ are separated and $\mathcal{N}^0(\lambda)=\{0\}$. Hence there exist at least two minimizers which are local minimizers for $J_{\lambda}$ on $\mathcal{N}(\lambda)$. These minimizers does not belong to $\mathcal{N}^{0}(\lambda)$. Moreover $J_{\lambda}(u_{\lambda}^{\pm}) = J_{\lambda}(|u_{\lambda}^{\pm}|)$ and $|u_{\lambda}^{\pm}| \in {\mathcal{N}}^{\pm}{(\lambda)}$, so we may assume $u_{\lambda}^{\pm} \geq 0$. By Lemma 3.1, $ u_{\lambda}^{\pm}$ are critical points of $J_{\lambda}$ on $X$ and hence are weak solutions of (1.1)-(1.2). Finally, by the Harnack inequality [18], we obtain that $u_{\lambda}^{\pm}$ are positive solutions of (1.1)-(1.2).

Finally in this section, we investigate the nature of $\mathcal{N}^{+}(\lambda)$ as $\lambda\rightarrow \lambda_1^{+}(f)$.

Theorem 5.4 Suppose $\displaystyle\int_{\Omega}g(\xi)\phi_1^r d\xi <0$, $\lambda_m\rightarrow \lambda_1^+(f)$ and $u_m\in \mathcal{N}^{+}(\lambda)$ is a critical point of $J_\lambda(u)$ corresponding to $\lambda=\lambda_m$ (we may assume that $u_m>0$). Then as $m\rightarrow\infty$,

(ⅰ) $u_m\rightarrow 0\, ;$

(ⅱ) $\frac{u_m}{\|u_m\|}\rightarrow \phi_1 \, \, \mbox{in}\, \, X.$

Proof (ⅰ) Since $u_m\in \mathcal{N}^{+}(\lambda)$ is a critical point of $J_{\lambda_m}(u)$, we have

$ \begin{equation*} \displaystyle\int_{\Omega} ( |\nabla_{\mathbb{H}} u_m |^p -\lambda_m f(\xi)|u_m|^p ) d\xi=\displaystyle\int_{\Omega} g(\xi)|u_m|^r d\xi <0\, . \end{equation*} $

By Lemma 5.1 and Theorem 5.2, we get $\mathcal{N}^{+}(\lambda)$ is bounded, and so is $\{u_m\}$. We may suppose that $u_m\rightharpoonup u_0$ in $X$. Suppose $u_m\not \to u_0$, then

$ \displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u_0 | ^p - \lambda_1 f|u_0|^p\right)} d\xi < \liminf\limits_{m\rightarrow \infty}\displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u_m | ^p - \lambda_m f|u_m|^p \right)} d\xi \leq 0, $

which is impossible because of (2.4). Hence $u_m\rightarrow u_0$ as $m\rightarrow\infty$. This, together with (f) and (g), implies that

$ \displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u_0 | ^p - \lambda_1 f|u_0|^p\right)} d\xi = \displaystyle\int_{\Omega} g|u_0|^r d\xi \leq 0\, . $

Combining with (2.4), it yields $\displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} u_0 | ^p - \lambda_1 f|u_0|^p\right)} d\xi=0$. Thus by Theorem 2.1, we have $u_0=k\phi_1$ for some $k$. But, as $\displaystyle\int_{\Omega}g(\xi)|\phi_1|^r d\xi <0$, it follows that $k=0$. Hence $u_m\rightarrow 0$ in $X$.

(ⅱ) Let ${v_m}=\frac{u_m}{\|u_m\|}$. We may assume that $v_m\rightharpoonup v_0$ in $X$. Clearly

$ \begin{equation*} \displaystyle\int_{\Omega} {\left(|\nabla_{\mathbb{H}} v_m | ^p - \lambda_m f(\xi)|v_m|^p\right)} d\xi =\|u_m\|^{r-p}\displaystyle\int_{\Omega} g(\xi)|v_m|^r d\xi \end{equation*} $

and so, since $\|u_m\|\rightarrow 0$, $\lim\limits_{m\rightarrow \infty}\displaystyle\int_{\Omega} {\left(|\nabla_{\mathbb{H}} v_m | ^p - \lambda_m f(\xi)|v_m|^p\right)} d\xi=0\, .$ Suppose $v_m \not \rightarrow v_0$, then $\|v_0\| <\lim\limits_{m\rightarrow \infty}\|v_m\|$ and therefore $\displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} v_0 | ^p - \lambda_1 f|v_0|^p\right)} d\xi < 0$ which gives us a contradiction. Hence $v_m\rightarrow v_0$, so $\|v_0\|=1$ and $\displaystyle\int_{\Omega} {\left(| \nabla_{\mathbb{H}} v_0 | ^p - \lambda_1 f|v_0|^p\right)} d\xi= 0$. It then follows from Theorem 2.1 that $v_0=\phi_1$ and the proof is completed.

The authors would like to thank Professor Wenyi Chen for his kind help and guidance. The first author would also like to thank Professor Xiaochun Liu for bringing Ref.[4, 5] to our attention.