The Fučik spectrum of the p-Laplace equations with different weights and its resonance problems

Abstract

In the present paper, our purpose is to construct the Fučik spectrum $C_{l}^{\pm}$ (cf. (2.13) and (2.17) below) with different weights for the p-Laplacian. As an application, we will discuss the existence of nontrivial solutions to the p-Laplace equations with resonance on the Fučik spectrum by making use of variational methods and Morse theory.

1 Introduction

In this paper, we consider the existence of nontrivial solutions to the Dirichlet boundary value problem

$$ \left \{ \textstyle\begin{array}{l@{\quad}l} - \Delta_{p}u = \lambda a(x)(u^{+})^{p -1} - \mu b(x)(u^{-})^{p -1} + g(x, u),& \mbox{in }\Omega, \\ u = 0, & \mbox{on }\partial\Omega, \end{array}\displaystyle \right . $$

(1.1)

where Ω is a bounded domain in $\mathbb{R}^{N}$ with smooth boundary ∂Ω, $\Delta_{p}$ denotes the p-Laplace operator, that is, $\Delta_{p}u=\nabla(|\nabla u|^{p-2}\nabla u)$, $u^{\pm} = \max\{\pm u(x), 0\}$, and $a(x), b(x) \in L^{r}(\Omega)$, where $r > \frac{N}{p}$ if $1 < p \leq N$ and $r =1$ if $p > N$. We assume that $g: \Omega\times\mathbb{R} \rightarrow \mathbb{R}$ is a Caratheodory function.

It is well known that the value of $(\lambda, \mu)$ plays an important role in the study of the solvability of (1.1). The approach here of course requires the preliminary study of weighted asymmetric eigenvalue problems of the form

$$ \left \{ \textstyle\begin{array}{l@{\quad}l} - \Delta_{p}u = \lambda a(x)(u^{+})^{p -1} - \mu b(x)(u^{-})^{p -1},& \mbox{in }\Omega, \\ u = 0,& \mbox{on }\partial\Omega. \end{array}\displaystyle \right . $$

(1.2)

The set $\Sigma_{p}(a, b)$, the Fučik spectrum of the p-Laplacian with the weights $a(x)$ and $b(x)$ on $W_{0}^{1, p}$, is defined by those $(\lambda,\mu)\in\mathbb{R}^{2}$ such that (1.2) has a nontrivial solution u.

The generalized notion of spectrum was introduced for $p=2$ (i.e. in the linear case) in the 1970s by Fučik [1] and Dancer [2] in connection with the study of the so-called jumping non-linearities. From then on, further work has been done in the study of $\Sigma_{2}(1,1)$; cf. [3–5]. The quasilinear case of $p \neq2$ and $N = 1$, in the situation where $a(x)$ and $b(x)$ are nonconstant and different, was investigated in [6–9] (for $a(x)$ and $b(x) > 0$) and [10] (for $a(x)$ and $b(x)$ are indefinite). From the above papers, $\Sigma_{p}(a, b)$ has the same general shape as in the linear ODE case.

In the quasilinear PDE case, when $\lambda a = \mu b$,

$$ \left \{ \textstyle\begin{array}{l@{\quad}l} - \Delta_{p}u = \lambda a(x) |u|^{p-2}u, & \mbox{in }\Omega, \\ u = 0, & \mbox{on }\partial\Omega. \end{array}\displaystyle \right . $$

(1.3)

It is well known that problem (1.3) has an unbound sequence of variational eigenvalues $\lambda_{l}(a)$ satisfying a standard min-max characterization, $\lambda_{l}(a) \rightarrow+\infty$ ($l\rightarrow\infty$). The first eigenvalue $\lambda_{1}(a)$ of $-\Delta_{p} u$ is positive, simple, and admits a positive eigenfunction $\varphi_{a}$ (see Lindqvist [11] for the case $a(x)=1$ and Cuesta [12] for the case where $a(x)$ is an indefinite weights), where $\lambda_{1}(a) := \min\{\int_{\Omega}| \nabla u|^{p}: u \in W_{0}^{1, p}(\Omega) \mbox{ and } \int_{\Omega} a|u|^{p} =1\}$ < $\lambda_{2}(a) := \min\{ \lambda\in \mathbb{R}: \lambda \mbox{ is the eigenvalue and } \lambda>\lambda_{1}(a )\}$. We denote $e_{a}(x) := \frac{\varphi_{a}}{\|\varphi_{a}\|_{1, p}}$. According to Arias et al. [13], we know that $\Sigma_{p}(a, b)$ contains the two lines $\lambda_{1}(a)\times\mathbb {R}$ and $\mathbb{R} \times\lambda_{1}(b)$.

As in [13], the first nontrivial eigenvalue which admits a sign-changing eigenfunction of the eigenvalue problem with weights $a(x)$ and $b(x)$ is defined as $c(a,b) := \inf_{\gamma\in\Gamma}\max_{t \in[-1, 1]}$ $A(\gamma(t))$, where $A(u) := \int_{\Omega}|\nabla u|^{p}\,dx$, $\Gamma:= \{\gamma\in C([-1,1], \mathscr{M}_{a, b}): \gamma(-1) = \varphi_{a} \mbox{ and } \gamma(1) = \varphi_{b} \}$, and $\mathscr{M}_{a, b} := \{u\in W_{0}^{1, p}(\Omega): \int_{\Omega}a(x)(u^{+})^{p} + b(x)(u^{-})^{p}\,dx =1\}$. Hence, the first nontrivial curve in $\Sigma_{p}(a, b)$ through $(c(a, b), c(a, b))$ asymptotic to $\lambda_{1}(a)\times\mathbb{R}$ and $\mathbb{R} \times\lambda_{1}(b)$ at infinity was constructed by Cuesta et al. [14] for the case $a = b = 1$, and by Arias et al. [13] for the case with two different nonconstant weights. More recently, unbounded curves $\{C_{l}^{\pm}\}$ in $\Sigma_{p}(a, b)$ have been constructed and variationally characterized by min-max procedures by Micheletti and Pistoia [15] for $p > 2$, $a = b = 1$, and by Perera [16] for $p > 1$, $a = b = 1$ and [17] for the fractional p-Laplace operator.

Related studies can also be found in [18–20].

It is our purpose in this paper to initiate the study of (1.2) and its relation with the solvability of (1.1) in the general case: $a(x), b(x)\geq0$ are possibly nonconstant and different. Hence we will discuss the existence of the Fučik spectrum for (1.2) besides $\lambda_{1}(a)\times\mathbb{R}$, $\mathbb {R}\times \lambda_{1}(b)$ and $\mathcal {C} := \{(\alpha(s), s\alpha(s)): \alpha(s) = c(a, sb), s > 0\}$ (in [13]), which is an open question raised in [13].

As an application, we will concern ourselves with the resonance type of the problem (1.1). The resonance problem has been studied by Perera [16] and Guo and Liu [21] for the case $(\lambda,\mu)\in\Sigma_{p}(a,b)$, with $a = b = 1$. The purpose of this paper is to discuss the general resonance case $(\lambda, \mu) \in\Sigma_{p}(a, b)$, with nonconstant and different weights $a(x)$ and $b(x)$. When $(\lambda, \mu) \in\lambda_{1}(a) \times \mathbb{R}$ ($(\lambda, \mu) \in\mathbb{R} \times\lambda_{1}(b)$), we obtain the existence of nontrivial solution for (1.1) by the Morse theorem. However, the usual Morse theory supposes that the functional $\Phi(u)$ satisfies the Palais-Smale condition ((P.S.) condition for short), which is not clear in the case where the nonlinear term $g(x, u)$ is asymptotic to $|u|^{p-2}u$ at infinity. To overcome this difficulty, Cerami introduced a weaker compactness condition (see [22]) that allows for more general results; see for example [23]. Fortunately, replacing the usual (P.S.) condition by Cerami’s weaker compactness condition (($C_{c}$) condition for short), we still have the deformation lemma (see [24]).

2 Fučik spectrum with different weights

In this section, we consider the eigenvalues problem of asymmetric p-Laplacian with weights (1.2). It is always assumed that $a(x), b(x)\in L^{r}(\Omega)$ with r as in the introduction, i.e. $r > \frac{N}{p}$ if $1 < p \leq N$ and $r = 1$ if $p > N$.

As is well known, the associated functional of the problem (1.2) is

$$I(u) = \int_{\Omega}|\nabla u|^{p} - \int_{\Omega}\lambda a(x) \bigl(u^{+}\bigr)^{p} + \mu b(x) \bigl(u^{-}\bigr)^{p}, \quad u\in W_{0}^{1, p}( \Omega). $$

We introduce two new functionals as follows:

$$\begin{aligned}& J_{s, t, s_{0}}^{+}(u) = \int_{\Omega}|\nabla u|^{p} - sa(x) \bigl(u^{+} \bigr)^{p}-(s_{0} + t)b(x) \bigl(u^{-}\bigr)^{p} + s_{0}a(x) \bigl(u^{-}\bigr)^{p}, \\& J_{s, t, s_{0}}^{-} (u) = \int_{\Omega}|\nabla u|^{p} - sb(x) \bigl(u^{-} \bigr)^{p} - (s_{0} + t)a(x) \bigl(u^{+}\bigr)^{p} + s_{0}b(x) \bigl(u^{+}\bigr)^{p}, \end{aligned}$$

where $u \in W_{0}^{1, p}(\Omega)$, $s_{0}$ is a positive constant to be determined later.

The Finsler manifold we need is defined as follows:

$$S(a) := \biggl\{ u \in W_{0}^{1, p}(\Omega) \Big| \int_{\Omega}a(x) \bigl\vert u(x)\bigr\vert ^{p} \,dx =1 \biggr\} . $$

$S(b)$ can be considered in a similar way. We will consider the functionals $J_{s, t, s_{0}}^{+}$ on the manifold $S(a)$ and $J_{s, t, s_{0}}^{-}$ on the manifold $S(b)$, respectively. Let us just deduce the case $J_{s, t, s_{0}}^{+}$ on $S(a)$, since the other case can be done similarly.

For each integer $l\geq1$, we define maps family ${\mathcal{F}}_{l}(a)$ as

$${\mathcal{F}}_{l}(a) := \bigl\{ A \subset S(a); \mbox{there exists an odd continuous surjection map } h: S^{l -1} \to A\bigr\} , $$

where $S^{l-1}$ is the unit sphere in $\mathbb{R}^{l}$.

As observed by Drábek and Robinson [25], we can define an unbounded sequence of functionals $J(u)$ by

$$\lambda_{l}(a) := \inf_{A \in{\mathcal{F}}_{l}} \max _{u \in A} J(u), $$

where $J(u) = \int_{\Omega}| \nabla u| \,dx = J_{0, 0, 0}(u)$. It is shown that $\lambda_{l}(a)$ is the lth eigenvalue of $-\Delta_{p}$ with weight $a(x)$, although it is not known whether this gives a complete list of eigenvalues. Furthermore,

$$\lambda_{l}(a) \to+ \infty, \quad \mbox{as } l \to\infty. $$

In the sequel we fix some $l \geq2$ and suppose that

$$ \lambda_{l - 1}(a) < \lambda_{l}(a). $$

(2.1)

For any $\varepsilon\in (0, \lambda_{l}(a) - \lambda_{l-1}(a) )$, we can choose an $A^{l - 2} \in{\mathcal{F}}_{l - 1}(a)$, such that

$$\max_{u \in A^{l - 2}}J(u) < \lambda_{l - 1}(a) + \varepsilon. $$

Let $h_{l - 2}: S^{l - 2} \to A^{l - 2}$ be any fixed continuous surjection and denote

$$\begin{aligned} {\mathcal{F}}_{l}^{+}(a) :=& \bigl\{ A_{+} \subset S(a); \mbox{there is a continuous surjection map }h: S_{+}^{l-1} \to A, \\ & \mbox{such that }h|_{S^{l-2}} = h_{l-2} \bigr\} , \end{aligned}$$

where $S_{+}^{l - 1}$ is the upper hemisphere of $S^{l - 1}$ with boundary $S^{l - 2}$. Then ${\mathcal{F}}_{l}^{+}(a)$ is a homotopy-stable family of compact subsets of $S(a)$ with closed boundary $A^{l - 2}$, that is,

• (i) every set $A_{+} \in{\mathcal{F}}_{l}^{+}(a)$ contains $A^{l - 2}$;

• (ii) for any set $A_{+} \in{\mathcal{F}}_{l}^{+}(a)$ and any deformation $\eta\in C([0, 1]\times S(a); S(a))$ satisfying $\eta(t, u) = u$ for all $(t, u) \in(\{ 0 \} \times S(a)) \cup([0, 1] \times A^{l - 2})$ we have $\eta(\{ 1 \} \times A) \in{\mathcal{F}}_{l}^{+}(a)$.

Now, we define

$$ c_{l}^{+}(s, t, s_{0}) := \inf _{A_{+}\in{\mathcal{F}}_{l}^{+}(a)}\max_{u\in A_{+}}J_{s, t, s_{0}}^{+}(u). $$

(2.2)

Similarly, we can define

$$ c_{l}^{-}(s, t, s_{0}) := \inf _{A_{-}\in{\mathcal{F}}_{l}^{-}(b)}\max_{u\in A_{-}}J_{s, t, s_{0}}^{-}(u). $$

(2.3)

We can make sure that $c_{l}^{\pm}(s, t, s_{0})$ is the critical value of $J_{s, t, s_{0}}^{\pm}(u)$, under some assumptions. Furthermore, the critical point of $J_{s, t, s_{0}}^{\pm}(u)$ corresponding to $c_{l}^{\pm}(s, t, s_{0})$ in which $s_{0}$ is determined in a special way, is just the solution to (1.2), that is, we will establish the following result.

Theorem 2.1

Suppose that the weights $a(x)$ and $b(x)$ are nonnegative and, for some fixed constants $k' \geq1 \geq k > 0$,

$$ 0 \leq ka(x) \leq b(x) \leq k'a(x),\quad \textit{a.e. }x \in\Omega, $$

(2.4)

where k and $k'$ satisfy

$$\biggl(\frac{1}{k} - \frac{1}{k'} \biggr) \lambda_{l}(a) < \bigl(\lambda _{l}(a) - \lambda_{l -1}(a)\bigr) (1 - \delta), $$

where $\delta\in(0, 1)$. Suppose that s and t are in some intervals as follows:

$$\begin{aligned}& t \in \biggl( {\lambda_{l}(a) \over k} - \bigl(\lambda_{l}(a) - \lambda_{l - 1}(a)\bigr), {\lambda_{l}(a) \over k'} - \delta\bigl( \lambda_{l}(a) - \lambda_{l-1}(a)\bigr) \biggr]; \end{aligned}$$

(2.5)

$$\begin{aligned}& s \in \bigl( \max \bigl\{ \lambda_{l-1}(a), (k-1) \bigl(\lambda_{l}(a)- \lambda _{l-1}(a)\bigr)+k't \bigr\} , \\& \hphantom{s \in{}}{}\min \bigl\{ k\bigl(\delta\bigl(\lambda_{l}(a)-\lambda _{l-1}(a)\bigr)+t\bigr),\lambda_{l}(a)-\delta\bigl( \lambda_{l}(a)-\lambda_{l-1}(a)\bigr) \bigr\} \bigr]. \end{aligned}$$

(2.6)

Then, for any $s_{0}$ in $[\delta(\lambda_{l}(a)-\lambda_{l-1}(a)), \lambda_{l}(a)-\lambda _{l-1}(a) )$, $c_{l}^{+}(s, t, s_{0})$ is a critical value of $J_{s, t, s_{0}}^{+}(u)$ on $S(a)$. In particular, the set of the Fučik spectrum of $-\Delta_{p}$ with weights $a(x)$ and $b(x)$ greater than $c(a(x),b(x))$ includes $(s+c_{l}^{+}(s, t, s_{0}), t+c_{l}^{+}(s, t, s_{0}))$ for s and t in the above intervals, and some well-defined $s_{0} \in [\delta(\lambda_{l}(a)-\lambda_{l-1}(a)),\lambda _{l}(a)-\lambda_{l-1}(a) )$ satisfying

$$c_{l}^{+}(s, t, s_{0})=s_{0}. $$

Lemma 2.1

$J_{s, t, s_{0}}^{+}(u)$ satisfies the (P.S.) condition on $S(a)$.

Proof

Let $\{u_{n}\}\subset S(a)$ and $t_{n}\in\mathbb{R}$ be sequence, such that, for some constant k,

$$ \bigl\vert J_{s, t, s_{0}}^{+}(u_{n})\bigr\vert \leq k $$

(2.7)

and

$$\begin{aligned}& \int_{\Omega} \bigl\{ |\nabla u_{n}|^{p-2}\nabla u_{n}\nabla w - \bigl[sa\bigl(u_{n}^{+}\bigr)^{p-1} -(s_{0}+t)b\bigl(u_{n}^{-}\bigr)^{p-1}+s_{0}a \bigl(u_{n}^{-}\bigr)^{p-1}\bigr]w \\& \quad {} -t_{n}a|u_{n}|^{p-2}u_{n}w \bigr\} \leq\varepsilon_{n}\|w\|_{1,p}, \end{aligned}$$

(2.8)

for all $w\in W_{0}^{1, p}(\Omega)$, where $\varepsilon_{n}\rightarrow0$, as $n \rightarrow\infty$ and $\|\cdot\|_{1, p}$ represents the norm of $W_{0}^{1, p}(\Omega)$. From (2.7) it follows that $\{ u_{n}\}$ remains bounded in $W_{0}^{1, p}(\Omega)$; consequently, for a subsequence, $u_{n}\rightarrow u$ weakly in $W_{0}^{1, p}(\Omega)$, strongly in $L^{p}(\Omega)$ and almost everywhere in Ω. Putting $w=u_{n} - u$ in equation (2.8), we also see that $t_{n}$ remains bounded. We then have

$$\begin{aligned} \int_{\Omega} | \nabla u_{n}|^{p-2} \nabla u_{n} (\nabla u_{n}-u) =& \int_{\Omega} \bigl[s a \bigl(u_{n}^{+} \bigr)^{p-1}-(s_{0} + t)b \bigl(u_{n}^{-} \bigr)^{p-1}+ s_{0} a \bigl(u_{n}^{-} \bigr)^{p-1} \bigr](u_{n}-u) \\ &{} + \int_{\Omega} t_{n} a|u_{n}|^{p-2}u_{n} (u_{n} - u) + o(\varepsilon_{n}), \end{aligned}$$

where the right-hand side goes to zero as $n\rightarrow\infty$, i.e.

$$ \int_{\Omega} |\nabla u_{n}|^{p-2} \nabla u_{n}(\nabla u_{n} - u) \rightarrow 0,\quad n \rightarrow \infty. $$

It is sufficient to obtain $u_{n} \rightarrow u$ strongly in $W_{0}^{1, p}$. □

Lemma 2.2

As a function of s, t, $s_{0}$, the min-max value

$$c_{l}^{+}(s, t, s_{0}) := \inf_{A_{+} \in{\mathcal{F}}_{l}^{+}(a)} \max _{u \in A_{+}}J_{s, t, s_{0}}^{+}(u) $$

is continuous on $s_{0}$.

Proof

Let $\{s_{n}\}\subset [\delta(\lambda_{l}(a)-\lambda_{l-1}(a)), \lambda_{l}(a)-\lambda_{l -1}(a)]$ be a sequence such that

$$s_{n} \rightarrow s_{0},\quad \mbox{as } n \rightarrow \infty. $$

Taking $A_{+} \in\mathcal{F}_{l}^{+}(a)$ in the following equation:

$$J_{s, t, s_{0}}^{+}(u)-J_{s, t, s_{n}}^{+}(u)=(s_{0}-s_{n}) \int_{\Omega }(a-b) \bigl(u^{-}\bigr)^{p}. $$

Then we can see that

$$\begin{aligned}& \begin{aligned}[b] &\max_{A_{+}}J_{s, t, s_{n}}^{+}(u)-\max _{A_{+}}(s_{n}-s_{0}) \int_{\Omega}(a-b) \bigl(u^{-}\bigr)^{p} \\ &\quad \leq\max_{A_{+}}J_{s, t, s_{0}}^{+}(u) \leq\max _{A_{+}}J_{s, t, s_{n}}^{+}(u) + \max_{A_{+}}(s_{0}-s_{n}) \int_{\Omega}(a-b) \bigl(u^{-}\bigr)^{p}. \end{aligned} \end{aligned}$$

(2.9)

Noting that $k a(x) \leq b(x) \leq k' a(x)$, we have

$$\begin{aligned} \max_{A_{+}}(s_{n}-s_{0}) \int_{\Omega}(a-b) \bigl(u^{-}\bigr)^{p} \leq& \vert s_{n} - s_{0}\vert \bigl(\max\bigl\{ \vert 1-k\vert , \bigl\vert k'-1\bigr\vert \bigr\} \bigr) \max _{A_{+}} \int_{\Omega }a\bigl(u^{-}\bigr)^{p} \\ \leq& \bigl(\max\bigl\{ \vert 1-k\vert ,\bigl\vert k'-1\bigr\vert \bigr\} \bigr)\vert s_{n}-s_{0}\vert := \varepsilon_{n}, \end{aligned}$$

where $\varepsilon_{n}\rightarrow0$, as $n\rightarrow\infty$. By (2.9) we have

$$ c_{l}^{+}(s, t, s_{n}) - \varepsilon_{n} \leq c_{l}^{+}(s, t, s_{0}) \leq c_{l}^{+}(s, t, s_{n}) + \varepsilon_{n}. $$

Letting $n\rightarrow\infty$, we have

$$ \limsup_{n\rightarrow\infty} c_{l}^{+}(s, t, s_{n}) \leq c_{l}^{+}(s, t, s_{0})\leq\liminf_{n\rightarrow\infty} c_{l}^{+}(s, t, s_{n}), $$

which shows that $\lim_{n\rightarrow\infty} c_{l}^{+}(s, t,s_{n}) = c_{l}^{+}(s, t, s_{0})$. The proof is completed. □

Proof of Theorem 2.1

We first claim that

$$ c_{l}^{+}(s, t, s_{0}) \geq \lambda_{l}(a) - s. $$

(2.10)

If it is not the case, that is, $c_{l}^{+}(s, t, s_{0}) < \lambda_{l}(a) - s$, then we can choose some $A_{+}\in{\mathcal{F}}^{+}_{l}(a)$, such that

$$\max_{u\in A_{+}}J_{s, t, s_{0}}^{+}(u)< \lambda_{l}(a)-s. $$

By setting $A=A_{+}\cup(-A_{+})$, we get an odd symmetric set A. The set A is thus in ${\mathcal{F}}_{l}(a)$. Since the functional $J(u) = J_{0,0, 0}(u)$ is even, from (2.4) it follows that

$$\begin{aligned} \lambda_{l}(a) \leq& \max_{A}J(u) = \max _{A_{+}}J(u) \\ \leq& \max_{A_{+}}J_{s, t, s_{0}}^{+}(u) +\max _{A_{+}} \int_{\Omega}sa\bigl(u^{+}\bigr)^{p}+(s_{0}+t)b \bigl(u^{-}\bigr)^{p}-s_{0}a\bigl(u^{-}\bigr)^{p} \\ \leq& \max_{A_{+}}J_{s, t, s_{0}}^{+}(u) + \max _{A_{+}} \biggl[ \int_{\Omega}sa\bigl(u^{+}\bigr)^{p}+ \bigl((t+s_{0})k'-s_{0}\bigr)a\bigl(u^{-} \bigr)^{p} \biggr] \\ \leq& \max_{A_{+}}J_{s, t, s_{0}}^{+}(u)+\max\bigl\{ s,tk'+\bigl(k'-1\bigr)s_{0}\bigr\} \int _{\Omega}a|u|^{p} \\ < & \lambda_{l}(a)-s+\max\bigl\{ s,tk'+ \bigl(k'-1\bigr) \bigl(\lambda_{l}(a) - \lambda_{l-1}(a)\bigr)\bigr\} \int_{\Omega}a|u|^{p} \\ \leq& \lambda_{l}(a), \end{aligned}$$

(2.11)

which leads to a contradiction. Thus, (2.10) follows.

The inequality (2.10) is crucial in proving the existence of critical value of $J_{s, t, s_{0}}^{+}(u)$ on the manifold $S(a)$. Since the assumption in the theorem implies that, for $\varepsilon\in(0, \lambda_{l}(a)-\lambda_{l-1}(a)-s_{0})$, we have

$$\begin{aligned} c^{+}_{l}(s, t, s_{0}) \geq& \lambda_{l}(a)-s \\ >& \lambda_{l-1}(a)+\varepsilon+s_{0}-s \\ >& \max_{A^{l-2}}J(u)+s_{0}-s \\ =& \max_{A^{l-2}}J(u) +\max_{A^{l-2}} \biggl[ \int_{\Omega} s_{0}a\bigl(u^{+}\bigr)^{p}+s_{0}a \bigl(u^{-}\bigr)^{p}-sa\bigl(u^{+}\bigr)^{p}-sa\bigl(u^{-} \bigr)^{p} \biggr] \\ \geq& \max_{A^{l-2}}J(u)+\max_{A^{l-2}} \biggl[ \int_{\Omega }s_{0}a\bigl(u^{-}\bigr)^{p}-s \int_{\Omega}a\bigl(u^{+}\bigr)^{p}-(s_{0}+t) \int_{\Omega} b\bigl(u^{-}\bigr)^{p} \biggr] \\ \geq& \max_{A^{l-2}}J_{s, t, s_{0}}^{+}(u). \end{aligned}$$

(2.12)

By Lemma 2.1, we know that $J_{s, t, s_{0}}^{+}$ satisfies the (P.S.) condition on $S(a)$. Then it follows from Theorem 3.2 of Ghoussoub [26] that $c^{+}_{l}(s, t, s_{0})$ is a critical value of $J_{s, t, s_{0}}^{+}(u)$, that is, according to the Lagrange multiple rule, there is some $u\in W_{0}^{1,p}(\Omega)$, such that

$$-\Delta_{p}u-sa\bigl(u^{+}\bigr)^{p-1}+(s_{0}+t)b \bigl(u^{-}\bigr)^{p-1}-s_{0}a\bigl(u^{-}\bigr)^{p-1} = c^{+}_{l}(s, t, s_{0})\bigl[a\bigl(u^{+}\bigr)^{p-1}-a \bigl(u^{-}\bigr)^{p-1}\bigr]. $$

The min-max value $c^{+}_{l}(s, t, s_{0})$ is also a function of $s_{0}$, and from Lemma 2.2, we know that $c^{+}_{l}(s, t, s_{0})$ is continuous on $s_{0}$. Set

$$\psi(s_{0})=c^{+}_{l}(s, t, s_{0})-s_{0}. $$

Next we claim that

$$\psi\bigl(\delta\bigl(\lambda_{l}(a)-\lambda_{l-1}(a)\bigr) \bigr)\geq0\quad \mbox{and}\quad \psi\bigl(\lambda_{l}(a)- \lambda_{l-1}(a)\bigr)\leq0. $$

In fact, for every $A_{+}\in{\mathcal{F}}^{+}_{l}(a)$, since $A=A_{+}\cup(-A_{+})\in{\mathcal{F}}_{l}(a)$, on A, we have

$$J(u)= \int_{\Omega}|\nabla u|^{p}\geq\lambda_{l}(a) \int_{\Omega}a|u|^{p}. $$

Hence, we can estimate the maximum of $J_{s, t, \delta(\lambda_{l}(a)-\lambda_{l-1}(a))}(u)$ on $A_{+}$ as follows:

$$\begin{aligned}& \max_{A_{+}}J_{s, t, \delta(\lambda_{l}(a)-\lambda_{l-1}(a))}^{+}(u) \\& \quad = \max_{A_{+}} \biggl\{ \int_{\Omega}|\nabla u|^{p}-sa\bigl(u^{+} \bigr)^{p} -\bigl(t+\delta\bigl(\lambda_{l}(a)-\lambda _{l-1}(a)\bigr)\bigr)b\bigl(u^{-}\bigr)^{p} \\& \qquad {}+ \delta\bigl(\lambda_{l}(a)-\lambda_{l-1}(a)\bigr) \int_{\Omega}a\bigl(u^{-}\bigr)^{p} \biggr\} \\& \quad \geq \max_{A_{+}} \biggl\{ \lambda_{l}(a) \int_{\Omega}a|u|^{p}-sa\bigl(u^{+}\bigr)^{p}- \bigl(t+\delta\bigl(\lambda_{l}(a)-\lambda_{l-1}(a)\bigr) \bigr)b\bigl(u^{-}\bigr)^{p} \\& \qquad {} + \delta\bigl(\lambda_{l}(a)-\lambda_{l-1}(a)\bigr) \int_{\Omega}a\bigl(u^{-}\bigr)^{p}\biggr\} \\& \quad \geq \max_{A_{+}}\biggl\{ \lambda_{l}(a) \int_{\Omega} a|u|^{p} \\& \qquad {}-\max\bigl\{ s,k'\bigl(t+\delta\bigl(\lambda_{l}(a)- \lambda_{l-1}(a)\bigr)\bigr) -\delta\bigl(\lambda_{l}(a)- \lambda_{l-1}(a)\bigr)\bigr\} \int_{\Omega}a|u|^{p} \biggr\} \\& \quad = \lambda_{l}(a)-\max\bigl\{ s,k'\bigl(t+\delta \bigl(\lambda_{l}(a)-\lambda_{l-1}(a)\bigr)\bigr) -\delta\bigl( \lambda_{l}(a)-\lambda_{l-1}(a)\bigr)\bigr\} . \end{aligned}$$

By the conditions (2.5) and (2.6), we have

$$ \left \{ \textstyle\begin{array}{l} \lambda_{l}(a) \geq s + \delta(\lambda_{l}(a)-\lambda_{l-1}(a)), \\ \lambda_{l}(a) \geq k'(t+\delta(\lambda_{l}(a)-\lambda_{l-1}(a))). \end{array}\displaystyle \right . $$

Then we see that $\psi(\delta(\lambda_{l}(a)-\lambda_{l-1}(a))) \geq0$.

Moreover, for any $\varepsilon> 0$, we pick up an $A \in{\mathcal{F}}_{l}(a)$, such that

$$A=h\bigl(S^{l-1}(a)\bigr),\qquad h|_{S^{l-2}(a)}=h_{l-2}, $$

and

$$\max_{u\in A}J(u)< \lambda_{l}(a)+\varepsilon. $$

Denote $A_{+}=h(S^{l-1}_{+}(a))$, and then $A_{+}\in{\mathcal{F}}^{+}_{l}(a)$. For every $u \in A_{+}$,

$$\begin{aligned}& J_{s, t, \lambda_{l}(a)-\lambda_{l-1}(a)}(u) \\& \quad = \int_{\Omega}|\nabla u|^{p}-sa\bigl(u^{+} \bigr)^{p}-\bigl(\lambda_{l}(a)-\lambda_{l-1}(a)+t \bigr)b\bigl(u^{-}\bigr)^{p}+\bigl(\lambda _{l}(a)- \lambda_{l-1}(a)\bigr)a\bigl(u^{-}\bigr)^{p} \\& \quad \leq \lambda_{l}(a) \int_{\Omega}a|u|^{p}+\varepsilon \\& \qquad {}+ \int_{\Omega}\bigl[-sa\bigl(u^{+}\bigr)^{p}-\bigl( \lambda_{l}(a)-\lambda _{l-1}(a)+t\bigr)b\bigl(u^{-} \bigr)^{p}+\bigl(\lambda_{l}(a)-\lambda_{l-1}(a) \bigr)a\bigl(u^{-}\bigr)^{p}\bigr] \\& \quad \leq \bigl(\lambda_{l}(a)-s\bigr) \int_{\Omega}a\bigl(u^{+}\bigr)^{p}+\varepsilon \\& \qquad {}+\bigl[2\lambda_{l}(a)-\bigl(\lambda_{l}(a)- \lambda_{l-1}(a)+t\bigr)k-\lambda _{l-1}(a)\bigr] \int_{\Omega}a\bigl(u^{-}\bigr)^{p} \\& \quad \leq \max\bigl\{ \lambda_{l}(a)-s,2\lambda_{l}(a)- \bigl(\lambda_{l}(a)-\lambda _{l-1}(a)+t\bigr)k- \lambda_{l-1}(a)\bigr\} +\varepsilon. \end{aligned}$$

We can deduce that $\psi(\lambda_{l}(a)-\lambda_{l-1}(a))\leq0$. Hence there exists some $s_{0} \in[ \delta(\lambda_{l}(a)-\lambda_{l-1}(a)),\lambda _{l}(a)-\lambda_{l-1}(a) )$, such that

$$\psi(s_{0}) = 0, $$

which means that if we take $s_{0}$ in $J_{s, t, s_{0}}(u)$ as above, then

$$s_{0}=c^{+}_{l}(s, t, s_{0}). $$

 □

We see that there exists some u which is also the solution, in the weak sense, to the following equation:

$$-\Delta_{p} u = \bigl(s+c^{+}_{l}(s, t, s_{0}) \bigr)a\bigl(u^{+}\bigr)^{p-1} - \bigl(t+c^{+}_{l}(s, t, s_{0})\bigr)b\bigl(u^{-}\bigr)^{p-1}. $$

Clearly,

$$ C_{l}^{+}:= \bigl\{ \bigl(s + c^{+}_{l}(s, t, s_{0}), t + c^{+}_{l}(s, t, s_{0})\bigr): s_{0} = c^{+}_{l}(s, t, s_{0})\bigr\} \subset\Sigma_{p}(a, b), $$

(2.13)

where s, t, $s_{0}$ and $c^{+}_{l}(s, t, s_{0})$ are as in Theorem 2.1.

In a similar way, suppose $\lambda_{l-1}(b)<\lambda_{l}(b)$ for some fix $l\geq2$, and then we can get a corresponding conclusion as follows.

Theorem 2.2

Suppose that the weights $a(x)$ and $b(x)$ are nonnegative and, for some fixed constants $k'\geq1 \geq k>0$,

$$ 0\leq kb(x)\leq a(x)\leq k'b(x),\quad \textit{a.e. }x\in\Omega, $$

(2.14)

where k and $k'$ satisfy

$$\biggl({1\over k}-{1\over k'} \biggr) \lambda_{l}(b)< \bigl(\lambda_{l}(b)-\lambda_{l-1}(b) \bigr) (1-\delta), $$

where $\delta\in(0,1)$. Suppose that s and t are in some intervals as follows:

$$\begin{aligned}& t \in \biggl({\lambda_{l}(b)\over k}-\bigl(\lambda_{l}(b)- \lambda_{l-1}(b)\bigr),{\lambda_{l}(b)\over k'}-\delta\bigl( \lambda_{l}(b)-\lambda_{l-1}(b)\bigr) \biggr]; \end{aligned}$$

(2.15)

$$\begin{aligned}& s \in \bigl(\max \bigl\{ \lambda_{l-1}(b),(k-1) \bigl(\lambda_{l}(b)- \lambda _{l-1}(b)\bigr)+k't \bigr\} , \\& \hphantom{s \in{}}{}\min \bigl\{ k\bigl(\delta\bigl(\lambda_{l}(b)-\lambda _{l-1}(b)\bigr)+t\bigr),\lambda_{l}(b)-\delta\bigl( \lambda_{l}(b)-\lambda _{l-1}(b)\bigr) \bigr\} \bigr]. \end{aligned}$$

(2.16)

Then, for any $s_{0}$ in $[ \delta(\lambda_{l}(b)-\lambda_{l-1}(b)),\lambda_{l}(b)-\lambda _{l-1}(b) )$, $c_{l}^{-}(s, t, s_{0})$ is a critical value of $J_{s, t, s_{0}}^{-}(u)$ on $S(b)$. In particular, the set of the Fučik spectrum of $-\Delta_{p}$ with weights $a(x)$ and $b(x)$ greater than $c(a(x),b(x))$ includes $(t+c_{l}^{-}(s, t, s_{0}),s+c_{l}^{-}(s, t, s_{0}))$ for s and t in the above intervals, and some well-defined $s_{0} \in [\delta(\lambda_{l}(b)-\lambda_{l-1}(b)),\lambda _{l}(b)-\lambda_{l-1}(b) )$ satisfies

$$c_{l}^{-}(s, t, s_{0})=s_{0}. $$

Hence,

$$ C_{l}^{-} := \bigl\{ \bigl(t + c^{-}_{l}(s, t, s_{0}),s + c^{-}_{l}(s, t, s_{0})\bigr): s_{0}=c^{-}_{l}(s, t, s_{0})\bigr\} \subset \Sigma_{p} (a, b), $$

(2.17)

where s, t, $s_{0}$ and $c^{-}_{l}(s, t, s_{0})$ are as in Theorem 2.2.

Remark 2.1

Theorem 2.1 and Theorem 2.2 contain the results in [16], in which $a(x)=b(x)=1$; so our results must be much more generic than those in [16].

Remark 2.2

Theorem 2.1 and Theorem 2.2 answer partly the open question raised in [13]: whether there exists any other Fučik spectrum for (1.2) besides $\lambda_{1}(a)\times\mathbb{R}$, $\mathbb{R}\times\lambda_{1}(b)$ and $\mathcal{C}:=\{(\alpha(s),s\alpha(s)):\alpha(s)=c(a,sb),s>0\}$.

3 Resonance problems with respect to the Fučik spectrum

As an application, we consider the resonance problems for the Dirichlet boundary problem (1.1) of the Fučik spectrum type. The first case: $(\lambda, \mu)$ is in the part of the two lines $\lambda _{1}(a) \times R$ and $R \times\lambda_{1}(b)$, which has been defined in [13]. The second case: $(\lambda, \mu) \in C_{l}^{\pm}$, which is defined in Section 2; see equations (2.13) and (2.17). Similar to Section 2, we always assumed that $a(x), b(x)\in L^{r}$ with r as in the introduction. We assume that $g: \Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Caratheodory function. In order to state our main results, we assume that $g(x,u)$ satisfies the following conditions:

(f₁):

$g(x,u)\in C(\bar{\Omega}\times \mathbb{R},\mathbb{R})$ with $g(x,0)=0$ and $\lim_{|s|\rightarrow \infty}\frac{g(x,s)}{|s|^{p-1}}=0$ uniformly;

(f₂):

$pG(x,s)-g(x,s)s > 0$ a.e. $x \in\Omega$, $\forall s \neq0$;

(f₃):

$\lim_{|s|\rightarrow\infty}(g(x,s)s - pG(x,s))= -\infty$;

(f₄):

for some $v\in(1,p)$, there are constants $m, a_{m}>0$ s.t. $G(x,u)\geq a_{m} |u|^{v}$ for $|u|\leq m$;

(F₁):

$|g(x,u)|\leq V(x)^{p-q}|u|^{q-1}+W(x)^{p-1}$ with $q \leq p$ and nonnegative $V,W\in L^{p}(\Omega)$;

(F₂):

$\exists\theta\in(0, 1)$ and $M > 0$ such that $G(x,u) - \theta^{p} G(x, \frac{u}{\theta}) \leqslant M$, for $|u| \geqslant M$.

As is well known, the weak solution to the problem (1.1) is a critical point of

$$ \Phi(u)= \int_{\Omega}|\nabla u|^{p} -\lambda a(x) \bigl(u^{+} \bigr)^{p} - \mu b(x) \bigl(u^{-}\bigr)^{p} - pG(x, u),\quad \forall u\in W_{0}^{1, p}(\Omega), $$

(3.1)

where $G(x, t)=\int_{0}^{t} g(x, s)\,ds$, and then we have the following.

Theorem 3.1

Suppose that conditions (f₁)-(f₄) hold, $a(x), b(x)\geq \epsilon_{0} > 0$, a.e. in Ω. Then (1.1) has at least one nontrivial weak solution in $W_{0}^{1,p}(\Omega)$ if one of the following conditions holds:

(f₅):

$\lambda=\lambda_{1}(a)$; and μ satisfies

$$c(a,a)a(x)>\mu b(x)\quad \textit{and}\quad \mu \int_{\Omega}b(x)e_{a}^{p}(x)\,dx\geq 1; $$

(f₆):

$\mu=\lambda_{1}(b)$; and λ satisfies

$$c(b,b)b(x)>\lambda a(x)\quad \textit{and}\quad \lambda \int_{\Omega}a(x)e_{b}^{p}(x)\,dx\geq1. $$

Theorem 3.2

Suppose that (F₁), (f₃) and (f₅) hold, $a(x), b(x)\geq \epsilon_{0} > 0$, a.e. in Ω and satisfy the conditions mentioned in Theorem  2.1 (or in Theorem  2.2), and then the problem (1.1) has a nontrivial solution if $(\lambda,\mu)\in C_{l}^{+}$ (or $(\lambda,\mu)\in C_{l}^{-}$).

In order to prove Theorem 3.1, we need to state the following conclusions about critical groups. Firstly, we introduce some notations. Let $E:= W_{0}^{1, p}(\Omega)$, $E^{*}$ represent the dual space to E, and J be a $C^{1}(E,\mathbb{R})$. Then the critical set of J on E is defined as

$$K=\bigl\{ u \in E, J'(u)=\theta\bigr\} . $$

For $c\in\mathbb{R}\cup\{+\infty\}$, we use

$$J^{c}=\bigl\{ u\in E, J(u)\leq c\bigr\} ,\qquad J^{c}=\bigl\{ u\in E, J(u)< c\bigr\} $$

standing for the closed and open sublevel sets of functional J on E. Recall that the critical groups of J at its isolated point u with a critical value $c=J(u)$ are defined by

$$C_{q}(J,u)=H_{q}\bigl(J^{c}\cap U, \bigl(J^{c}\cap U\bigr)\setminus\{u\}\bigr), \quad \mbox{for all } q \geq0, $$

where U is an isolated neighborhood of u, and $H_{q}(\cdot,\cdot)$ is the qth singular relative homology groups with the integer coefficients $\mathbb{Z}$. Moreover, it is well known that $C_{q}(J,u)$ is independent of the choice of U and due to the excision property of singular homology theory. Suppose that $-\infty<\inf J(K)$, and we choose $c<\inf J(K)$. The critical groups of J at infinity are defined by

$$C_{q}(J,\infty)=H_{q}\bigl(E,J^{c}\bigr), \quad \mbox{for all } q \geq0. $$

From the deformation lemma, we see that $C_{q}(J,\infty)$ is independent of the choice of $c<\inf J(K)$ while J satisfies the (P.S.) condition. For details of the topological notation mentioned here and throughout this paper refer to [27].

Lemma 3.1

Suppose that conditions (f₁)-(f₃) hold and $\lambda =\lambda_{1}(a)$ (or $\mu=\lambda_{1}(b)$); then the functional Φ satisfies ($C_{c}$) condition for all $c \in \mathbb{R}$.

Proof

We just give the proof of the case when $\lambda=\lambda_{1}(a)$, and the other case of $\mu=\lambda_{1}(b)$ can be treated similarly.

Suppose that $\{u_{n}\}\in W_{0}^{1, p}$ is a sequence such that

$$\begin{aligned}& \Phi(u_{n})\rightarrow c, \end{aligned}$$

(3.2)

$$\begin{aligned}& \bigl(1+\Vert u_{n}\Vert _{1,p}\bigr) \bigl\Vert \Phi'(u_{n})\bigr\Vert _{E^{*}} \rightarrow0. \end{aligned}$$

(3.3)

It is sufficient to obtain that $\{u_{n}\}$ is bounded. Assume the contrary, that is, $\|u_{n}\|\rightarrow\infty$. Let $v_{n}=\frac{u_{n}}{\|u_{n}\|_{1,p}}$. Then, up to a subsequence, we still denote the subsequence by $\{v_{n}\}$. There is a $v_{0}$ such that $\{v_{n}\}$ converges weakly to $v_{0}$ in $W_{0}^{1,p}(\Omega)$, $v_{n}\rightarrow v_{0}$ strongly in $L^{p}(\Omega)$, and $v_{n}(x)\rightarrow v_{0}(x)$ almost everywhere in Ω. In (3.3) divided by $\|u_{n}\|_{1,p}^{p-1}$, and then for every $w \in W_{0}^{1,p}$, we have

$$ \int_{\Omega} \biggl[|\nabla v_{n}|^{p-2}\nabla v_{n} \nabla w -\lambda a(x) \bigl(v_{n}^{+} \bigr)^{p-1}w + \mu b(x) \bigl(v_{n}^{-}\bigr)^{p-1}w - \frac{g(x,u_{n})w}{\|u_{n}\|_{1,p}^{p-1}} \biggr] \,dx \rightarrow0. $$

(3.4)

In (3.4), first we claim that $\|u_{n}\|_{1,p}^{1-p}\int_{\Omega}g(x,u_{n})w\,dx \rightarrow0$. Indeed, it follows from (f₁) that, for any $\epsilon > 0$, there exists $C_{\epsilon} > 0$ such that

$$\begin{aligned} \bigl\vert g(x,u)\bigr\vert \leq C_{\epsilon}+\epsilon \vert u\vert ^{p-1}, \quad \forall (x,u)\in\bar{\Omega}\times\mathbb{R}. \end{aligned}$$

By Hölder’s inequality and Sobolev’s inequality, we get

$$ \biggl\vert \int_{\Omega} \frac{g(x,u_{n})w}{\|u_{n}\|_{1,p}^{p-1}} \,dx \biggr\vert \leq \int_{\Omega} \frac{C_{\epsilon}+\epsilon|u_{n}|^{p-1}}{\|u_{n}\|_{1,p}^{p-1}}|w| \,dx \leq\biggl( \frac{C_{1}}{\|u_{n}\|_{1,p}^{p-1}}+\epsilon C_{2}\biggr)\|w\| \rightarrow0, \quad n \rightarrow\infty. $$

Hence from (3.4) we have

$$ \int_{\Omega}\bigl[|\nabla v_{n}|^{p-2}\nabla v_{n}\nabla w -\lambda a(x) \bigl(v_{n}^{+} \bigr)^{p-1}w +\mu b(x) \bigl(v_{n}^{-}\bigr)^{p-1}w \bigr]\,dx \rightarrow0,\quad n\rightarrow\infty. $$

(3.5)

Putting $w=v_{n}-v_{0}$, it is sufficient to obtain

$$ \int_{\Omega}|\nabla v_{n}|^{p-2}\nabla v_{n}\nabla(v_{n} -v_{0}) \,dx \rightarrow0, \quad n\rightarrow\infty. $$

And consequently

$$ \int_{\Omega}\bigl(|\nabla v_{n}|^{p-2}\nabla v_{n} - |\nabla v_{0}|^{p-2}\nabla v_{0}\bigr)\nabla(v_{n} -v_{0}) \,dx \rightarrow0,\quad n\rightarrow\infty. $$

Using the inequality

$$ |\xi-\eta|^{p} \leq c\bigl[\bigl(|\xi|^{p-2}\xi-| \eta|^{p-2}\eta\bigr) (\xi- \eta)\bigr]^{s/2}\bigl[| \xi|^{p}+|\eta|^{p}\bigr]^{1-s/2}, $$

where $\xi, \eta\in\mathbb{R}^{N}$, $c=c(p)>0$, and $s=2$ if $p\geq 2$, $s=p$ if $1< p<2$, one easily obtains $v_{n} \rightarrow v_{0}$ in $W_{0}^{1, p}$ with $\|v_{0}\|_{1, p}=1$.

Hence from (3.5) we see that

$$ \int_{\Omega}|\nabla v_{0}|^{p-2}\nabla v_{0}\nabla w = \int_{\Omega}\bigl[\lambda a(x) \bigl(v_{0}^{+} \bigr)^{p-1}w +\mu b(x) \bigl(v_{0}^{-}\bigr)^{p-1}w \bigr]\,dx, \quad \forall w\in W_{0}^{1,p}. $$

(3.6)

Taking $w=v_{0}^{\pm}$, respectively, we get

$$ \int_{\Omega}\bigl\vert \nabla v_{0}^{+}\bigr\vert ^{p} =\lambda_{1}(a) \int_{\Omega}a(x)\bigl\vert v_{0}^{+}\bigr\vert ^{p} $$

(3.7)

and

$$ \int_{\Omega}\bigl\vert \nabla v_{0}^{-}\bigr\vert ^{p} =\mu \int_{\Omega}b(x)\bigl\vert v_{0}^{-}\bigr\vert ^{p}, $$

(3.8)

respectively. From the above equality (3.7), we know either $v_{0}^{+} = 0$ or $v_{0} = e_{a}$, where $e_{a}$ is the eigenfunction corresponding to $\lambda= \lambda_{1}(a)$. If $v_{0} = e_{a}$, then $u_{n}(x) = \|u_{n}\|_{1, p} e_{a} \rightarrow+\infty$ for almost everywhere in Ω. If $v_{0}^{+} = 0$, then we must have $v_{0} = -e_{b} < 0$ where $e_{b}$ is the eigenfunction corresponding to $\mu= \lambda_{1}(b)$, since $v_{0} \neq0$ and equality (3.8), then $u_{n}(x) =- \|u_{n}\|_{1, p} e_{b} \rightarrow-\infty$ for almost everywhere in Ω.

From assumption (f₃), we have

$$ \lim_{n\rightarrow\infty}\bigl(g(x, u_{n})u_{n} - pG(x, u_{n})\bigr)= -\infty,\quad \mbox{a.e. } x \in\Omega, \mbox{as } n\rightarrow\infty. $$

(3.9)

However, by (3.2) and (3.3), one has

$$ p\Phi(u_{n})-\bigl\langle \Phi'(u_{n}),u_{n} \bigr\rangle \rightarrow pc,\quad n\rightarrow\infty, $$

which implies that

$$ \int_{\Omega}\bigl(g(x,u_{n})u_{n} - pG(x,u_{n})\bigr)\,dx\rightarrow pc, \quad \mbox{as } n\rightarrow \infty. $$

This contradicts (3.9). Thus $\{u_{n}\}$ is bounded, and the proof completed. □

Lemma 3.2

Suppose that conditions (f₁)-(f₃) and (f₅) (or (f₆)) hold, $a(x), b(x)\geq\epsilon_{0} > 0$, a.e. in Ω. Then we have

$$ C_{1}(\Phi,\infty)\neq0. $$

Proof

We just give the proof of the case (f₅), and the other case (f₆) can be treated in a similar way.

By (f₁), we know that Φ is of class $C^{1}$ in $W_{0}^{1,p}(\Omega)$. We consider the decomposition of $W_{0}^{1,p}=V \oplus W$, where $V=\operatorname{span} \{e_{a}\}$ is the one-dimensional eigenspace associated with $\lambda_{1}(a)$, with $e_{a} > 0$ in Ω and $\|e_{a}\|_{1,p}=1$, and $W := \{w\in W_{0}^{1,p}(\Omega): \int_{\Omega}a(x)we_{a}^{p-1}\,dx=0\}$ is a complementary subspace of V in $W_{0}^{1,p}(\Omega)$. Therefore (see [28]), there exists $\bar{\lambda} >\lambda_{1}(a)$ such that

$$ \int_{\Omega}|\nabla u|^{p}\,dx \geq\bar{\lambda} \int_{\Omega}a(x)|u|^{p} \,dx, \quad \forall u\in W. $$

We claim that $\bar{\lambda}=c(a,a)$, if it is not the case, that is, there exists $u\in W$, such that

$$ \lambda_{1}(a) \int_{\Omega}a(x)|u|^{p} < \int_{\Omega}|\nabla u|^{p} < c(a,a) \int_{\Omega}a(x)|u|^{p}. $$

Hence there exist $\lambda_{1}(a)<\theta<c(a,a)$ and $u\in W$, such that

$$ \int_{\Omega}|\nabla u|^{p}=\theta \int_{\Omega}a(x)|u|^{p}. $$

Hence, θ is the eigenvalue of problem (1.3), which contradicts Theorem 11 in [13] stating that problem (1.3) does not admit any eigenvalue in the open interval $( \lambda_{1}(a), c(a, a) )$. Then we have

$$ \int_{\Omega}|\nabla u|^{p}\,dx \geq c(a,a) \int_{\Omega}a(x)|u|^{p} \,dx,\quad \forall u\in W. $$

(3.10)

On the other hand, (f₁) implies that

$$ \lim_{|s|\rightarrow\infty}\frac{pG(x,s)}{|s|^{p}}=0. $$

And from (f₃), we obtain: for any $M > 0$, there is some $R > 0$ such that

$$ g(x,s)s-pG(x,s)\leq-M,\quad |s| > R, \mbox{a.e. } x\in\Omega. $$

Integrating the equality

$$ \frac{d}{\,dx} \biggl[\frac{G(x,s)}{|s|^{p}} \biggr]=\frac {g(x,s)s-pG(x,s)}{|s|^{p+1}}, $$

over the interval $[t,T]\subset[R,+\infty)$, we have

$$ \frac{G(x,T)}{|T|^{p}}-\frac{G(x,t)}{|t|^{p}}\leq\frac{M}{p}\biggl( \frac {1}{T^{p}}-\frac{1}{t^{p}}\biggr). $$

(3.11)

Noting that $\lim_{T\rightarrow\infty}T^{-p}G(x,T)=0$, and letting $T\rightarrow\infty$ in (3.11), we have

$$ G(x,t)\geq\frac{M}{p}, \quad \forall t \geq R, \mbox{a.e. } x\in \Omega. $$

(3.12)

Similarly, it is shown that $G(x,t)\geq\frac{M}{p}$, $\forall t \leq-R$, a.e. $x\in\Omega$. Hence

$$ \lim_{|t|\rightarrow\infty}G(x,t)=+\infty,\quad \mbox{a.e. } x\in \Omega. $$

Now, letting $v=\xi e_{a}\in V$ ($\xi\in\mathbb{R}$), from the assumption (f₅) we have

$$\begin{aligned} \Phi(v) =& \frac{1}{p} \int_{\Omega}|\nabla v|^{p} \,dx-\frac{1}{p} \int_{\Omega} \bigl[\lambda_{1}(a)a(x) \bigl(v^{+} \bigr)^{p}+\mu b(x) \bigl(v^{-}\bigr)^{p} \bigr]\,dx- \int_{\Omega}G(x,v)\,dx \\ =& \left \{ \textstyle\begin{array}{l@{\quad}l} \frac{1}{p}\xi^{p}\int_{\Omega}|\nabla e_{a}|^{p}\,dx - \frac{1}{p}\lambda_{1}(a)\xi^{p}\int_{\Omega}a(x)e_{a}^{p}\,dx - \int_{\Omega}G(x,\xi e_{a}), &\xi\geq0, \\ \frac{1}{p}|\xi|^{p}\int_{\Omega}|\nabla e_{a}|^{p}\,dx - \frac{1}{p}\mu|\xi|^{p}\int_{\Omega}b(x)e_{a}^{p}\,dx - \int_{\Omega}G(x,\xi e_{a}), &\xi< 0 \end{array}\displaystyle \right . \\ =& \left \{ \textstyle\begin{array}{l@{\quad}l} -\int_{\Omega}G(x,\xi e_{a}),& \xi\geq0, \\ \frac{1}{p}|\xi|^{p}[1 - \mu\int_{\Omega}b(x)e_{a}^{p}\,dx] - \int _{\Omega}G(x,\xi e_{a}),& \xi< 0 \end{array}\displaystyle \right . \\ \leq& - \int_{\Omega}G(x,\xi e_{a}). \end{aligned}$$

It follows from Lebesgue’s theorem that

$$ \Phi(v)\rightarrow-\infty,\quad \mbox{for } v\in V \quad \mbox{and}\quad \|v \|_{1, p}\rightarrow\infty. $$

Now we turn to proving

$$ \inf_{W}\Phi=m > -\infty. $$

(3.13)

Indeed, by (f₁),we obtain that, for any $\epsilon > 0$, there exists $C_{\epsilon} > 0$ such that

$$\begin{aligned} \bigl\vert G(x,u)\bigr\vert \leq C_{\epsilon}+\epsilon \vert u\vert ^{p}, \quad \forall (x,u)\in\bar{\Omega}\times \mathbb{R}. \end{aligned}$$

(3.14)

By (3.10) and (3.14), we have, for any $w \in W$,

$$\begin{aligned} p\Phi(w) =& \int_{\Omega}|\nabla w|^{p}\,dx - \int_{\Omega}\bigl[\lambda_{1}({a})a(x) \bigl(w^{+} \bigr)^{p}+\mu b(x) \bigl(w^{-}\bigr)^{p}+pG(x,w)\bigr]\,dx \\ \geq& c(a,a) \int_{\Omega}a(x)|w|^{p}\,dx- \int_{\Omega}\bigl[\lambda _{1}({a})a(x) \bigl(w^{+} \bigr)^{p}+\mu b(x) \bigl(w^{-}\bigr)^{p}+pG(x,w)\bigr]\,dx \\ =& \bigl(c(a,a)-\lambda_{1}(a)\bigr) \int_{\Omega}a(x) \bigl(w^{+}\bigr)^{p}\,dx \\ &+ \int_{\Omega}\bigl[c(a,a)a(x)-\mu b(x)\bigr]\bigl(w^{-} \bigr)^{p}\,dx-p \int_{\Omega}G(x,w)\,dx \\ \geq& \int_{\Omega}\bigl[\bigl(c(a,a)-\lambda_{1}(a) \bigr)a(x)-p\epsilon\bigr]\bigl(w^{+}\bigr)^{p}\,dx \\ &{}+ \int_{\Omega}\bigl(c(a,a)a(x)-\mu b(x)-p\epsilon\bigr) \bigl(w^{-} \bigr)^{p}\,dx-pC_{\epsilon}|\Omega|. \end{aligned}$$

With the condition (f₅), we can choose ϵ small enough such that $(c(a,a)-\lambda_{1}(a))a(x)-p\epsilon\geq0$, $c(a,a)a(x)-\mu b(x)-p\epsilon\geq0$. Then fix ϵ, and we have $\Phi(w)\geq-pC_{\epsilon}|\Omega|$ where $-pC_{\epsilon}|\Omega|$ is a constant. Obviously (3.13) holds true.

We suppose that Φ has only a finite number of critical points. Then, for sufficiently small number c, we have by [24], that

$$C_{1}(\Phi,\infty)=H_{1}\bigl(E,\Phi^{c}\bigr) \neq0. $$

 □

Lemma 3.3

Theorem 2.1 in [21]

Suppose that conditions (f₁)-(f₄) hold, and let zero be an isolated critical point of Φ. Then the Morse critical groups for Φ at zero are trivial, that is,

$$ C_{q}(\Phi,0)= 0, \quad \textit{for all } q. $$

Proof of Theorem 3.1

In view of Lemma 3.1 and Lemma 3.2, we have $C_{1}(\Phi,\infty)\neq0$, and according to the Morse equality, the functional Φ has at least one critical point u such that $C_{1}(\Phi,u)\neq0$. Since $C_{1}(\Phi,0)= 0$ by Lemma 3.3, we conclude that $u\neq0$. The proof is complete. □

Proof of Theorem 3.2

Here we only consider the case $(\lambda, \mu)\in C_{l}^{+}$, since the case $(\lambda, \mu)\in C_{l}^{-}$ can be done similarly. Let $(\lambda, \mu)=(s+c_{l}^{+}(s, t, s_{0}), t+c_{l}^{+}(s, t, s_{0}))$, where $s_{0} = c_{l}^{+}(s, t, s_{0})$. We introduce a new functional as follows:

$$\begin{aligned} \Phi_{j}(u) =&\Phi(u)+\frac{1}{j} \int_{\Omega}a|u|^{p} \\ =& \int_{\Omega}|\nabla u|^{p}-sa\bigl(u^{+} \bigr)^{p}-\bigl(t+c_{l}^{+}(s, t, s_{0})\bigr)b \bigl(u^{-}\bigr)^{p}+c_{l}^{+}(s, t, s_{0})a\bigl(u^{-} \bigr)^{p} \\ &{}-\biggl(c_{l}^{+}(s, t, s_{0})-\frac{1}{j} \biggr)a|u|^{p}-pG(x,u). \end{aligned}$$

(3.15)

First we will show that, for sufficiently large j, there is some $u_{j}\in W_{0}^{1,p}(\Omega)$ such that

$$ \Vert u_{j}\Vert \bigl\Vert \Phi_{j}'(u_{j}) \bigr\Vert \rightarrow0, \qquad \inf\Phi_{j}(u_{j})> - \infty. $$

(3.16)

Let $A^{l-2} \in{\mathcal{F}}_{l-1}^{+}(a)$ be as in Section 2. By (2.12), there exists $j_{0}$, for any $j \geqslant j_{0}$, and we have

$$ \max_{u\in A^{l-2}}J_{s, t, s_{0}}^{+}(u)\leq c_{l}^{+}(s, t, s_{0})-\frac{2}{j }. $$

For such $j, u \in A^{l-2}$ and $R>0$, by (F₁) and $a(x) \geqslant \epsilon_{0}$ (a.e. in Ω) we will have

$$\begin{aligned} \Phi_{j}(Ru) =&R^{p}\biggl[J_{s, t, s_{0}}^{+} - \biggl(c_{l}^{+}(s, t, s_{0})-\frac{1}{j}\biggr) \biggr]-p \int_{\Omega}G(x,Ru) \\ \leq& -\frac{R^{p}}{j} + p\bigl(\|V\|_{p}^{p-q} \|u \|_{p}^{q} R^{q} + \|W\| _{p}^{p-1} \|u\|_{p} R\bigr) \\ \leq&-\frac{R^{p}}{j}+C\bigl(\|V\|_{p}^{p-q}R^{q}+ \|W\|_{p}^{p-1}R\bigr) \\ \leq& - \frac{R^{p}}{j} + C\bigl(R^{q} + 1\bigr). \end{aligned}$$

Then we see that

$$ \max_{A^{l-2}}\Phi_{j}(Ru)\rightarrow- \infty,\quad \mbox{as }R\rightarrow \infty. $$

(3.17)

Let

$$ F=\biggl\{ u\in W_{0}^{1,p}( \Omega):J_{s, t, s_{0}}^{+}\geq c_{l}^{+}(s, t, s_{0}) \int_{\Omega}a|u|^{p}\biggr\} . $$

(3.18)

For $u \in F$, by (3.18) and (F₁), we can deduce the following from (3.15):

$$\begin{aligned} \Phi_{j}(u) \geq& \frac{\int_{\Omega}a|u|^{p}}{j} - p\bigl(\|V \|_{p}^{p-q}\|u\|_{p}^{q}+\| W \|_{p}^{p-1}\|u\|_{p}\bigr) \\ \geq& \frac{\int_{\Omega}a|u|^{p}}{j} - C \biggl(\|V\|_{p}^{p-q}\biggl( \int _{\Omega}a|u|^{p}\biggr)^{\frac{q}{p}}+\|W \|_{p}^{p-1}\biggl( \int_{\Omega }a|u|^{p}\biggr)^{\frac{1}{p}} \biggr) \\ \geq& \frac{\int_{\Omega}a|u|^{p}}{j} - C \biggl( \biggl( \int_{\Omega }a|u|^{p}\biggr)^{\frac{q}{p}} + 1 \biggr). \end{aligned}$$

Denote $r := (\int_{\Omega}a |u|^{p})^{\frac{1}{p}}$, $k(r) := \frac {r^{p}}{j} - C (r^{q}+1)$. It is sufficient to deduce that $k(r)$ ($r \in[0, +\infty)$) takes its minimum at point $r_{m} = (\frac {Cqj}{p})^{\frac{1}{p-q}}$, and its minimum is

$$ \max_{r \geqslant0} k(r) = k\biggl(\biggl(\frac{Cqj}{p} \biggr)^{\frac{1}{p-q}}\biggr) = \biggl(\frac{qj}{p}\biggr)^{\frac{q}{p-q}} C^{\frac{p}{p-q}} \biggl(\frac{q}{p} - 1\biggr) -C. $$

Thus, we have

$$ \inf_{u\in F}\Phi_{j}(u)\geq C_{j} := \min_{r \geq 0} k(r) = \biggl(\frac{qj}{p} \biggr)^{\frac{q}{p-q}} C^{\frac{p}{p-q}} \biggl(\frac {q}{p} - 1\biggr) -C >- \infty. $$

(3.19)

Now using (3.17), for any $j \geqslant j_{0}$, we can choose sufficiently high $R_{j} > 0$, thus

$$ \max\Phi_{j}(B)< C_{j}, $$

(3.20)

where $B=\{R_{j}u: u \in A^{l-2}\}$. In fact, (3.20) holds only if $R_{j}$ is larger than the positive solution of the following equation:

$$ - \frac{R^{p}}{j} + C\bigl(R^{q} + 1\bigr) = C_{j} = \biggl(\frac{qj}{p}\biggr)^{\frac{q}{p-q}} C^{\frac{p}{p-q}} \biggl( \frac{q}{p} - 1\biggr) -C. $$

It is sufficient to find that (3.20) holds as $R_{j} \geqslant C j^{\frac{1}{p - q}}$. As a matter of convenience, and without loss of generality, we suppose $R_{j_{0}}$ is large enough and satisfies

$$ R_{j +1} = \frac{1}{\theta} R_{j}, \quad j = j_{0}, j_{0} + 1, \ldots. $$

(3.21)

Then we will consider the homotopy-stable family of compact subsets of $W_{0}^{1, p}(\Omega)$ with boundary B given by

$$\begin{aligned} \begin{aligned} &\mathcal{F}_{j}^{+} = \bigl\{ A\subset W_{0}^{1, p}: \mbox{there is a continuous surjection } h:S_{+}^{l-1}\rightarrow A \\ &\hphantom{\mathcal{F}_{j}^{+} ={}}{}\mbox{such that } h|_{S^{l-2}}=R_{j} h_{l-2 } \bigr\} , \end{aligned} \end{aligned}$$

where $h_{l-2}:S^{l-2}\to A^{l-2}$ is as in Section 2. We claim that the set F is dual to the class $\mathcal{F}_{j}^{+}$, i.e.,

$$ F\cap B=\emptyset,\qquad F\cap A\neq\emptyset,\quad \forall A\in \mathcal {F}_{j}^{+}. $$

(3.22)

It is clear from (3.19) and (3.20) that $F\cap B=\emptyset$. Let $A \in\mathcal{F}_{j}^{+}$. If $0\in A$, then we are done. Otherwise, we denote by π the radial projection onto S, $\pi(A)\in\mathcal{F}_{l}^{+}$, and we have

$$ \max_{u\in\pi(A)}J_{s, t, s_{0}}^{+}(u)\geq c_{l}^{+}(s, t, s_{0}). $$

So $F\cap\pi(A)\neq\emptyset$, and we obtain $F\cap A\neq\emptyset$.

Now let us define a minimax sequence on the class $\mathcal{F}^{+}$:

$$ c_{j}:=\inf_{A\in\mathcal{F}_{j}^{+}}\max _{u\in A} \Phi_{j}(u). $$

(3.23)

Noting the truth that the set F is dual to the class $\mathcal {F}_{j}^{+}$, and by (3.19), we have

$$ c_{j} \geqslant C_{j} > \max _{u\in R_{j} A^{l-2}} \Phi_{j}(u). $$

(3.24)

Hence it follows from a deformation argument of Cerami [22] that, for the above j, there exists a sequence ${u_{j}}$ such that

$$ \Vert u_{j}\Vert \bigl\Vert \Phi_{j}'(u_{j}) \bigr\Vert < \frac{1}{j}, \qquad \bigl\vert \Phi_{j}(u_{j})-c_{j} \bigr\vert < \frac{1}{j}. $$

(3.25)

It is easy to prove that $\{c_{j}\}_{j_{0}}^{\infty}$ is a non-decreasing sequence. According to (3.24) and (3.25), for $j \geqslant j_{0}$ we can get

$$\begin{aligned} c_{j_{0}} + \frac{1}{j_{0}} \geqslant& c_{j} + \frac{1}{j} \\ \geqslant& \Phi_{j} (u_{j}) \\ \geqslant& c_{j} - \frac{1}{j} \\ \geqslant& C_{j} - \frac{1}{j} \\ =& \biggl(\frac{qj}{p}\biggr)^{\frac{q}{p-q}} C^{\frac{p}{p-q}} \biggl( \frac{q}{p} - 1\biggr) -C. \end{aligned}$$

(3.26)

We claim that the sequence $\{c_{j}\}_{j_{0}}^{\infty}$ is bounded. Assume the contrary, that is, $c_{j} \rightarrow-\infty$ as $j \rightarrow \infty$, and then we will prove that $\{c_{j}\}_{j_{0}}^{\infty}$ has some fixed growth rate, which contradicts (3.26).

In fact, for a fixed number $j \geqslant j_{0}$ and $\forall\epsilon> 0$, there exists $A \in\mathcal{F}_{j}^{+}$ and a continuous surjection h, such that $A = h(S_{+}^{l - 1})$ and

$$ \max_{u \in S_{+}^{l - 1}} \Phi_{j} \bigl(h(u)\bigr) < c_{j} +\epsilon. $$

Noting that $H(\cdot) := \frac{R_{j + 1}}{R_{j}} h(\cdot) = \theta ^{-1}h(\cdot) \in\mathcal{F}_{j + 1}^{+}$. Therefore, for all $u \in S_{+}^{l - 1}$, we have

$$\begin{aligned} c_{j} + \epsilon >& \Phi_{j}\bigl(h(u)\bigr) = \Phi \bigl(h(u)\bigr) + \frac{1}{j} \int _{\Omega} a \bigl\vert h(u)\bigr\vert ^{p} \\ =& \int_{\Omega} \bigl\vert \nabla h(u)\bigr\vert ^{p} - \bigl(s + c_{l}^{+}(s, t, s_{0})\bigr) \bigl(h(u)^{+} \bigr)^{p} - \bigl(t + c_{l}^{+}(s, t, s_{0})\bigr) \bigl(h(u)^{-}\bigr)^{p} \\ &{}- p G\bigl(x, h(u)\bigr) + \frac{1}{j} \int_{\Omega} a \bigl\vert h(u)\bigr\vert ^{p} \\ =& \biggl( \frac{R_{j}}{R_{j + 1}} \biggr)^{p} \biggl( \int_{\Omega} \bigl\vert \nabla H(u)\bigr\vert ^{p} - \bigl(s + c_{l}^{+}(s, t, s_{0})\bigr) \bigl(H(u)^{+} \bigr)^{p} - \bigl(t + c_{l}^{+}(s, t, s_{0})\bigr) \bigl(H(u)^{-}\bigr)^{p} \\ &{}- p G\bigl(x, H(u)\bigr) + \frac{1}{j + 1} \int_{\Omega} a \bigl\vert H(u)\bigr\vert ^{p} \biggr) \\ &{}+ \biggl( \frac{R_{j}}{R_{j + 1}} \biggr)^{p} \int_{\Omega} p G\bigl(x, H(u)\bigr) - \int_{\Omega} p G\bigl(x, h(u)\bigr) + \frac{1}{j(j + 1)} \int _{\Omega} a \bigl\vert h(u)\bigr\vert ^{p} \\ =& \theta^{p} \Phi_{j + 1}\bigl(H(u)\bigr) + \theta^{p} \int_{\Omega} p G\biggl(x, \frac{h(u)}{\theta}\biggr) - \int_{\Omega} p G\bigl(x, h(u)\bigr) + \frac{1}{j(j + 1)} \int_{\Omega} a \bigl\vert h(u)\bigr\vert ^{p} \\ \geq& \theta^{p} \Phi_{j + 1}\bigl(H(u)\bigr) - C_{M}, \end{aligned}$$

where $C_{M}$ is a constant and only related to the condition (F₂). Noting the arbitrariness of $u \in S_{+}^{l - 1}$, we have

$$\begin{aligned} c_{j} + \epsilon \geqslant& \theta^{p} \max _{u \in S_{+}^{l - 1}} \Phi_{j + 1}\bigl(H(u)\bigr) - C_{M} \\ \geqslant& \theta^{p} c_{j + 1} - C_{M}. \end{aligned}$$

According to the arbitrariness of ϵ, we have

$$ c_{j} \geqslant\theta^{p} c_{j + 1} - C_{M}. $$

From the assumption that $c_{j} \rightarrow- \infty$ as $j \rightarrow \infty$, there will exist $j_{1} \geqslant j_{0}$, such that $c_{j_{1}} < 0$, and then we can get

$$ |c_{j} | \leqslant\theta^{p} |c_{j + 1}| + C_{M}, \quad j = j_{1}, j_{1} +1, \ldots. $$

Suppose that $|c_{j} |$ is large enough, and then we can translate the last formula into the following:

$$ |c_{j} | \leqslant\theta|c_{j + 1}|,\quad j = j_{1}, j_{1} +1, \ldots. $$

Then we have

$$ |c_{j + 1} | \geqslant\theta^{j_{1} - j - 1} |c_{j_{1}}|, \quad j = j_{1}, j_{1} +1, \ldots, $$

contradicting (3.26). Thus, $\{c_{j}\}_{j_{0}}^{\infty}$ is bounded. According to (3.25), we obtain (3.16).

We complete the proof by showing that a subsequence of $\{u_{j}\}$ mentioned in (3.16) converges to a solution to (1.1). It is sufficient to prove that $\{u_{j}\}$ is bounded. Assume the contrary, that is, $\|u_{j}\|\rightarrow\infty$. Setting $v_{j}:=\frac{u_{j}}{\|u_{j}\|}$, then up to a subsequence, we still denote the subsequence by $\{v_{j}\}$. There is a $v_{0}$ such that $v_{j}\rightharpoonup v_{0}$ weakly in $W_{0}^{1,p}(\Omega)$, strongly in $L^{p}(\Omega)$, and almost everywhere in Ω. We have

$$\begin{aligned} \int_{\Omega}|\nabla v_{j}|^{p-2}\nabla v_{j}\nabla (v_{j}-v_{0}) =&\frac{(\Phi_{j}'(u_{j}),v_{j}-v_{0})}{p\|u_{j}\|_{1,p}^{p-1}} + \int_{\Omega}\biggl[\biggl(\lambda-\frac{1}{j}\biggr) a \bigl(v_{j}^{+}\bigr)^{p-1} \\ &{}+\biggl(\mu b-\frac{1}{j}a\biggr) \bigl(v_{j}^{-} \bigr)^{p-1} -\frac{g(x, u_{j})}{\|u_{j}\|_{1,p}^{p-1}}\biggr](v_{j}-v_{0}) \rightarrow0. \end{aligned}$$

It is sufficient to obtain

$$ \int_{\Omega}|\nabla v_{j}|^{p-2}\nabla v_{j}\nabla(v_{j} -v_{0}) \,dx \rightarrow0, \quad j\rightarrow\infty. $$

Hence, we can deduce that $v_{j}\rightarrow v_{0}$ strongly in $W_{0}^{1,p}$, and $\|v_{0}\|=1$. For each $w \in W_{0}^{1, p}$, we have

$$\begin{aligned} \frac{(\Phi_{j}'(u_{j}),w)}{p\|u_{j}\|_{1,p}^{p-1}} =& \int_{\Omega }|\nabla v_{j}|^{p-2}\nabla v_{j}\nabla w - \int_{\Omega}\biggl[\biggl(\lambda-\frac{1}{j}\biggr) a \bigl(v_{j}^{+}\bigr)^{p-1} \\ &{} +\biggl(\mu b-\frac{1}{j}a\biggr) \bigl(v_{j}^{-} \bigr)^{p-1} -\frac{g(x,u_{j})}{\|u_{j}\|_{1,p}^{p-1}}\biggr] w. \end{aligned}$$

Going to the limit as $j \rightarrow\infty$, we get

$$ \int_{\Omega}|\nabla v_{0}|^{p-2}\nabla v_{0}\nabla w - \bigl[\lambda a\bigl(v_{0}^{-} \bigr)^{p-1}-\mu b\bigl(v_{0}^{+}\bigr)^{p-1}\bigr]w = 0. $$

So $v_{0}\in E_{\lambda a,\mu b}$. Thus,

$$ \frac{(\Phi_{j}'(u_{j}),u_{j})}{p}-\Phi_{j}(u_{j})= \int_{\Omega }\bigl[pG(x,u_{j})-g(x,u_{j})u_{j} \bigr]\rightarrow +\infty, $$

contradicting (3.16). Hence, $\{u_{j}\}$ is bounded. There exists a subsequence of $\{u_{j}\}$ converging to a weak solution to (1.1). □