Open Access

Multiplicity results for nonlinear Neumann boundary value problems involving p-Laplace type operators

Boundary Value Problems20162016:95

https://doi.org/10.1186/s13661-016-0603-x

Received: 3 December 2015

Accepted: 3 May 2016

Published: 9 May 2016

Abstract

We consider the existence of at least two or three distinct weak solutions for the nonlinear elliptic equations
$$ \textstyle\begin{cases} {-}\operatorname{div}(\varphi(x,\nabla u))+{|u|}^{p-2}u= \lambda f(x,u) &\mbox{in } \Omega,\\ \varphi(x,\nabla u) \frac{\partial u}{\partial n}= \lambda g(x,u) & \mbox{on }\partial\Omega. \end{cases} $$
Here the function \(\varphi(x,v)\) is of type \(|v|^{p-2}v\) and the functions f, g satisfy a Carathéodory condition. To do this, we give some critical point theorems for continuously differentiable functions with the Cerami condition which are extensions of the recent results in Bonanno (Adv. Nonlinear Anal. 1:205-220, 2012) and Bonanno and Marano (Appl. Anal. 89:1-10, 2010) by applying Zhong’s Ekeland variational principle.

Keywords

p-Laplace type operator weak solutions Cerami condition multiple critical points

MSC

35B38 35D30 35J20 35J60 35J66

1 Introduction

In the present paper, we are concerned with multiple solutions for the nonlinear Neumann boundary value problem associated with p-Laplacian type,
$$ \textstyle\begin{cases} -\operatorname{div}(\varphi(x,\nabla u))+{|u|}^{p-2}u= \lambda f(x,u) &\mbox{in } \Omega, \\ \varphi(x,\nabla u) \frac{\partial u}{\partial n}= \lambda g(x,u) & \mbox{on }\partial\Omega, \end{cases} $$
(P)
where the function \(\varphi(x,v)\) is of type \(\vert v\vert ^{p-2}v\) with a real constant \(p>1\), Ω is a bounded domain in \(\Bbb {R}^{N}\) with Lipschitz boundary Ω, \(\frac{\partial u}{\partial n}\) denotes the outer normal derivative of u with respect to Ω, and the functions f, g satisfy a Carathéodory condition. Concerning elliptic equations with nonlinear boundary conditions, we refer to [37].

Ricceri’s three-critical-points theorems which are important to obtain the existence of at least three weak solutions for nonlinear elliptic equations have been extensively studied by various researchers; see [811]. It is well known that Ricceri’s theorems in [12, 13] gave no further information on the size and location of an interval of values λ in \(\Bbb {R}\) for the existence of at least three critical points.

Based on [1, 2], Bonanno and Chinnì [14] obtained the existence of at least two or three distinct weak solutions for nonlinear elliptic equations with the variable exponents whenever the parameter λ belongs to a precise positive interval. To obtain the existence of two distinct weak solutions for this problem, they assumed that the nonlinear term f satisfies the Ambrosetti and Rabinowitz condition (the (AR) condition, for short) in [15]:
  1. (AR)
    There exist positive constants M and θ such that \(\theta>p\) and
    $$0< \theta F(x,t)\le f(x,t)t, \quad\mbox{for } x\in\Omega \mbox{ and } \vert t \vert \ge M, $$
    where \(F(x,t)=\int_{0}^{t}f(x,s)\,ds\) and Ω is a bounded domain in \(\Bbb {R}^{N}\).
     
Moreover, in order to determine the precise positive interval of the parameter for the existence of multiple solutions of the nonlinear elliptic equations, they observed an embedding constant of the variable exponent Sobolev spaces into the variable exponent Lebesgue spaces by using Talenti’s inequality (see [16]).

The goal of the present paper is to establish the existence of at least two or three weak solutions for the problem (P) whenever the parameter λ belongs to a precise positive interval. To do this, we give a theorem which is an extension of the recent critical point theorem in [1] by considering Zhong’s Ekeland variational principle. Roughly speaking, we give this result under the Cerami condition which is another compactness condition of the Palais-Smale type introduced by Cerami [17]. First we show the existence of at least two weak solutions for (P) without assuming that f satisfies the (AR) condition. In recent years, some authors in [1823] have tried to drop the (AR) condition that is crucial to guarantee the boundedness of the Palais-Smale sequence of the Euler-Lagrange functional which plays a decisive role in applying the critical point theory. In this respect, we observe that the energy functional associated with (P) satisfies the Cerami condition when the nonlinear term f does not satisfies (AR) condition. This together with the best Sobolev trace constant given in [3] yields the existence of at least two weak solutions for (P). Finally, as an application of the recent three-critical-points theorem introduced by [2], we show that the problem (P) has at least three weak solutions provided that λ is suitable.

This paper is organized as follows. In Section 2, by using Zhong’s Ekeland variational principle, we state some critical point theorems for continuously differentiable functions with the Cerami condition. In Section 3, we state some basic results for the integral operators corresponding to the problem (P) under certain conditions on φ, f, and g. In Sections 4 and 5, we establish the existence of at least two or three distinct weak solutions for the problem (P) using some properties in Sections 2 and 3.

2 Abstract critical point theorem

Let \((X, \|\cdot\|)\) be a real Banach space. We denote the dual space of X by \(X^{*}\), while \(\langle\cdot,\cdot\rangle\) stands for the duality pairing between \(X^{*}\) and X. A function \(I:X\rightarrow\Bbb {R}\) is called locally Lipschitz when, with every \(u\in X\), there corresponds a neighborhood U of u and a constant \(L\ge0\) such that
$$ \bigl\vert I(v)-I(w)\bigr\vert \le L\|v-w\|_{X} \quad\mbox{for all } v,w\in U. $$
If \(u,v\in X\), the symbol \(\langle J'(u), v\rangle\) indicates the generalized directional derivative of I at point u along direction v, namely
$$ \bigl\langle J'(u), v\bigr\rangle :=\limsup_{w\rightarrow u, t\rightarrow0^{+}} \frac{I(w+tv)-I(w)}{t}. $$
The generalized gradient of the function I at u, denoted by \(\partial I(u)\), is the set
$$ \partial I(u):= \bigl\{ u^{*}\in X^{*} : \bigl\langle u^{*},v\bigr\rangle \le\bigl\langle J'(u), v\bigr\rangle \mbox{ for all } v\in X \bigr\} . $$
A function \(I:X\rightarrow\Bbb {R}\) is called Gâteaux differentiable at \(u\in X\) if there is \(\varphi\in X^{*}\)(denoted by \(I'(u)\)) such that
$$ \lim_{t\rightarrow0^{+}}\frac{I(u+tv)-I(u)}{t}=I'(u) (v) $$
for any \(v\in X\). It is called continuously Gâteaux differentiable if it is Gâteaux differentiable for any \(u\in X\) and the function \(u \rightarrow I'(u)\) is a continuous map from X to its dual \(X^{*}\). We recall that if I is continuously Gâteaux differentiable then it is locally Lipschitz and one has \(\langle J'(u), v\rangle=I'(u)(v)\) for all \(u,v\in X\).

Definition 2.1

Let X be a real Banach space and let \(I:X {\rightarrow }\Bbb {R}\) be a Gâteaux differentiable function.
  1. (i)

    I satisfies the Palais-Smale condition (\((PS)\)-condition for short), if any sequence \(\{u_{n} \}\subset X\) such that \(\{I(u_{n})\}\) is bounded and \(\|I^{\prime}(u_{n})\|_{X^{*}}\to0\) as \(n\to\infty\) has a convergent subsequence.

     
  2. (ii)

    I satisfies the Cerami condition (the \((C)\)-condition for short), if any sequence \(\{u_{n} \}\subset X\) such that \(\{I(u_{n})\}\) is bounded and \(\|I^{\prime}(u_{n})\|_{X^{*}}(1+\|u_{n}\|_{X})\to 0\) as \(n\to\infty\) has a convergent subsequence.

     
Now, let \(\Phi, \Psi: X\rightarrow\Bbb {R}\) be two continuously Gâteaux differentiable functions; put
$$ I=\Phi-\Psi $$
and fix \(\mu\in[-\infty, +\infty]\), we say that the function I verifies the Cerami condition cut off upper at μ (in short, the \({(C)^{[\mu]}}\)-condition) if any sequence \(\{u_{n}\}\) such that I satisfies the Cerami condition and \(\Phi(u_{n})<\mu\), for any \(n\in \Bbb {N}\), has a convergent subsequence.

As a key tool, recall the following lemma, which is a generalization of Ekeland’s variational principle [24] due to Zhong in [25], Theorem 2.1.

Lemma 2.2

[26]

Let \(h:[0,\infty)\rightarrow[0,\infty)\) be a continuous nondecreasing function such that \(\int_{0}^{\infty}{1/(1+h(r))}\,dr=+\infty\). Let \({\mathcal {M}}\) be a complete metric space and \(x_{0}\) be a fixed point of \({\mathcal {M}}\). Suppose that \(f:{\mathcal {M}}\rightarrow \mathbb {R}\cup\{+\infty\}\) is a lower semicontinuous function, not identically +∞, bounded from below. Then, for every \({\varepsilon }>0\) and \(y\in{\mathcal {M}}\) such that
$$f(y)< \inf_{{\mathcal {M}}}{f}+{\varepsilon }, $$
and every \(\lambda>0\), there exists some point \(z\in{\mathcal {M}}\) such that
$$ f(z) \le f(y),\quad d(z,x_{0})\le\bar{r}+r_{0}, $$
and
$$ f(x)\ge f(z)- \frac{{\varepsilon }}{\lambda(1+h(d(x_{0},z)))} {d(x,z)} \quad \textit{for all } x\in{\mathcal {M}}, $$
where \(d(x,y)\) is the distance of two points \(x,y\in\mathcal {M}\), \(r_{0}=d(x_{0},y)\), and is such that
$$\int_{r_{0}}^{r_{0}+\bar{r}}{\frac{1}{1+h(r)}}\,dr\ge\lambda. $$

Remark 2.3

[26]

To employ Lemma 2.2, we give a specific function h. If we define the function h by \(h(r):=r+d(x_{0},0)\), then it is continuous and nondecreasing. Setting \(x_{0}:=y\) and \(d(y,z):=\|y-z\|_{X}\). Then we see that \(h(d(y,z))=d(y,z)+d(y,0)=\|y-z\|_{X}+\|y\|_{X}\ge\|z\|_{X}\) and hence
$$-\frac{{\varepsilon }}{\lambda(1+h(d(y,z)))}\ge -\frac{{\varepsilon }}{\lambda(1+\|z\|_{X})}. $$
Since \(r_{0}=0\) and
$$\lambda\le \int_{0}^{\bar{r}}{\frac{1}{1+h(r)}}\,dr= \int_{0}^{\bar {r}}{\frac{1}{1+r+\|y\|_{X}}}\,dr= \bigl[ \ln\bigl|1+r+ \|y\|_{X}\bigr| \bigr]_{0}^{\bar{r}}=\ln{\biggl\vert \frac{1+\bar{r}+\|y\|_{X}}{1+\|y\| _{X}}\biggr\vert }, $$
if we take, for each \(n\in \mathbb {N}\),
$$\lambda_{n}:=\ln \biggl(1+\frac{1}{n(1+\|y\|_{X})} \biggr)>0, $$
then we can get \(\bar{r}_{n}=1/{n}\). Also,
$$\frac{{\varepsilon }_{n}}{\lambda_{n}} {\rightarrow }0 \quad\mbox{as } n {\rightarrow }\infty, $$
where \({\varepsilon }_{n}:=\lambda_{n}^{2}>0\).

We point out the following consequence of Zhong’s Ekeland variational principle in Lemma 2.2.

Lemma 2.4

Let X be a real Banach space and let \(I : X\rightarrow\Bbb {R}\) be a locally Lipschitz function bounded from below. Then, for all minimizing sequences of I, \(\{u_{n}\}_{n\in\Bbb {N}}\subseteq X\), there exists a minimizing sequence of I, \(\{v_{n}\}_{n\in\Bbb {N}}\subseteq X\), such that for any \(n\in\Bbb {N}\)
$$ I(v_{n})\le I(u_{n}) \quad \textit{and} \quad \bigl\langle I'(v_{n}), h \bigr\rangle \ge\frac{-{\varepsilon }_{n}\|h\|_{X}}{1+\|v_{n}\|_{X}} $$
for all \(h\in X\), and \(n\in\Bbb {N}\), where \({\varepsilon }_{n}{\rightarrow }0^{+}\).

Using Lemma 2.4, we obtain the following result; see [27] for the case of \((PS)\)-condition cut off upper at μ. The proof of this theorem proceeds in the analogous way to that of Theorem 3.1 in [27].

Theorem 2.5

Let X be a real Banach space and let \(\Phi, \Psi: X\rightarrow \Bbb {R}\) be two continuously Gâteaux differentiable functions with Φ bounded from below. Put
$$I=\Phi-\Psi $$
and assume that there are \(x_{0}\in X\) and \(r \in\Bbb {R}\), with \(\mu>\Phi(x_{0})\), such that
$$ \sup_{u\in\Phi^{-1}((-\infty, \mu))} \Psi(u) \le \mu-\Phi(x_{0})+ \Psi(x_{0}). $$
(2.1)
Moreover, assume that I satisfies \({(C)^{[\mu]}}\)-condition. Then there is a \(u_{0}\in\Phi^{-1}((-\infty, \mu))\) such that \(I(u_{0})\le I(u)\) for all \(u\in\Phi^{-1}((-\infty, \mu))\) and \(I'(u_{0})=0\).

Proof

Put
$$\begin{aligned} &M=\mu-\Phi(x_{0})+\Psi(x_{0}), \\ &\Psi_{M}(u)= \textstyle\begin{cases} \Psi(u) &\mbox{if } \Psi(u)< M, \\ M &\mbox{if } \Psi(u)\ge M, \end{cases}\displaystyle \\ &I_{M}=\Phi-\Psi_{M}. \end{aligned}$$
(2.2)
Then it is obvious that \(I_{M}\) is locally Lipschitz and bounded from below. Now, given a sequence \(\{u_{n}\}_{n\in\Bbb {N}}\) in X satisfying \(\lim_{n\rightarrow\infty}I_{M}(u_{n})=\inf_{X}I_{M}\), according to Lemma 2.4 there is a sequence \(\{v_{n}\}_{n\in\Bbb {N}}\) in X such that
$$\lim_{n\rightarrow\infty}I_{M}(v_{n})=\inf _{X}I_{M} \quad\mbox{and} \quad \bigl\langle I_{M}'(v_{n}), h\bigr\rangle \ge - \frac{\varepsilon_{n}\|h\|_{X}}{1+\|v_{n}\|_{X}} $$
for all \(h\in X\) and for all \(n\in\Bbb {N}\), where \(\varepsilon_{n}\rightarrow0^{+}\). If \(I_{M}(x_{0})=\inf_{X}I_{M}\) then \(x_{0}\) verifies the consequence. Indeed, if \(u\in\Phi^{-1}((-\infty, \mu))\), the inequality (2.1) implies that \(\Psi(u)\le M\) and \(I_{M}(u)=I(u)\) for all \(u\in\Phi^{-1}((-\infty, \mu))\), and hence \(I(x_{0})=I_{M}(x_{0})\le I_{M}(u)=I(u)\) for all \(u\in\Phi^{-1}((-\infty, \mu))\). So, we assume \(\inf_{X}I_{M} < I_{M}(x_{0})\). Therefore, there is an \(N \in \Bbb {N}\) such that \(I_{M}(v_{n})< I_{M}(x_{0})\) for all \(n>N\). Now, we claim that \(\Phi(v_{n})<\mu\) for all \(n>N\). Since \(\Phi(v_{n})-\Psi_{M}(v_{n})<\Phi(x_{0})-\Psi_{M}(x_{0})\), we assert that
$$\Phi(v_{n})< \Psi_{M}(v_{n})+ \Phi(x_{0})-\Psi(x_{0})\le M+\Phi(x_{0})- \Psi(x_{0})=\mu, $$
as claimed.
Hence, one has \(I_{M}(v_{n})=I(v_{n})\) and \(\langle I_{M}'(v_{n}), h\rangle=I'(v_{n})(h)\) for all \(n>N\). Therefore it follows from Lemma 2.4 that
$$\lim_{n\rightarrow\infty}I(v_{n})=\lim_{n\rightarrow \infty}I_{M}(v_{n})= \inf_{X}I_{M} \quad\mbox{and}\quad I'(v_{n}) (h)\ge -\frac{\varepsilon_{n}\|h\|_{X}}{1+\|v_{n}\|_{X}}, $$
that is, \(\lim_{n\rightarrow\infty}(1+\|v_{n}\|_{X})\|I'(v_{n})\|_{X^{*}}=0\). Since I satisfies \({(C)^{[\mu]}}\)-condition, \(\{v_{n}\}\) admits a subsequence strongly converging to \(v^{*}\) in X as \(n \rightarrow \infty\). So, \(I(v^{*})=\inf_{X}I_{M}\le I_{M}(u)=I(u)\) for all \(u\in \Phi^{-1}((-\infty, \mu))\), that is,
$$ I\bigl(v^{*}\bigr)\le I(u) $$
(2.3)
for all \(u\in\Phi^{-1}((-\infty, \mu))\). Since \(\Phi(v_{n})<\mu\) for all \(n>N\), the continuity of Φ implies that \(v^{*}\in \Phi^{-1}((-\infty, \mu])\).
If \(v^{*}\in\Phi^{-1}((-\infty, \mu))\), by the inequality (2.3) the conclusion holds. If \(\Phi(v^{*})=\mu\), first we observe that \(\Psi(v^{*})\le M\). In fact, taking into account that \(I(v^{*})=I_{M}(v^{*})\), we have \(\mu-\Psi(v^{*})=\mu-\Psi_{M}(v^{*})\) and \(\Psi(v^{*})=\Psi_{M}(v^{*})\le M\). Next, we prove that \(I(v^{*})=I(x_{0})\). Indeed, suppose that \(I(v^{*})< I(x_{0})\); from (2.2) we have
$$I\bigl(v^{*}\bigr)=\mu-\Psi\bigl(v^{*}\bigr)\ge\mu-M=\Phi(x_{0})- \Psi(x_{0})=I(x_{0}), $$
that is, \(I(v^{*})\ge I(x_{0})\) and this contradicts with the assumption. Hence, from the inequality (2.3) we have \(I(x_{0})\le I(u)\) for all \(u\in\Phi^{-1}((-\infty, \mu))\) and also in this case the conclusion is achieved. □

The next result is an immediate consequence of Theorem 2.5. This is crucial to get the existence of at least two distinct weak solutions for the problem (P) in the next section.

Theorem 2.6

Let X be a real Banach space, \(\Phi, \Psi:X\to\Bbb {R}\) be two continuously Gâteaux differentiable functionals such that Φ is bounded from below and \(\Phi(0)=\Psi(0)=0\). Fix \(\mu>0\) and assume that, for each
$$\lambda\in \biggl(0, \frac{\mu}{\sup_{u\in\Phi^{-1}((-\infty ,\mu))}\Psi(u)} \biggr), $$
the functional \(I_{\lambda}:=\Phi-\lambda\Psi\) satisfies \((C)\)-condition for all \(\lambda>0\) and it is unbounded from below. Then, for each
$$\lambda\in \biggl(0, \frac{\mu}{\sup_{u\in\Phi^{-1}((-\infty ,\mu))}\Psi(u)} \biggr), $$
the functional \(I_{\lambda}\) admits two distinct critical points.

Proof

Fix λ as in the conclusion. One has \(\frac{\sup_{u\in\Phi^{-1}((-\infty,\mu))}\Psi(u)}{\mu}<\frac {1}{\lambda}\), so there is an element \(x_{0}\) in \(\Phi^{-1}((-\infty,\mu))\) such that
$$ \frac{\sup_{u\in\Phi^{-1}((-\infty,\mu))}\Psi(u)-\Psi(x_{0})}{\mu -\Phi(x_{0})} < \frac{\sup_{u\in\Phi^{-1}((-\infty,\mu))}\Psi(u)}{\mu}< \frac {1}{\lambda}. $$
This implies
$$ \sup_{u\in\Phi^{-1}((-\infty,\mu))}\lambda \Psi(u)< \mu-\Phi(x_{0})+\lambda \Psi(x_{0}). $$
Hence, it follows from Theorem 2.5 that \(I_{\lambda}\) admits a local minimum. Since \(I_{\lambda}\) is unbounded from below, it is not strictly global and the mountain pass theorem ensures the conclusion. □

Combining [27], Remark 7.1, with [2], Theorem 3.6, we get the following assertion.

Theorem 2.7

Let X be a reflexive real Banach space, \(\Phi:X\to\Bbb {R}\) be a coercive, continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional, \(\Psi:X\to\Bbb {R}\) be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that
$$\inf_{u\in X}\Phi(u)=\Phi(0)=\Psi(0)=0. $$
Assume that the functional \(I_{\lambda}\) satisfies \((C)\)-condition for all \(\lambda>0\) and that there exist a positive constant μ and an element \(\tilde{u}\in X\), with \(\mu<\Phi(\tilde{u})\), such that
  1. (A1)

    \(\frac{\sup_{\Phi(u)\le\mu}\Psi(u)}{\mu}<\frac {\Psi(\tilde{u})}{\Phi(\tilde{u})}\);

     
  2. (A2)

    for each \(\lambda\in\Lambda_{\mu}:= (\frac {\Phi(\tilde{u})}{\Psi(\tilde{u})}, \frac{\mu}{\sup_{\Phi (u)\le\mu}\Psi(u)} )\), the functional \(I_{\lambda}\) is coercive.

     
Then, for each \(\lambda\in\Lambda_{\mu}\), the functional \(I_{\lambda}\) has at least three distinct critical points in X.

Proof

The proof is essentially the same as in that of [2]. □

This is an immediate result of Theorem 2.7. This plays an important role in obtaining the fact that the problem (P) admits at least three distinct weak solutions.

Corollary 2.8

Let X be a reflexive real Banach space, \(\Phi:X\to\Bbb {R}\) be a coercive, continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on \(X^{*}\), \(\Psi:X\to\Bbb {R}\) be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that
$$\inf_{u\in X}\Phi(u)=\Phi(0)=\Psi(0)=0. $$
Assume that there exist a positive constant μ and an element \(\tilde{u}\in X\), with \(\mu<\Phi(\tilde{u})\), such that
  1. (A1)

    \(\frac{\sup_{\Phi(u)\le\mu}\Psi(u)}{\mu}<\frac {\Psi(\tilde{u})}{\Phi(\tilde{u})}\);

     
  2. (A2)

    for each \(\lambda\in\Lambda_{\mu}\), the functional \(I_{\lambda}\) is coercive.

     
Then, for each \(\lambda\in\Lambda_{\mu}\), the functional \(I_{\lambda}\) has at least three distinct critical points in X.

Proof

Since Gâteaux derivative of Φ admits a continuous inverse and \(I_{\lambda}\) is coercive, \(I_{\lambda}\) satisfies \((C)\)-condition. Hence, applying Theorem 2.7 to the function \(I_{\lambda}\) the conclusion is obtained. □

Corollary 2.9

Let X be a reflexive real Banach space, \(\Phi:X\to\Bbb {R}\) be a coercive, continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional, \(\Psi:X\to\Bbb {R}\) be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that
$$\inf_{u\in X}\Phi(u)=\Phi(0)=\Psi(0)=0. $$
If \(I_{\lambda}\) is bounded from below and satisfies \((C)\)-condition for any \(\lambda>0\), and there exist \(\mu>0\) and \(\tilde{u}\in X\), with \(\mu<\Phi(\tilde{u})\), such that
$$ \frac{\sup_{\Phi(u)\le\mu}\Psi(u)}{\mu}< \frac{\Psi(\tilde {u})}{\Phi(\tilde{u})} $$
then, for each \(\lambda\in\Lambda_{\mu}\), the functional \(I_{\lambda}\) has at least three distinct critical points in X.

Proof

Since \(I_{\lambda}\) is bounded from below and satisfies \((C)\)-condition, \(I_{\lambda}\) is coercive; see [28]. Hence, by Corollary 2.8 the conclusion is obtained. □

3 Basic concepts and preliminary results

In this section, we first collect some preliminary properties that will be used later. Throughout this paper, consider the Sobolev space \(X:=W^{1,p}(\Omega)\) with the usual norm
$$\|u\|_{X}= \biggl( \int_{\Omega} \vert \nabla u\vert ^{p}+\vert u \vert ^{p}\,dx \biggr)^{\frac{1}{p}}. $$

Lemma 3.1

[8, 29]

Let \(\Omega\subset\Bbb {R}^{N}\) be an open bounded domain with Lipschitz boundary. Let \(p\in[1,N) \) be a constant. Then there is a continuously embedding \(X\hookrightarrow L^{p^{*}}(\Omega)\) where \(p^{*}=\frac{Np}{N-p}\). Moreover, for every \(q\in[1, p^{*})\) the embedding \(X\hookrightarrow L^{q}(\Omega)\) is compact.

Lemma 3.2

[8]

Let \(\Omega\subset\Bbb {R}^{N}\) be an open bounded domain with Lipschitz boundary. Let \(p\in[1,N) \) be a constant. Then there is a continuous boundary trace embedding \(X\hookrightarrow L^{p^{\partial}}(\partial\Omega)\) where \(p^{\partial}=\frac{(N-1)p}{N-p}\). Moreover, for every \(r\in[1, p^{\partial})\) the trace embedding \(X\hookrightarrow L^{r}(\partial\Omega)\) is compact.

Definition 3.3

We say that \(u\in X\) is a weak solution of the problem (P) if
$$\int_{\Omega} \varphi(x,\nabla u) \cdot\nabla v \,dx + \int_{\Omega}{ \vert u\vert ^{p-2}uv}\,dx=\lambda \int_{\Omega }f(x,u) v\,dx +\lambda \int_{\partial\Omega}{g(x,u)v}\,dS $$
for all \(v \in X\), where dS is the measure on the boundary.
We assume that \(\varphi: \Omega\times\Bbb {R}^{N} \to\Bbb {R}^{N}\) is the continuous derivative with respect to v of the mapping \(\Phi_{0}:\Omega\times\Bbb {R}^{N} \to\Bbb {R}\), \(\Phi_{0}=\Phi_{0}(x,v)\), that is, \(\varphi(x,v)=\frac{d}{dv}\Phi_{0}(x,v)\). Suppose that φ and \(\Phi_{0}\) satisfy the following assumptions:
  1. (J1)
    The equality
    $$\Phi_{0}(x,\textbf{0})=0 $$
    holds for almost all \(x\in\Omega\).
     
  2. (J2)
    There is a function \(a\in L^{p'}(\Omega)\) and a nonnegative constant b such that
    $$ \bigl|\varphi(x,v)\bigr|\le a(x)+b |v|^{p-1} $$
    for almost all \(x\in\Omega\) and for all \(v\in\Bbb {R}^{N}\) where \(1/p+1/{p'}=1\).
     
  3. (J3)

    \(\Phi_{0}(x,\cdot)\) is strictly convex in \(\Bbb {R}^{N}\) for all \(x\in\Omega\).

     
  4. (J4)
    The relations
    $$d|v|^{p}\le\varphi(x,v)\cdot v \quad \mbox{and} \quad d|v|^{p}\le p \Phi_{0}(x,v) $$
    hold for all \(x\in\Omega\) and \(v\in\Bbb {R}^{N}\), where d is a positive constant.
     
  5. (J5)
    There exists a constant \(\mu_{1}\ge0\) such that
    $$\mathcal {H}(x,sv)\leq\mathcal {H}(x,v)+\mu_{1} $$
    for \(v \in\Bbb{R}^{N}\) and \(s\in[0,1]\), where \(\mathcal {H}(x,v) = p \Phi_{0}(x,v)-\varphi(x,v)\cdot v\) for almost all \(x\in\Omega\).
     
Let us define the functional \(\Phi: X \to\Bbb {R}\) by
$$\Phi(u)= \int_{\Omega}\Phi_{0}(x,\nabla u)\,dx+ \int_{\Omega}{\frac{1}{p}\vert u\vert ^{p}} \,dx. $$
Under assumptions (J1)-(J2) and (J4), it follows from [30] that the functional Φ is well defined on X, \(\Phi \in C^{1}(X,\Bbb {R})\), and its Fréchet derivative is given by
$$ \bigl\langle {\Phi^{\prime}(u),v}\bigr\rangle = \int_{\Omega}{\varphi (x,\nabla u)\cdot\nabla v}\,dx + \int_{\Omega}{ \vert u\vert ^{p-2}uv}\,dx, $$
for any \(\varphi\in X\).

Example 3.4

Let us consider
$$\varphi(x,v)= \biggl(1+\frac{\vert v\vert ^{p}}{\sqrt{1+\vert v\vert ^{2p}}} \biggr)\vert v\vert ^{p-2}v $$
and
$$\Phi_{0}(x,v)=\frac{1}{p} \bigl(\vert v\vert ^{p}+\sqrt{1+\vert v\vert ^{2p}}-1 \bigr) $$
for all \(v\in\Bbb {R}^{N}\). Then the direct calculation shows that \(\mathcal {H}(x,sv)\leq\mathcal {H}(x,v)+1 \) for all \(s\in[0,1]\), and so the assumption (J5) holds for \(\mu_{1}\ge1\).

The following assertion can be found in [31]; see also [32].

Lemma 3.5

[31]

Assume that (J1)-(J4) hold. Then the functional \(\Phi:X\to\Bbb {R}\) is convex and weakly lower semicontinuous on X. Moreover, the operator \(\Phi^{\prime}\) is a mapping of type \((S_{+})\), i.e., if \(u_{n}\rightharpoonup u\) in X as \(n\to\infty\) and \(\lim\sup_{n\to \infty} \langle\Phi^{\prime}(u_{n})-\Phi^{\prime}(u), u_{n}-u \rangle\le0\), then \(u_{n}\to u\) in X as \(n\to\infty\).

Corollary 3.6

Assume that (J1)-(J4) hold. Then the operator \(\Phi^{\prime}:X\to X^{*}\) is strictly monotone, coercive and hemicontinuous on X. Furthermore, the operator \(\Phi^{\prime}\) is a homeomorphism onto \(X^{*}\).

Proof

It is obvious that the operator \(\Phi^{\prime}\) is strictly monotone, coercive, and hemicontinuous on X. By the Browder-Minty theorem, the inverse operator \((\Phi^{\prime})^{-1}\) exists; see Theorem 26.A in [33]. Since \(\Phi^{\prime}\) is a mapping of type \((S_{+})\), by Lemma 3.5, the proof of continuity of the inverse operator \((\Phi^{\prime})^{-1}\) is obvious. □

Next we need the following assumptions for f and g. Denoting \(F(x,t)=\int_{0}^{t}f(x,s)\,ds\) and \(G(x,t)=\int_{0}^{t}g(x,s)\,ds\), then we assume that:
  1. (F1)

    \(f: \Omega\times\Bbb {R} \to\Bbb {R}\) satisfies the Carathéodory condition in the sense that \(f(\cdot,t)\) is measurable for all \(t\in\Bbb {R}\) and \(f(x,\cdot)\) is continuous for almost all \(x\in\Omega\).

     
  2. (F2)
    \(f:\overline{\Omega}\times\Bbb {R} \to\Bbb {R}\) is a continuous function and there exist two constants \(d_{1} \geq0\) and \(d_{2}>0\) such that
    $$ \bigl\vert f(x,t)\bigr\vert \leq d_{1} + d_{2} \vert t\vert ^{\alpha-1}, $$
    for all \(x\in\Omega\) and for all \(t\in\Bbb {R}\), where \(p<\alpha<p^{*}\).
     
  3. (G1)
    \(g: \partial\Omega\times\Bbb {R} \to\Bbb {R}\) satisfies the Carathéodory condition and there exist two constants \(d_{3} \geq0\) and \(d_{4}>0\) such that
    $$ \bigl\vert g(x,t)\bigr\vert \leq d_{3} + d_{4} \vert t\vert ^{\beta-1}, $$
    for all \(x\in\partial\Omega\) and for all \(t\in\Bbb {R}\), where \(p<\beta<p^{\partial}\).
     
Under assumptions (F1), (F2), and (G1), we define the functionals \(\Psi_{1}, \Psi_{2}:X\to\Bbb {R}\) by
$$\Psi_{1}(u)= \int_{\Omega}F(x,u)\,dx ,\qquad \Psi_{2}(u)= \int_{\partial\Omega}G(x,u)\,dS \quad \mbox{and}\quad \Psi(u)= \Psi_{1}(u)+\Psi_{2}(u). $$
Then it is easy to check that \(\Psi_{1}, \Psi_{2} \in C^{1}(X,\Bbb {R})\) and these Fréchet derivatives are
$$ \bigl\langle \Psi_{1}^{\prime}(u),v \bigr\rangle = \int_{\Omega }f(x,u)v\,dx \quad\mbox{and} \quad \bigl\langle \Psi_{2}^{\prime}(u),v \bigr\rangle = \int_{\partial\Omega}g(x,u)v\,dS $$
for any \(u,v \in X\). Next we define the functional \(I_{\lambda}:X\to \Bbb {R}\) by
$$I_{\lambda}(u)=\Phi(u)-\lambda\Psi(u). $$
Then it follows that the functional \(I_{\lambda} \in C^{1}(X,\Bbb {R})\) and its Fréchet derivative is
$$\bigl\langle I^{\prime}_{\lambda}(u),v \bigr\rangle = \int_{\Omega }\varphi(x,\nabla u)\cdot\nabla v\,dx + \int_{\Omega}{ \vert u\vert ^{p-2}uv}\,dx - \lambda \int_{\Omega}f(x,u)v\,dx-\lambda \int_{\partial\Omega}g(x,u)v\,dS $$
for any \(u,v\in X\).

Lemma 3.7

Assume that (F1)-(F2) and (G1) hold. Then Ψ and \(\Psi^{\prime}\) are weakly-strongly continuous on X.

Proof

Proceeding like the analogous argument in [34], it follows that functionals Ψ and \(\Psi^{\prime}\) are weakly-strongly continuous on X. □

To localize precisely the intervals of λ for which the problem (P) has at least two or three distinct weak solutions, we consider the following eigenvalue problem:
$$ \textstyle\begin{cases} -\operatorname{div}(\vert \nabla u\vert ^{p-2}\nabla u)+\vert u\vert ^{p-2}u=\lambda m(x)\vert u\vert ^{q-2}u & \mbox{in } \Omega,\\ \frac{\partial u}{\partial n}= 0 & \mbox{on }\partial\Omega. \end{cases} $$
(E)

Definition 3.8

We say that \(\lambda\in\Bbb {R}\) is an eigenvalue of the eigenvalue problem (E) if
$$\int_{\Omega} \vert \nabla u\vert ^{p-2}\nabla u \cdot \nabla v \,dx + \int_{\Omega}{ \vert u\vert ^{p-2}uv}\,dx=\lambda \int_{\Omega}m(x) \vert u\vert ^{q-2}uv\,dx $$
holds for any \(v \in X\) and \(p< q< p^{*}\). Then u is called an eigenfunction associated with the eigenvalue λ.

Now we obtain the positivity of the infimum of all eigenvalues for the problem (E). Although the idea of the proof is completely the same as in that of Lemma 3.1 in [29], for the sake of convenience, we give the proof of the following proposition.

Proposition 3.9

Assume that
  1. (H1)

    \(m(x)>0\) for all \(x\in\Omega\) and \(m\in L^{\frac {\gamma}{\gamma-q}}(\Omega)\) with some γ satisfying \(q<\gamma<p^{*}\).

     
Then the eigenvalue problem (E) has a pair \((\lambda_{1}, u_{1})\) of a principal eigenvalue \(\lambda_{1}\) and an eigenfunction \(u_{1}\) with \(\lambda_{1}>0\) and \(0< u_{1}\in X\).

Proof

Set
$$\lambda_{1}=\inf \biggl\{ \int_{\Omega} \vert \nabla v\vert ^{p}+\vert v \vert ^{p}\,dx \biggr\} $$
the infimum being taken over all v such that \(\int_{\Omega}m(x)|v|^{q}\,dx=1\). We shall prove that \(\lambda_{1}\) is the least eigenvalue of (E). The expression for \(\lambda_{1}\) presented above will be referred to as its variational characterization. Obviously \(\lambda_{1}\ge0\). Let \(\{v_{n}\}_{n=1}^{\infty}\) be the minimizing sequence for \(\lambda_{1}\), i.e.,
$$ \int_{\Omega}m(x)|v_{n}|^{q}\,dx=1 \quad \mbox{and}\quad \int_{\Omega} \vert \nabla v_{n}\vert ^{p}+\vert v_{n}\vert ^{p}\, dx= \lambda_{1}+\delta_{n} $$
(3.1)
with \(\delta_{n} \rightarrow0^{+}\) for \(n\rightarrow\infty\). It follows from (3.1) that \(\|v_{n}\|_{X}\le c\) for some constant \(c>0\). The reflexivity of X yields the weak convergence \(v_{n} \rightharpoonup u_{1}\) in X for some \(u_{1}\) (at least for some subsequence of \(\{v_{n}\}\)). The compact embedding \(X \hookrightarrow L^{\gamma}(\Omega)\) implies the strong convergence \(v_{n} \rightarrow u_{1}\) in \(L^{\gamma}(\Omega)\). It follows from (H1), (3.1), and the Minkowski and Hölder inequalities that
$$\begin{aligned} 1&= \lim_{n\rightarrow \infty} \biggl( \int_{\Omega}m(x)\vert v_{n}\vert ^{q} \,dx \biggr)^{\frac{1}{q}} \\ & \le\lim_{n\rightarrow \infty} \biggl( \int_{\Omega}m(x)\vert v_{n}-u_{1} \vert ^{q}\,dx \biggr)^{\frac {1}{q}}+ \biggl( \int_{\Omega}m(x)\vert u_{1}\vert ^{q} \,dx \biggr)^{\frac{1}{q}} \\ &\le\lim_{n\rightarrow \infty} \biggl( \int_{\Omega}\bigl(m(x)\bigr)^{\frac{\gamma}{\gamma -q}}\,dx \biggr)^{\frac{\gamma-q}{q\gamma}} \biggl( \int_{\Omega} \vert v_{n}-u_{1}\vert ^{\gamma}\,dx \biggr)^{\frac{1}{\gamma}}+ \biggl( \int _{\Omega}m(x)\vert u_{1}\vert ^{q} \,dx \biggr)^{\frac{1}{q}} \\ &= \biggl( \int_{\Omega}m(x)\vert u_{1}\vert ^{q} \,dx \biggr)^{\frac {1}{q}}, \end{aligned}$$
and analogously
$$\begin{aligned}[b] \biggl( \int_{\Omega}m(x)\vert u_{1}\vert ^{q} \,dx \biggr)^{\frac {1}{q}} \le{}& \lim_{n\rightarrow \infty} \biggl( \int_{\Omega}\bigl(m(x)\bigr)^{\frac{\gamma}{\gamma -q}}\,dx \biggr)^{\frac{\gamma-q}{q\gamma}} \biggl( \int_{\Omega} \vert u_{1}-v_{n}\vert ^{\gamma}\,dx \biggr)^{\frac{1}{\gamma}} \\ &{}+\lim_{n\rightarrow\infty} \biggl( \int_{\Omega}m(x)\vert v_{n}\vert ^{q} \,dx \biggr)^{\frac{1}{q}}=1. \end{aligned} $$
Hence
$$\int_{\Omega}m(x)\vert u_{1}\vert ^{q} \,dx=1. $$
In particular, \(u_{1}\not\equiv0\). The weak lower semicontinuity of the norm in X yields
$$\begin{aligned} \lambda_{1} &\le \int_{\Omega} \vert \nabla u_{1}\vert ^{p}+\vert u_{1}\vert ^{p}\,dx= \|u_{1}\|_{X}^{p}\le\liminf _{n\rightarrow \infty} \|v_{n}\|_{X}^{p} \\ &=\liminf_{n\rightarrow\infty} \biggl\{ \int_{\Omega} \vert \nabla v_{n}\vert ^{p}+\vert v_{n}\vert ^{p}\,dx \biggr\} = \liminf_{n\rightarrow\infty} (\lambda_{1}+\delta_{n})= \lambda_{1}, \end{aligned}$$
i.e.,
$$ \lambda_{1}= \int_{\Omega} \vert \nabla u_{1}\vert ^{p}+\vert u_{1}\vert ^{p}\,dx. $$
(3.2)
It follows from (3.2) that \(\lambda_{1}>0\) and it is easy to see that \(\lambda_{1}\) is the least eigenvalue of (E) with the corresponding eigenfunction \(u_{1}\).

Moreover, if u is an eigenfunction corresponding to \(\lambda_{1}\) then \(|u|\) is also an eigenfunction corresponding to \(\lambda_{1}\). Hence we can suppose that \(u_{1}> 0\) a.e. in Ω. □

4 Existence of two weak solutions

In this section, we present the existence of at least two distinct weak solutions for the problem (P). To do this, we assume that
  1. (F3)

    \(\lim_{\vert t\vert \to\infty}{\frac{F(x,t)}{\vert t\vert ^{p}}}=\infty\) uniformly for almost all \(x\in\Omega\).

     
  2. (F4)
    There is a constant \(\mu_{2}>0\) such that
    $${\mathcal {F}}(x,t) \leq{\mathcal {F}}(x,s)+\mu_{2} $$
    for any \(x\in\Omega\), \(0< t< s\) or \(s< t<0\), where \({\mathcal {F}}(x,t)=tf(x,t)-pF(x,t)\).
     
  3. (F5)

    \(\limsup_{s\to0}{\frac{\vert f(x,s)\vert }{m(x)\vert s\vert ^{\xi_{1}-1}}}<\infty\) uniformly for almost all \(x\in\Omega\), where \(\xi_{1}\in\mathbb{R}\) with \(q<\xi_{1}<p^{*}\).

     
  4. (F6)

    \(\limsup_{\vert s\vert \to\infty}{ \Big(\operatorname {ess\,sup}_{x\in\Omega}{\frac{\vert f(x,s)\vert }{\vert s\vert ^{q-1}}} \Big)}<\infty\), where \(q \in\mathbb{R}\) satisfies \(p< q< p^{*}\).

     
  5. (G2)

    \(\lim_{\vert t\vert \to\infty}{\frac{G(x,t)}{\vert t\vert ^{p}}}=\infty\) uniformly for almost all \(x\in\partial \Omega\).

     
  6. (G3)
    There is a constant \(\mu_{3}>0\) such that
    $$ {\mathcal {G}}(x,t)\le{\mathcal {G}}(x,s)+\mu_{3}, $$
    for any \(x\in\partial\Omega\), \(0< t< s\) or \(s< t<0\), where \({\mathcal {G}}(x,t)=tg(x,t)-pG(x,t)\).
     
  7. (G4)

    \(\limsup_{s\to0}{\frac{\vert g(x,s)\vert }{\vert s\vert ^{\xi _{2} -1}}}<\infty\) uniformly for almost all \(x\in\partial\Omega\), where \(\xi_{2}\in\mathbb{R}\) with \(q<\xi_{2}<p^{\partial}\).

     
  8. (G5)

    \(\limsup_{\vert s\vert \to\infty}{ \Big(\operatorname {ess\,sup}_{x\in\partial\Omega}{\frac{\vert g(x,s)\vert }{\vert s\vert ^{r-1}}} \Big)}<\infty\), where \(r \in\mathbb{R}\) satisfies \(p< r< p^{\partial}\).

     

With the help of Lemmas 3.5 and 3.7, we prove that the energy functional \(I_{\lambda}\) satisfies the \((C)\)-condition for any \(\lambda>0\). This plays an important role in obtaining our first main result.

Lemma 4.1

Assume that (J1)-(J5), (F1)-(F4), and (G1)-(G3) hold. Then the functional \(I_{\lambda}\) satisfies the \((C)\)-condition for any \(\lambda>0\).

Proof

Note that \(\Psi_{1}^{\prime}\) and \(\Psi_{2}^{\prime}\) are the mapping of type \((S_{+})\) by Lemma 3.7. Let \(\{u_{n}\}\) be a \((C)\)-sequence in X, i.e., \(\{I_{\lambda}(u_{n})\}\) is bounded and \(\|I_{\lambda}^{\prime}(u_{n})\|_{X^{*}}(1+\|u\|_{X})\to0\) as \(n\to\infty\), so that \(\sup \vert I_{\lambda}(u_{n})\vert \le M\) for some \(M>0\) and \(\langle I'_{\lambda}(u_{n}), u_{n}\rangle=o(1)\). Since X is reflexive and \(I_{\lambda}^{\prime}\) is the mapping of type \((S_{+})\), it suffices to verify that \(\{u_{n}\}\) is bounded in X. Indeed, if \(\{u_{n} \}\) is unbounded in X, we may assume that \(\|u_{n}\|_{X}>1\) and \(\|u_{n}\|_{X}\to\infty\) as \(n\to\infty\). We define a sequence \(\{w_{n} \}\) by \(w_{n}={u_{n}}/{\|u_{n}\|_{X}}\), \(n=1,2,\ldots\) . It is clear that \(\{w_{n} \} \subset X\) and \(\|w_{n}\|_{X}=1\) for any n. Therefore, up to a subsequence, still denoted by \(\{w_{n} \}\), we see that \(\{w_{n} \}\) converges weakly to \(w\in X\) and, by Lemmas 3.1 and 3.2, we have
$$ \begin{aligned} &w_{n}(x) \to w(x)\quad \mbox{a.e. in }\Omega\mbox{ and }\partial\Omega,\\ & w_{n} \to w \quad\mbox{in }L^{\alpha}( \Omega)\mbox{ and } L^{\beta}(\partial\Omega) \quad\mbox{as } n\to\infty. \end{aligned} $$
(4.1)
Let \(\Omega_{0}= \{ x\in\overline{\Omega}: w(x)\neq0 \}\). If \(x\in\Omega_{0}\cap\Omega\), then it follows from (4.1) that \(\vert u_{n}(x)\vert =\vert w_{n}(x)\vert \|u_{n}\|_{X}\to \infty\) as \(n\to\infty\). Similarly we know by (4.1) that \(\vert u_{n}(x)\vert \to\infty\) as \(n\to\infty\) for \(x\in\Omega_{0}\cap\partial\Omega\). According to (F3) and (G2), we have
$$ \lim_{n\to\infty}{\frac{F(x,u_{n}(x))}{\|u_{n}\|^{p}_{X}}} = \lim _{n\to\infty}{\frac{F(x,u_{n}(x))}{\vert u_{n}(x)\vert ^{p}}\bigl\vert w_{n}(x)\bigr\vert ^{p} } = \infty, \quad x\in\Omega\cap\Omega_{0}, $$
(4.2)
and
$$ \lim_{n\to\infty}{\frac{G(x,u_{n}(x))}{\|u_{n}\|^{p}_{X}}} = \lim _{n\to\infty}{\frac{G(x,u_{n}(x))}{\vert u_{n}(x)\vert ^{p}}\bigl\vert w_{n}(x)\bigr\vert ^{p} } = \infty, \quad x\in\partial\Omega\cap \Omega_{0}. $$
(4.3)
In addition, the condition (F3) implies that there exists \(t_{0}>1\) such that \({F(x,t)}>{\vert t\vert ^{p}}\) for all \(x\in \Omega\) and \(\vert t\vert >t_{0}\). Since \(F(x,t)\) is continuous on \(\overline{\Omega}\times[-t_{0},t_{0}]\) by (F2), there exists a positive constant \(C_{1}\) such that \(\vert F(x,t)\vert \leq C_{1}\) for all \((x,t)\in\overline{\Omega}\times[-t_{0},t_{0}]\). Therefore we can choose a real number \(C_{2}\) such that \(F(x,t)\geq C_{2}\) for all \((x,t)\in\overline{\Omega}\times\Bbb {R}\), and thus
$$\frac{F(x,u_{n}(x))-C_{2}}{\|u_{n}\|^{p}_{X}}\geq0, $$
for all \(x\in\Omega\) and for all \(n\in\Bbb {N}\). Similarly, using the assumption (G2), we see that there exists a constant \(C_{3}\in\Bbb {R}\) such that
$$ \frac{G(x,u_{n}(x))-C_{3}}{\|u_{n}\|^{p}_{X}}\geq0, $$
for all \(x\in\partial\Omega\) and for all \(n\in\Bbb {N}\). Also, using the assumption (J2), we get
$$\begin{aligned} I_{\lambda}(u_{n})={}& \int_{\Omega}{\Phi_{0}}(x,\nabla u_{n}) \,dx + \int_{\Omega}{\frac{1}{p}\vert u_{n}\vert ^{p}}\,dx - \lambda \int _{\Omega}{F(x,u_{n})}\,dx - \lambda \int_{\partial\Omega }G(x,u_{n})\,dS \\ \leq{}& \int_{\Omega}{a(x)\vert \nabla u_{n}\vert }\,dx + \frac {b}{p} \int _{\Omega}{ \vert \nabla u_{n}\vert ^{p}}\,dx + \frac{1}{p} \int _{\Omega }{\vert u_{n}\vert ^{p}} \,dx - \lambda \int_{\Omega}{F(x,u_{n})}\,dx \\ &{}- \lambda \int_{\partial\Omega}G(x,u_{n})\,dS \\ \leq{}& \bigl(2\|a\|_{L^{p^{\prime}}(\Omega)}+b+1 \bigr)\|u_{n}\|_{X}^{p} - \lambda \int_{\Omega}{F(x,u_{n})}\,dx -\lambda \int_{\partial\Omega}G(x,u_{n})\,dS. \end{aligned}$$
It follows that
$$ 2\|a\|_{L^{p^{\prime}}(\Omega)}+b+1 \geq \frac{1}{\|u_{n}\|_{X}^{p}} \biggl(I_{\lambda}(u_{n})+\lambda \int _{\Omega}{F(x,u_{n})}\,dx+\lambda \int_{\partial\Omega}G(x,u_{n})\,dS \biggr) $$
(4.4)
for n large enough. We claim that \(\vert \Omega_{0}\vert =0\). In fact, if \(\vert \Omega_{0}\vert \neq0\), then by (4.2), (4.3), (4.4), and the Fatou lemma, we have
$$\begin{aligned} &2\|a\|_{L^{p^{\prime}}(\Omega)}+b+1 \\ &\quad\geq\liminf_{n\to\infty }{ \frac{1}{\|u_{n}\|_{X}^{p}} \biggl(I_{\lambda}(u_{n})+\lambda \int _{\Omega}{F(x,u_{n})}\,dx+\lambda \int_{\partial\Omega}G(x,u_{n})\,dS \biggr)} \\ &\quad= {\liminf_{n\to\infty} \int_{\Omega}{\frac {F(x,u_{n}(x))-C_{2}}{\|u_{n}\|^{p}_{X}}}}\,dx + {\liminf _{n\to \infty} \int_{\partial\Omega}{\frac{G(x,u_{n}(x))-C_{3}}{\|u_{n}\| ^{p}_{X}}}}\,dS \\ &\quad\geq \int_{\Omega\cap\Omega_{0}}{\liminf_{n\to\infty}{ \frac {F(x,u_{n}(x))-C_{2}}{\|u_{n}\|^{p}_{X}}}}\,dx + \int_{\partial \Omega\cap\Omega_{0}}{\liminf_{n\to\infty}{ \frac {G(x,u_{n}(x))-C_{3}}{\|u_{n}\|^{p}_{X}}}}\,dS \\ &\quad= \int_{\Omega\cap\Omega_{0}}{\liminf_{n\to\infty}{\frac {F(x,u_{n}(x))}{\vert u_{n}(x)\vert ^{p}}} \bigl\vert w_{n}(x)\bigr\vert ^{p}}\,dx- \int _{\Omega\cap\Omega_{0}}{\limsup_{n\to\infty}{ \frac{C_{2}}{\| u_{n}\|^{p}_{X}}}}\,dx \\ &\qquad{} + \int_{\partial\Omega\cap\Omega_{0}}{\liminf_{n\to\infty}{\frac{G(x,u_{n}(x))}{\vert u_{n}(x)\vert ^{p}}} \bigl\vert w_{n}(x)\bigr\vert ^{p}}\,dS- \int_{\partial\Omega\cap \Omega_{0}}{\limsup_{n\to\infty}{ \frac{C_{3}}{\|u_{n}\|^{p}_{X}}}}\,dS \\ &\quad=\infty, \end{aligned}$$
which is a contradiction. This shows that \(\vert \Omega_{0}\vert =0\) and thus \(w(x)=0\) almost everywhere in Ω̅.
Since \(I_{\lambda}(tu_{n})\) is continuous in \(t\in[0,1]\), for each \(n\in\Bbb {N}\) there exists \(t_{n}\) in [0,1] such that
$$ I_{\lambda}(t_{n}u_{n}):= \max _{t\in[0,1]}{I_{\lambda}(tu_{n})}. $$
Let \(\{R_{k} \}\) be a positive sequence of real numbers such that \(\lim_{k\to\infty}{R_{k}}=\infty\) and \(R_{k}>1\) for any k. Then \(\|R_{k}w_{n}\|_{X}=R_{k}>1\) for any k and n. For fixed k, we derive from the continuity of the Nemytskii operator that \(F(x,R_{k}w_{n}) \rightarrow0\) in \(L^{1}(\Omega)\) and \(G(x,R_{k}w_{n}) \rightarrow0\) in \(L^{1}(\partial\Omega)\) as \(n\to\infty\), respectively. Hence we assert that
$$ \lim_{n\to\infty}{ \int_{\Omega}{F(x,R_{k}w_{n})}}\,dx=0 \quad\mbox{and}\quad \lim_{n\to\infty}{ \int_{\partial\Omega}{G(x,R_{k}w_{n})}}\,dS=0. $$
(4.5)
Since \(\|u_{n}\|_{X}\to\infty\) as \(n\to\infty\), we also have \(\|u_{n}\|_{X}>R_{k}\) or \(0 < \frac{R_{k}}{\|u_{n}\|_{X}} < 1\) for n large enough. Hence, using the assumption (J4) and (4.5) it follows that
$$\begin{aligned} I_{\lambda}(t_{n}u_{n}) \geq{}& I_{\lambda} \biggl(\frac{R_{k}}{\|u_{n}\|_{X}}u_{n} \biggr)=I_{\lambda}(R_{k}w_{n}) \\ ={}& \int_{\Omega}{\Phi_{0}\bigl(x,\vert \nabla R_{k}w_{n}\vert \bigr)}\,dx + \int _{\Omega}{\frac{1}{p}\vert R_{k}w_{n} \vert ^{p}}\,dx - \lambda \int _{\Omega}{F(x,R_{k}w_{n})}\,dx \\ &{}- \lambda \int_{\partial\Omega }{G(x,R_{k}w_{n})}\,dS \\ &\geq\frac{d}{p} \int_{\Omega}{ \vert \nabla R_{k}w_{n} \vert ^{p}}\,dx + \frac{1}{p} \int_{\Omega}{ \vert R_{k}w_{n}\vert ^{p}}\,dx -\lambda{ \int _{\Omega}{F(x,R_{k}w_{n})}}\,dx\\ &{}- \lambda \int_{\partial\Omega }{G(x,R_{k}w_{n})}\,dS \\ \geq{}& C_{4}\|R_{k}w_{n}\|^{p}_{X}- \lambda \int_{\Omega }{F(x,R_{k}w_{n})}\,dx - \lambda \int_{\partial\Omega }{G(x,R_{k}w_{n})}\,dS \\ \geq{}&\frac{C_{4}}{2}R_{k}^{p} \end{aligned}$$
for some positive constant \(C_{4}\) and for any n large enough. Then letting n and k tend to infinity, it follows that
$$ \lim_{n\to\infty}{I_{\lambda}(t_{n}u_{n})}= \infty. $$
(4.6)
Since \(I_{\lambda}(0)=0\) and \(\vert I_{\lambda}(u_{n})\vert \le M\), it is obvious that \(t_{n}\in(0,1)\) and also \(\langle I_{\lambda}^{\prime}(t_{n}u_{n}),t_{n}u_{n} \rangle=o(1)\). Therefore, due to the assumptions (J5), (F4) and (G3), for n large enough, we deduce that
$$\begin{aligned} I_{\lambda}(t_{n}u_{n})={}& I_{\lambda}(t_{n}u_{n}) - \frac {1}{p} \bigl\langle I_{\lambda}^{\prime}(t_{n}u_{n}),t_{n}u_{n} \bigr\rangle +o(1) \\ ={}& \int_{\Omega}{\Phi_{0}(x,t_{n}\nabla u_{n})}\,dx + \int_{\Omega }{\frac{1}{p}\vert t_{n}u_{n} \vert ^{p}}\,dx \\ &{}- \lambda \int_{\Omega}{F(x,t_{n}u_{n})}\,dx - \lambda \int _{\partial\Omega}{G(x,t_{n}u_{n})}\,dS - \frac{1}{p} \int_{\Omega }{\varphi(x,t_{n}\nabla u_{n}) \cdot (t_{n}\nabla u_{n} )}\, dx \\ &{} - \frac{1}{p} \int_{\Omega}{ \vert t_{n}u_{n}\vert ^{p}}\, dx + \frac{\lambda}{p} \int_{\Omega}{f(x,t_{n}u_{n})t_{n}u_{n}} \,dx + \frac{\lambda}{p} \int_{\partial\Omega }{g(x,t_{n}u_{n})t_{n}u_{n}} \,dS +o(1) \\ \leq{}&\frac{1}{p} \int_{\Omega}{\mathcal {H}(x,t_{n} \nabla u_{n})}\,dx + \int_{\Omega}{\frac{1}{p}\vert t_{n}u_{n} \vert ^{p}}\,dx \\ &{} - \frac{1}{p} \int_{\Omega}{ \vert t_{n}u_{n}\vert ^{p}}\, dx + \frac{\lambda}{p} \int_{\Omega}{{\mathcal {F}}(x,t_{n}u_{n})} \,dx +\frac{\lambda}{p} \int_{\partial\Omega}{{\mathcal {G}}(x,t_{n}u_{n})} \,dS+o(1) \\ \leq{}&\frac{1}{p} \int_{\Omega} \bigl(\mathcal {H}(x,\nabla u_{n} )+\mu _{1} \bigr)\,dx + \int_{\Omega}{\frac{1}{p}\vert u_{n}\vert ^{p}}\,dx \\ &{} - \frac{1}{p} \int_{\Omega}{ \vert u_{n}\vert ^{p}} \,dx + \frac{\lambda}{p} \int_{\Omega} \bigl({\mathcal {F}}(x,u_{n})+\mu _{2} \bigr)\,dx + \frac{\lambda}{p} \int_{\partial\Omega} \bigl({\mathcal {G}}(x,u_{n})+ \mu_{3} \bigr)\,dS+o(1) \\ \leq{}& \int_{\Omega}{\Phi_{0}(x, \nabla u_{n} )} \,dx + \int_{\Omega }{\frac{1}{p}\vert u_{n}\vert ^{p}}\,dx - \lambda \int_{\Omega }{F(x,u_{n})}\,dx - \lambda \int_{\partial\Omega}{G(x,u_{n})}\,dS \\ &{}- \frac{1}{p} \biggl( \int_{\Omega}{\varphi(x, \nabla u_{n})\cdot\nabla u_{n}}\,dx - \int_{\Omega}{ \vert u_{n}\vert ^{p}} \,dx - \lambda \int_{\Omega}{f(x,u_{n})u_{n}}\,dx \\ &{}- \lambda \int_{\partial\Omega}{g(x,u_{n})u_{n}}\,dS \biggr) \\ &{} + \frac{\lambda}{p} (\mu_{1}+\mu_{2} )\vert \Omega \vert + \frac{\lambda}{p}{\mu_{3}}\vert \partial\Omega \vert +o(1) \\ ={}& I_{\lambda}(u_{n})-\frac{1}{p} \bigl\langle I_{\lambda}^{\prime }(u_{n}),u_{n} \bigr\rangle + \frac{\lambda}{p} (\mu_{1}+\mu_{2} )\vert \Omega \vert + \frac{\lambda}{p}{\mu_{3}}\vert \partial\Omega \vert +o(1) \\ \le{}& M + \frac{\lambda}{p} (\mu_{1}+\mu_{2} )\vert \Omega \vert + \frac{\lambda}{p}{\mu_{3}}\vert \partial\Omega \vert \quad\mbox{as } n\to\infty, \end{aligned}$$
which contradicts with (4.6). This completes the proof. □

Remark 4.2

It is easily to confirm that the conditions (F1) and (F5) imply that \(f(x,0)=0\) for almost all \(x\in\Omega\). Otherwise there exists \(A\subset\Omega\), \(|A|>0\) such that \(|f(x,0)|>0\) for all \(x\in A\). Hence \(\lim_{s\to 0}\frac{|f(x,s)|}{m(x)|s|^{\xi_{1}-1}}=\infty\) for all \(x\in A\), contradicting (F5). In addition, we get \(\limsup_{s\to 0}\frac{|F(x,s)|}{m(x)|s|^{\xi_{1}}}<\infty\) uniformly almost everywhere in Ω, by the L’Hôpital rule. Define the crucial value
$$ {\mathcal {C}}_{f}=\mathop{\operatorname {ess\,sup}}\limits_{s\neq0,x\in\Omega}{ \frac{\vert f(x,s)\vert }{m(x)\vert s\vert ^{q-1}}}. $$
(4.7)
Then it follows from the analogous arguments in [10] that \({\mathcal {C}}_{f}\) is a positive constant. However, since the conditions of f are slightly different from those of [10], we discuss this fact. Indeed, \(\mathcal{C}_{f}>0\) having \(f\not\equiv0\) and \(\mathcal{C}_{f}<\infty\), since first, by (F7)
$$\lim_{s\to0}\frac{|f(x,s)|}{m(x)|s|^{q-1}}=\lim_{s\to 0} \biggl(\frac{|f(x,s)|}{m(x)|s|^{\xi_{1}-1}} \biggr)|s|^{\xi_{1}-q}=0 $$
uniformly almost everywhere in Ω, having \(q<\xi_{1}\). This, together with the assumption (F6) yields \(\mathcal{C}_{f}<\infty\). Similarly, we assert that \(g(x,0)=0\) for almost all \(x\in\Omega\) and the crucial value
$$ {\mathcal {C}}_{g}=\mathop{\operatorname {ess\,sup}}\limits_{s\neq0,x\in \partial\Omega} \frac{\vert g(x,s)\vert }{\vert s\vert ^{p-1}} $$
(4.8)
is a positive constant. Furthermore, the following relations hold:
$$ \mathop{\operatorname {ess\,sup}}\limits_{s\neq0, x\in\Omega}{\frac{\vert F(x,s)\vert }{m(x)\vert s\vert ^{q}}}=\frac{{\mathcal {C}}_{f}}{q} $$
and
$$ \mathop{\operatorname {ess\,sup}}\limits_{s\neq0, x\in\partial\Omega}{\frac{\vert G(x,s)\vert }{\vert s\vert ^{p}}}=\frac {{\mathcal {C}}_{g}}{p}. $$

Remark 4.3

[3, 8, 16]

From the embeddings in Lemmas 3.1 and 3.2, for any \(u\in X\), the following inequalities hold:
$$ s\|u\|_{L^{q}(\Omega)}\leq\|u\|_{X}\quad \mbox{and} \quad \tilde{s}\|u\|_{L^{r}(\partial\Omega)}\leq\|u\|_{X} $$
for every \(q\in[1,p^{*})\) and \(r\in[1, p^{\partial})\). The best constants for these embeddings are the largest constants s and such that the above inequalities hold, that is,
$$ {s}_{q}=\inf_{u\in X\setminus\{0\}}\frac{\|u\|_{X}}{\|u\|_{L^{q}(\Omega)}} $$
and
$$ \tilde{s}_{r}=\inf_{u\in X\setminus\{0\}} \frac{\|u\|_{X}}{\|u\|_{L^{r}(\partial \Omega)}}. $$
Moreover, since these embeddings are compact by Lemmas 3.1 and 3.2, we have the existence of extremals, namely, functions where the infimum is attained.

Theorem 4.4

Assume (H1), (J1)-(J5), (F1)-(F6) and (G1)-(G5) hold. Then
  1. (i)

    there exists a positive constant \(\ell_{*}=\min\{d, 1 \} \lambda_{1}/({\mathcal {C}}_{f} + \lambda_{1} {\mathcal {C}}_{g}\tilde{s}_{p}^{-1})\) such that the problem (P) has only the trivial solution for all \(\lambda\in[0,\ell_{*})\);

     
  2. (ii)

    there exists a positive constant λ̃ such that the problem (P) admits at least two distinct weak solutions in X for each \(\lambda\in(\ell_{*},\tilde{\lambda})\).

     

Proof

We prove the assertion (i). Let \(u\in X\) be a nontrivial weak solution of the problem (P). Then it is clear that
$$ \int_{\Omega} \varphi(x,\nabla u) \cdot\nabla v \,dx + \int_{\Omega} |u|^{p-2} u v \,dx =\lambda \int_{\Omega}f(x,u) v\,dx+\lambda \int_{\partial\Omega}g(x,u)v\,dS $$
for any \(v\in X\). If we put \(v=u\), then it follows from (J4) and the definitions of \({\mathcal {C}}_{f}\) and \({\mathcal {C}}_{g}\) that
$$\begin{aligned} &\min \{d, 1 \} \lambda_{1} \biggl( \int_{\Omega} \vert \nabla u\vert ^{p}\,dx + \int_{\Omega} \vert u\vert ^{p}\,dx \biggr) \\ &\quad\leq\lambda_{1} \biggl( \int_{\Omega}{\varphi(x,\nabla u)\cdot \nabla u}\,dx + \int_{\Omega} \vert u\vert ^{p}\,dx \biggr) \\ &\quad= \lambda_{1}\lambda \biggl( \int_{\Omega}{f(x,u)u}\,dx+ \int _{\partial\Omega}g(x,u)u\,dS \biggr) \\ &\quad\leq\lambda_{1}\lambda \biggl( \int_{\Omega}{\frac {f(x,u)}{m(x)\vert u\vert ^{q-1}}m(x)\vert u\vert ^{q}}\,dx+ \int_{\partial\Omega}{\frac{g(x,u)}{\vert u\vert ^{p-1}}\vert u\vert ^{p}} \, dS \biggr) \\ &\quad\leq\lambda_{1}\lambda \biggl({\mathcal {C}}_{f} \int_{\Omega }{m(x)\vert u\vert ^{q}}\,dx+ {\mathcal {C}}_{g} \int_{\partial \Omega}{ \vert u\vert ^{p}}\,dS \biggr) \\ &\quad\leq\lambda{\mathcal {C}}_{f} \biggl( \int_{\Omega} \vert \nabla u\vert ^{p}\,dx + \int_{\Omega} \vert u\vert ^{p}\,dx \biggr) + \lambda_{1}\lambda{\mathcal {C}}_{g} \int_{\partial\Omega} \vert u\vert ^{p}\,dS \\ &\quad\leq\lambda\bigl( {\mathcal {C}}_{f} + \lambda_{1} { \mathcal {C}}_{g}\tilde{s}_{p}^{-1} \bigr) \biggl( \int_{\Omega} \vert \nabla u\vert ^{p}\,dx + \int_{\Omega} \vert u\vert ^{p}\,dx \biggr). \end{aligned}$$
Thus if u is a nontrivial weak solution of the problem (P), then necessarily \(\lambda\geq\ell_{*}=\min\{d, 1 \} \lambda_{1}/({\mathcal {C}}_{f} + \lambda_{1} {\mathcal {C}}_{g}\tilde{s}_{p}^{-1})\), as claimed.
Next we prove the assertion (ii). It is obvious that Φ is bounded from below and \(\Phi(0)=\Psi(0)=0\). By the conditions (F3) and (G2), for any \(C(M)>0\), there exists a constant \(\delta>0\) such that
$$ F(x,\eta)\ge C(M)\vert \eta \vert ^{p} \quad \mbox{and}\quad G(x,\eta )\ge C(M)\vert \eta \vert ^{p} $$
(4.9)
for \(\vert \eta \vert >\delta\) and for almost all \(x \in\Omega\). Take \(v\in X\setminus \{0 \}\). Then the relation (4.9) implies that
$$\begin{aligned} I_{\lambda}(tv)&= \int_{\Omega}\Phi_{0}(x,t\nabla v)\,dx+ \int _{\Omega}\frac{1}{p}\vert tv\vert ^{p} \,dx-\lambda \biggl( \int _{\Omega}F(x,tv)\,dx+ \int_{\partial\Omega}G(x,tv)\,dS \biggr) \\ &\le \int_{\Omega}\frac{d}{p}\vert t \nabla v\vert ^{p}\,dx+ \int _{\Omega }\frac{1}{p}\vert tv\vert ^{p} \,dx-\lambda \biggl( \int_{\Omega }F(x,tv)\, dx+ \int_{\partial\Omega}G(x,tv)\,dS \biggr) \\ &\le t^{p} \biggl( \int_{\Omega}\frac{d}{p}\vert \nabla v\vert ^{p}\,dx+ \int_{\Omega}\frac{1}{p}\vert v\vert ^{p} \,dx-\lambda C(M) \biggl( \int_{\Omega} \vert v\vert ^{p}\,dx+ \int_{\partial \Omega} \vert v\vert ^{p}\,dS \biggr) \biggr) \end{aligned}$$
for sufficiently large \(t>1\). If \(C(M)\) is large enough, then we assert that \(I_{\lambda}(tv)\to-\infty\) as \(t \to\infty\). Hence the functional \(I_{\lambda}\) is unbounded from below. Proposition 3.9 and relations (4.7) and (4.8) imply that
$$\begin{aligned} \Psi(u) &= \int_{\Omega} F(x,u)\,dx+ \int_{\partial\Omega} G(x,u)\, dS \\ &\le \int_{\Omega}\frac{\vert F(x,u)\vert }{m(x)\vert u\vert ^{q}}m(x)\vert u\vert ^{q}\,dx+ \int_{\partial\Omega}\frac {\vert G(x,u)\vert }{\vert u\vert ^{p}}\vert u\vert ^{p}\, dS \\ &\le\frac{\mathcal {C}_{f}}{q} \int_{\Omega}m(x)\vert u\vert ^{q}\, dx+ \frac {\mathcal {C}_{g}}{p} \int_{\partial\Omega} \vert u\vert ^{p}\,dS \\ &\le \biggl(\frac{\mathcal {C}_{f}}{\lambda_{1}q}+\frac{\mathcal {C}_{g}}{p\tilde{s}_{p}} \biggr) \biggl( \int_{\Omega} \vert \nabla u\vert ^{p}\,dx + \int_{\Omega} \vert u\vert ^{p}\,dx \biggr) \\ &\le\frac{1}{\min \{d,1 \}} \biggl(\frac{\mathcal {C}_{f}p}{\lambda_{1}q}+\frac{\mathcal {C}_{g}}{\tilde{s}_{p}} \biggr) \biggl( \int_{\Omega}{\Phi_{0}(x,\nabla u)}\,dx+ \int_{\Omega }\frac{1}{p}\vert u\vert ^{p} \,dx \biggr) . \end{aligned}$$
For each \(u \in\Phi^{-1}((-\infty,\mu))\), it follows that
$$ \Phi(u)= \int_{\Omega}{\Phi_{0}(x,\nabla u)}\,dx+ \int_{\Omega}\frac{1}{p}\vert u\vert ^{p} \,dx < \mu. $$
Denote
$$ \tilde{\lambda}=\min \{d,1 \} \biggl(\frac{\mathcal {C}_{f}p}{\lambda_{1}q}+\frac{\mathcal {C}_{g}}{\tilde{s}_{p}} \biggr)^{-1}. $$
Hence we assert that
$$ \frac{1}{\mu}\sup_{u \in\Phi^{-1}((-\infty,\mu))} \Psi(u)< \frac{1}{\min \{d,1 \}} \biggl(\frac{\mathcal {C}_{f}p}{\lambda_{1}q}+\frac{\mathcal {C}_{g}}{\tilde{s}_{p}} \biggr) = \frac{1}{\tilde{\lambda}} < \frac{1}{\lambda}. $$

According to Lemma 4.1, we see that the functional \(I_{\lambda}\) satisfies \((C)\)-condition for each \(\lambda\in(\ell_{*},\tilde{\lambda})\). Therefore, all assumptions of Theorem 2.6 are satisfied, so that, for each \(\lambda\in (\ell_{*},\tilde{\lambda})\), the problem (P) admits at least two distinct weak solutions in X. This completes the proof. □

5 Existence of three weak solutions

Now, we deal with the existence of at least three weak solutions for the problem (P). We start from the following conditions:
  1. (F7)
    There exist a real number \(s_{0}\) and a positive constant \(r_{0}\) so small that
    $$\int_{B_{N}(x_{0},r_{0})}{F\bigl(x,\vert s_{0}\vert \bigr)} \,dx>0, $$
    and \(F(x,t)\ge0\) for almost all \(x\in B_{N}(x_{0},r_{0})\setminus B_{N}(x_{0},\sigma r_{0})\) with \(\sigma\in(0,1)\) and for all \(0\le t \le \vert s_{0}\vert \), where \(B_{N}(x_{0},r_{0})= \{x\in \Omega: \vert x-x_{0}\vert \le r_{0} \}\).
     
  2. (F8)
    f satisfies the following growth condition: for all \((x,s) \in\Omega\times\Bbb {R}\),
    $$\bigl|f(x,s)\bigr| \le a_{1}(x)+b_{1}(x)\vert s\vert ^{\gamma_{1}-1}, $$
    where \(a_{1} \in L^{q^{\prime}}(\Omega)\) with \(p< q< p^{*}\), \(\gamma_{1}< p\), \(b_{1}\in L^{\nu_{1}}(\Omega)\) with \(\nu_{1}>1\) and there exists \(t_{1}\in\mathbb{R}\) such that
    $$p\le t_{1} \le p^{*} \quad\mbox{and}\quad \frac{1}{\nu_{1}}+\frac{\gamma_{1}}{t_{1}}=1. $$
     
  3. (G6)
    There exists a positive constant d such that
    $$G(x,s)\ge0 $$
    for almost all \(x\in\partial\Omega\) and for all \(s\in[0,d]\).
     
  4. (G7)
    g satisfies the Carathéodory condition and the following growth condition holds: for all \((x,s) \in\partial\Omega\times\Bbb {R}\),
    $$\bigl|g(x,s)\bigr| \le a_{2}(x)+b_{2}(x)\vert s\vert ^{\gamma_{2}-1}, $$
    where \(a_{2} \in L^{q^{\prime}}(\partial\Omega)\) with \(p< q< p^{\partial}\), \(\gamma_{2}< p\), \(b_{2}\in L^{\nu_{2}}(\partial\Omega)\) with \(\nu_{2}>1\), and there exists \(t_{2}\in\mathbb{R}\) such that
    $$p\le t_{2} \le p^{\partial} \quad\mbox{and}\quad \frac{1}{\nu_{2}}+\frac{\gamma_{2}}{t_{2}}=1. $$
     

Lemma 5.1

Assume that (J1)-(J4), (F1), (F5), (F8), (G4), and (G7) hold. Then
$$ \lim_{\mu\to0^{+}}{\frac{\sup_{\Phi(u)\le\mu}\Psi(u)}{\mu}}=0. $$

Proof

By the conditions (F5), (F8), (G4), and (G7), there exists a positive constant \(\eta\in(0,1]\) such that
$$ F(x,s)< M_{1}\vert s\vert ^{\xi_{1}} \quad \mbox{and}\quad G(x,s)< N_{1}\vert s\vert ^{\xi_{2}} $$
(5.1)
for positive constants \(M_{1}\), \(N_{1}\), for almost all \(x\in\Omega\) and for all \(s\in[-\eta,\eta]\). Let us consider some positive constants \(M_{2}\), \(M_{3}\), \(N_{2}\), and \(N_{3}\) given by
$$\begin{aligned}& M_{2}=\sup_{\vert s\vert >1}{\frac{C_{7}(\vert s\vert +\vert s\vert ^{{\gamma _{1}}})}{\vert s\vert ^{{\xi_{1}}}}} \quad \mbox{and}\quad M_{3}=\sup_{\eta< \vert s\vert < 1}{ \frac{C_{7}(\vert s\vert +\vert s\vert ^{{\gamma _{1}}})}{\vert s\vert ^{{\xi_{1}}}}}, \\& N_{2}=\sup_{\vert s\vert >1}{\frac{C_{8}(\vert s\vert +\vert s\vert ^{{\gamma _{2}}})}{\vert s\vert ^{{\xi_{2}}}}} \quad \mbox{and}\quad N_{3}=\sup_{\eta< \vert s\vert < 1}{ \frac{C_{8}(\vert s\vert +\vert s\vert ^{{\gamma _{2}}})}{\vert s\vert ^{{\xi_{2}}}}}, \end{aligned}$$
for some positive constants \(C_{7}\) and \(C_{8}\). Then it follows from (5.1), (F8) and (G7) that
$$F(x,s)< M\vert s\vert ^{{\xi_{1}}} \quad\mbox{and}\quad G(x,s)< N\vert s \vert ^{{\xi_{2}}} $$
for almost all \(x\in\Omega\) and for all \(s\in\Bbb {R}\), where \(M=\max{ \{M_{1},M_{2},M_{3} \}}\) and \(N=\max\{N_{1}, N_{2}, N_{3} \}\).
If μ satisfies \((\min \{d,1 \}/p )\|u\|_{X}^{p}\le\mu<1\), where d is the positive constant from (J4), then by Lemmas 3.1 and 3.2, for some positive constants \(C_{9}\) and \(C_{10}\), we have
$$\begin{aligned} \Psi(u)&= \int_{\Omega}{F(x,u)}\,dx+ \int_{\partial\Omega}{G(x,u)}\, dS \\ &< M \int_{\Omega}{ \vert u\vert ^{{\xi_{1}}}}\,dx+N \int_{\partial \Omega }{\vert u\vert ^{{\xi_{2}}}}\,dS \\ &\le C_{9}\|u\|_{X}^{\xi_{1}}+C_{10}\|u \|_{X}^{\xi_{2}}\le C_{9}\mu^{\frac{\xi_{1}}{p}}+C_{10} \mu^{\frac{\xi_{2}}{p}}, \end{aligned}$$
where \(\xi_{1}>p\) and \(\xi_{2}>p\). It follows that
$$\lim_{\mu\to 0^{+}}{\frac{\sup_{\frac{\min \{d,1 \}}{p}\|u\|_{X}^{p}\le \mu }\Psi(u)}{\mu}}=0. $$
For any \(u\in\Phi^{-1}((-\infty,\mu])\) with \(\mu <\min \{d,1 \}/p\), we know that \(\Phi(u)\le\mu\) and so using the assumption (J4), we get
$$\frac{\min \{d,1 \}}{p} \biggl( \int_{\Omega}{ \vert \nabla u\vert ^{p}}\,dx+ \int_{\Omega} \vert u\vert ^{p}\,dx \biggr) \le \int_{\Omega}{\Phi_{0}(x,\nabla u)}\,dx+ \int_{\Omega}\frac{1}{p}\vert u\vert ^{p} \,dx\le\mu. $$
Hence we deduce that
$$\int_{\Omega}{ \vert \nabla u\vert ^{p}}\,dx+ \int_{\Omega }{\vert u\vert ^{p}}\,dx\le \frac{p}{\min \{d,1 \}}\mu< 1. $$
This inequality implies that \(\|u\|_{X}<1\). It follows that
$$\frac{\min \{d,1 \}}{p}\|u\|_{X}^{p}\le \int_{\Omega }{\Phi _{0}(x,\nabla u)}\,dx+ \int_{\Omega}\frac{1}{p}\vert u\vert ^{p} \,dx \le\mu. $$
So we can get
$$\Phi^{-1}\bigl((-\infty,\mu]\bigr)\subset \biggl\{ u\in X:\frac{\min \{ d,1 \}}{p}\|u \|_{X}^{p}\le\mu \biggr\} . $$
Then it follows that
$$ 0\le\lim_{\mu\to0^{+}}{\frac{\sup_{\Phi(u)\le\mu}\Psi(u)}{\mu }}\le \lim _{\mu\to 0^{+}}{\frac{\sup_{\frac{\min \{d,1 \}}{p}\|u\|_{X}^{p}\le \mu }\Psi(u)}{\mu}}=0 $$
and therefore the conclusion holds. □

Theorem 5.2

Assume that (J1)-(J4), (F1), (F7), (F8), and (G6)-(G7) hold. Then for each \(\lambda\in\Lambda_{\mu}\), the problem (P) has at least three distinct weak solutions in X for each \(\lambda\in \Lambda_{\mu}= \Big( \frac{\Phi(u_{\varrho})}{\Psi(u_{\varrho })}, \frac{\mu}{\sup_{\Phi(u)\in\mu}\Psi(u)} \Big)\).

Proof

All assumptions in Corollary 2.8 except the conditions (A1) and (A2) hold by Corollary 3.6 and a similar argument to Lemma 3.7. Note that \(s_{0}\neq0\) be from (F7). For \(\varrho\in (0,1 )\), define
$$ u_{\varrho}(x)= \textstyle\begin{cases} 0 &\mbox{if } x\in\overline{\Omega} \setminus B_{N}(x_{0},r_{0}),\\ \vert s_{0}\vert &\mbox{if } x\in B_{N}(x_{0},\varrho r_{0}),\\ \frac{\vert s_{0}\vert }{r_{0}(1-\varrho)}(r_{0}-\vert x-x_{0}\vert )&\mbox{if } x\in B_{N}(x_{0},r_{0})\setminus B_{N}(x_{0},\varrho r_{0}). \end{cases} $$
(5.2)
It is clear that \(0\le u_{\varrho}(x)\le \vert s_{0}\vert \) for all \(x\in \overline{\Omega}\), and so \(u_{\varrho}\in X\). Moreover, we have
$$\|u_{\varrho}\|_{X}^{p}=\frac{ {\vert s_{0}\vert }^{p}(1-\varrho^{N})}{(1-\varrho )^{p}}r_{0}^{N-p} \omega_{N}>0, $$
where \(\omega_{N}\) is the volume of \(B_{N}(0,1)\). Also, by using the assumption (F7), we get
$$\begin{aligned} \Psi(u_{\varrho})&= \int_{B_{N}(x_{0},\varrho r_{0})}{F\bigl(x,\vert s_{0}\vert \bigr)} \,dx+ \int _{B_{N}(x_{0},r_{0})\setminus B_{N}(x_{0},\varrho r_{0})}F \biggl(x,\frac{\vert s_{0}\vert }{r_{0}(1-\varrho)}\bigl(r_{0}- \vert x-x_{0}\vert \bigr) \biggr)\,dx \\ &> 0. \end{aligned}$$
Let us check the assumption (A1) in Corollary 2.8. Fix a real number \(\mu_{0}\) such that
$$0< \mu< \mu_{0}< \frac{1}{p}\min \{d,1 \}\min \bigl\{ \| u_{\varrho}\|_{X}^{p},1 \bigr\} \le \frac{1}{p}\min \{d,1 \}, $$
where \(u_{\varrho}\) was defined in (5.2). By Lemma 3.2 and (J4), we have
$$\begin{aligned} \Phi(u_{\varrho})&= \int_{\Omega}{\Phi_{0}(x,\nabla u_{\varrho})} \,dx+ \int_{\Omega}{\frac{1}{p}\vert u_{\varrho }\vert ^{p}}\,dx \\ &\ge \int_{\Omega}{\frac{d}{p}\vert \nabla u_{\varrho} \vert ^{p}}\,dx+ \int_{\Omega}{\frac{1}{p}\vert u_{\varrho} \vert ^{p}}\,dx\\ &\ge \frac{\min \{d,1 \}}{p}\|u_{\varrho} \|_{X}^{p}\ge\mu _{0}>\mu. \end{aligned}$$
From Lemma 5.1, we obtain
$$ \sup_{u\in\Phi^{-1}((-\infty,\mu])} {\Psi(u)} \le \frac{\mu}{2} \frac{\Psi(u_{\varrho})}{\Phi(u_{\varrho})}< \mu \frac{\Psi(u_{\varrho})}{\Phi(u_{\varrho})}, $$
that is,
$$\sup_{u\in\Phi^{-1}((-\infty,\mu])}{\Psi(u)}< \mu\frac{\Psi (u_{\varrho})}{\Phi(u_{\varrho})} $$
and hence the condition (A1) of Corollary 2.8 is fulfilled. For \(\|u\|_{X}\) large enough and for all \(\lambda\in \Bbb {R}\), it follows from the conditions (J4), (F8), and (G7) that
$$\begin{aligned} I_{\lambda}(u)={}& \int_{\Omega}{\Phi_{0}(x,\nabla u)}\,dx+ \int _{\Omega}\frac{1}{p}\vert u\vert ^{p} \,dx-\lambda \int_{\Omega }{F(x,u)}\, dx-\lambda \int_{\partial\Omega}{G(x,u)}\,dS \\ \ge{}&\frac{d}{p} \int_{\Omega}{ \vert \nabla u\vert ^{p}}\,dx + \frac{1}{p} \int_{\Omega}{ \vert u\vert ^{p}}\,dx -\vert \lambda \vert \int_{\Omega}{\bigl\vert a_{1}(x)\bigr\vert \vert u\vert }\,dx -\vert \lambda \vert \int_{\Omega}{\frac{1}{\gamma_{1}}\bigl\vert b_{1}(x) \bigr\vert \vert u\vert ^{\gamma_{1}}}\,dx \\ &{} -\vert \lambda \vert \int_{\partial\Omega}{\bigl\vert a_{2}(x)\bigr\vert \vert u\vert }\,dS -\vert \lambda \vert \int_{\partial\Omega}{\frac{1}{\gamma _{2}}\bigl\vert b_{2}(x) \bigr\vert \vert u\vert ^{\gamma_{2}}}\,dS \\ \ge{}&\frac{d}{p} \int_{\Omega}{ \vert \nabla u\vert ^{p}}\,dx + \frac{1}{p} \int_{\Omega}{ \vert u\vert ^{p}}\,dx -2\vert \lambda \vert \|a_{1}\|_{L^{q^{\prime}}(\Omega)}\|u\| _{L^{q}(\Omega)}\\ &{} - \frac{2\vert \lambda \vert }{{\gamma_{1}}}\|b_{1}\|_{L^{\nu _{1}}(\Omega)}\bigl\| \vert u\vert ^{\gamma_{1}}\bigr\| _{L^{\frac {t_{1}}{\gamma _{1}}}(\Omega)} \\ &{} -2\vert \lambda \vert \| a_{2}\|_{L^{q^{\prime}}(\partial\Omega)}\|u \|_{L^{q}(\partial \Omega)}-\frac{2\vert \lambda \vert }{{\gamma_{2}}}\|b_{2}\| _{L^{\nu _{2}}(\partial\Omega)}\bigl\| \vert u\vert ^{\gamma_{2}}\bigr\| _{L^{\frac {t_{2}}{\gamma_{2}}}(\partial\Omega)} \\ \ge{}&\frac{\min\{d,1\}}{p} \biggl( \int_{\Omega}{ \vert \nabla u\vert ^{p}}\,dx + \int_{\Omega}{ \vert u\vert ^{p}}\,dx \biggr) -2 \vert \lambda \vert C_{11}\|u\|_{X}-\frac{2\vert \lambda \vert C_{12}}{{\gamma _{1}}} \|u\|_{L^{t_{1}}(\Omega)}^{{\gamma_{1}}} \\ &{}-2\vert \lambda \vert C_{13}\|u\|_{X}- \frac{2\vert \lambda \vert C_{14}}{{\gamma _{2}}}\|u\|_{L^{t_{2}}(\partial\Omega)}^{{\gamma_{2}}} \\ ={}& \frac{\min\{d,1\}}{p}\|u\|_{X}^{p}-2\vert \lambda \vert C_{11}\|u\|_{X}-\frac{2\vert \lambda \vert C_{12}}{{\gamma _{1}}}\|u \|_{L^{t_{1}}(\Omega)}^{{\gamma_{1}}}\\ &{}-2\vert \lambda \vert C_{13}\|u \|_{X}-\frac{2\vert \lambda \vert C_{14}}{{\gamma_{2}}}\|u\|_{L^{t_{2}}(\partial\Omega)}^{{\gamma _{2}}} \\ \ge{}&\frac{\min\{d,1\}}{p}\|u\|_{X}^{p}-2\vert \lambda \vert C_{11}\|u\|_{X}-\frac{2\vert \lambda \vert C_{15}}{{\gamma_{1}}}\|u \|_{X}^{{\gamma_{1}}} \\ &{}-2\vert \lambda \vert C_{13}\|u \|_{X}-\frac{2\vert \lambda \vert C_{16}}{{\gamma_{2}}}\|u\|_{X}^{{\gamma_{2}}} \end{aligned}$$
for some constants \(C_{11}\), \(C_{12}\), \(C_{13}\), \(C_{14}\), \(C_{15}\), and \(C_{16}\). Since \(p>{\gamma_{1}}\) and \(p>{\gamma_{2}}\), we deduce that
$$\lim_{\|u\|_{X}\to\infty}{I_{\lambda}(u)}=\infty $$
for all \(\lambda\in\Bbb {R}\). Hence the functional \(I_{\lambda}\) is coercive for all \(\lambda\in\Lambda_{\mu}\). Consequently, Corollary 2.8 implies that the problem (P) has at least three distinct weak solutions in X for each \(\lambda\in\Lambda_{\mu}\). □

Theorem 5.3

Assume that (J1)-(J4), (F1), (F5), (F8), (G4), and (G7) hold. If furthermore f satisfies the following assumption:
  1. (F9)
    There exist \(\mu>0\) and \(s_{0}\in\Bbb {R}\) with \(\mu< (\min \{d,1 \}/p)\vert 2s_{0}\vert ^{p} (1- 2^{-N} )r_{0}^{N-p}\omega_{N}\) such that (F7) holds and
    $$2 \biggl(\frac{p}{\min \{d,1 \}} \biggr)^{\frac {1}{p}}\mathcal{M} \bigl(2 \mu^{\frac{1}{p}-1}+\mu^{\frac{\gamma _{1}}{p}-1}+ \mu^{\frac{\gamma_{2}}{p}-1} \bigr)< \frac{pr_{0}^{N-\frac{N-p}{p}} \operatorname {ess\,inf}_{B_{N}(x_{0},\frac{r_{0}}{2})}F(x,\vert s_{0}\vert )}{ 2^{N}\vert s_{0}\vert (2p\|a\|_{L^{p\prime}(\Omega)}+b\vert s_{0}\vert ^{p-1}+1 )}, $$
     
where \(\mathcal{M}=\max \{{s}_{q}^{-1}\|a_{1}\|_{L^{q\prime}(\Omega )},{s}_{t_{1}}^{-1}\|b_{1}\|_{L^{\nu_{1}}(\Omega)}, {\tilde{s}}_{q}^{-1}\|a_{2}\|_{L^{q\prime}(\partial\Omega )},{\tilde{s}}_{t_{2}}^{-1}\|b_{2}\|_{L^{\nu_{2}}(\partial\Omega)}\}\), then the problem (P) has at least distinct three solutions for every
$$\begin{aligned} \lambda\in\tilde{\Lambda}:={}& \biggl(\frac{2^{N}\vert s_{0}\vert (2p\|a\|_{L^{p\prime }(\Omega)}+b\vert s_{0}\vert ^{p-1}+1 )}{p r_{0}^{N-\frac {N-p}{p}} \operatorname {ess\,inf}_{B_{N}(x_{0},\frac{r_{0}}{2})}F(x,\vert s_{0}\vert )} ,\\ &{} \frac{1}{2} \biggl(\frac{\min \{d,1 \} }{p} \biggr)^{\frac{1}{p}}\mathcal{M}^{-1} \bigl(2\mu^{\frac {1}{p}-1}+\mu^{\frac{\gamma_{1}}{p}-1}+ \mu^{\frac{\gamma _{2}}{p}-1} \bigr)^{-1} \biggr) \\ \subset{}& \biggl(\frac{\Phi(\tilde{u}_{1})}{\Psi(\tilde{u}_{1})},\frac{\mu}{\sup_{\Phi(u)\le\mu}\Psi(u)} \biggr). \end{aligned}$$

Proof

By Corollary 3.6 and a similar argument to Lemma 3.7, all assumptions in Corollary 2.8 except the conditions (A1) and (A2) are fulfilled.

Define
$$ \tilde{u}_{1}(x)= \textstyle\begin{cases} 0 &\mbox{if } x\in\Omega\setminus B_{N}(x_{0},r_{0}),\\ \vert s_{0}\vert &\mbox{if } x\in B_{N}(x_{0},\frac{r_{0}}{2}),\\ \frac{2\vert s_{0}\vert }{r_{0}}(r_{0}-\vert x-x_{0}\vert )&\mbox{if } x\in B_{N}(x_{0},r_{0})\setminus B_{N}(x_{0},\frac{r_{0}}{2}). \end{cases} $$
Then it is clear that \(0\le\tilde{u}_{1}(x)\le \vert s_{0}\vert \) for all \(x\in\overline{\Omega}\), and so \(\tilde{u}_{1}\in X\). Moreover, it follows from (J4) that
$$ \Phi(\tilde{u}_{1})\ge \frac{\min \{d,1 \}}{p}\vert 2s_{0} \vert ^{p} \biggl(1-\frac{1}{2^{N}} \biggr)r_{0}^{N-p} \omega_{N}>0 $$
and
$$\begin{aligned} \Phi(\tilde{u}_{1})&= \int_{\Omega}\Phi_{0}(x,\nabla\tilde {u}_{1})\,dx+ \int_{\Omega}\frac{1}{p}\vert \tilde{u}_{1} \vert ^{p}\,dx \\ &\le \int_{\Omega}a(x)\vert \nabla\tilde{u}_{1}\vert + \frac {b}{p}\vert \nabla\tilde{u}_{1}\vert ^{p} \,dx+ \int_{\Omega}\frac {1}{p}\vert \tilde{u}_{1} \vert ^{p}\,dx \\ &\le 2\|a\|_{L^{p'}(\Omega)}\vert s_{0}\vert \omega_{N}r_{0}^{\frac {N}{p}-1}+\frac{b\vert s_{0}\vert ^{p}}{p} \omega_{N} \biggl(1-\frac {1}{2^{N}} \biggr)r_{0}^{N-p} +\frac{\vert s_{0}\vert }{2^{N}p}\omega_{N}r_{0}^{N} \\ &\le \frac{1}{p}\vert s_{0}\vert \omega_{N}r_{0}^{\frac{N-p}{p}} \bigl(2p\|a\|_{L^{p\prime}(\Omega)}+b\vert s_{0}\vert ^{p-1}+1 \bigr). \end{aligned}$$
Owing to the assumption (F7), we deduce that
$$\begin{aligned} \Psi(\tilde{u}_{1}) &\ge \int_{B_{N}(x_{0},\frac {r_{0}}{2})}F(x,\tilde{u}_{1})\,dx \\ &\ge\mathop{\operatorname {ess\,inf}}\limits_{B_{N}(x_{0},\frac{r_{0}}{2})}F\bigl(x,\vert s_{0}\vert \bigr) \biggl(\frac{\omega_{N}r_{0}^{N}}{2^{N}} \biggr), \end{aligned}$$
and thus
$$ \frac{\Psi(\tilde{u}_{1})}{\Phi(\tilde{u}_{1})} \ge\frac{pr_{0}^{N-\frac{N-p}{p}} \operatorname {ess\,inf}_{B_{N}(x_{0},\frac {r_{0}}{2})}F(x,\vert s_{0}\vert )}{ 2^{N}\vert s_{0}\vert (2p\|a\|_{L^{p\prime}(\Omega)}+b\vert s_{0}\vert ^{p-1}+1 )}. $$
(5.3)
From \(\mu< \frac{\min \{d,1 \}}{p}\vert 2s_{0}\vert ^{p} (1-\frac{1}{2^{N}} )r_{0}^{N-p}\omega_{N}\), one has \(\mu< \Phi(\tilde{u}_{1})\). For each \(u\in\Phi^{-1}((-\infty,\mu])\), it follows from (F8) and (G7) that
$$\begin{aligned} \Psi(u) ={}& \int_{\Omega}{F(x,u)}\,dx+\lambda \int_{\partial\Omega}{G(x,u)}\, dS \\ \le{}& \int_{\Omega}{\bigl\vert a_{1}(x)\bigr\vert \vert u\vert }\,dx + \int_{\Omega}{\frac{1}{\gamma_{1}}\bigl\vert b_{1}(x) \bigr\vert \vert u\vert ^{\gamma _{1}}}\,dx+ \int_{\partial\Omega}{\bigl\vert a_{2}(x)\bigr\vert \vert u\vert }\,dS \\ &{}+ \int_{\partial\Omega}{\frac{1}{\gamma_{2}}\bigl\vert b_{2}(x) \bigr\vert \vert u\vert ^{\gamma_{2}}}\,dS \\ \le{}&2\|a_{1}\|_{L^{q^{\prime}}(\Omega)}\|u\|_{L^{q}(\Omega)} + \frac{2}{{\gamma_{1}}}\|b_{1}\|_{L^{\nu_{1}}(\Omega)}\bigl\| \vert u\vert ^{\gamma_{1}}\bigr\| _{L^{\frac{t_{1}}{\gamma_{1}}}(\Omega)}\\ &{}+2\| a_{2}\|_{L^{q^{\prime}}(\partial\Omega)}\|u \|_{L^{q}(\partial \Omega)} +\frac{2}{{\gamma_{2}}}\|b_{2}\|_{L^{\nu_{2}}(\partial\Omega )}\bigl\| \vert u\vert ^{\gamma_{2}}\bigr\| _{L^{\frac{t_{2}}{\gamma _{2}}}(\partial \Omega)} \\ \le{}&2{s}_{q}^{-1}\|a_{1}\|_{L^{q^{\prime}}(\Omega)}\|u \|_{X}+\frac{2\| b_{1}\|_{L^{\nu_{1}}(\Omega)}}{{\gamma _{1}}}\|u\|_{L^{t_{1}}(\Omega)}^{{\gamma_{1}}}\\ &{}+ 2 \tilde {s}_{q}^{-1}\|a_{2}\|_{L^{q^{\prime}}(\partial\Omega)}\|u \|_{X}+\frac {2\|b_{2}\|_{L^{\nu_{2}}(\partial\Omega)}}{{\gamma _{2}}}\|u\|_{L^{t_{2}}(\partial\Omega)}^{{\gamma_{2}}} \\ \le{}& 2{s}_{q}^{-1}\|a_{1}\|_{L^{q^{\prime}}(\Omega)}\|u \|_{X}+\frac {2\|b_{1}\|_{L^{\nu_{1}}(\Omega)}}{{\gamma_{1}}{s}_{t_{1}}}\|u\| _{X}^{{\gamma_{1}}} \\ &{}+2\tilde{s}_{q}^{-1}\|a_{2}\|_{L^{q^{\prime}}(\partial\Omega )}\|u \|_{X}+\frac{2 \|b_{2}\|_{L^{\nu_{2}}(\partial\Omega)}}{{\gamma_{2}}\tilde {s}_{t_{2}}}\|u\|_{X}^{{\gamma_{2}}} \\ \le{}& 2 \biggl(\frac{p}{\min \{d,1 \}} \biggr)^{\frac {1}{p}} \biggl({s}_{q}^{-1} \|a_{1}\|_{L^{q\prime}(\Omega)}\mu^{\frac {1}{p}}+\frac{\|b_{1}\|_{L^{\nu_{1}}(\Omega)}}{\gamma_{1}{s}_{t_{1}}}\mu ^{\frac{\gamma_{1}}{p}}\\ &{}+ \tilde{s}_{q}^{-1}\|a_{2} \|_{L^{q\prime }(\partial\Omega)}\mu^{\frac{1}{p}}+\frac{\|b_{2}\|_{L^{\nu _{2}}(\partial\Omega)}}{\gamma_{2}\tilde{s}_{t_{2}}}\mu^{\frac{\gamma _{2}}{p}} \biggr) \\ \le{}& 2 \biggl(\frac{p}{\min \{d,1 \}} \biggr)^{\frac {1}{p}}\mathcal{M} \bigl(2 \mu^{\frac{1}{p}}+\mu^{\frac{\gamma _{1}}{p}}+ \mu^{\frac{\gamma_{2}}{p}} \bigr), \end{aligned}$$
and so
$$ \sup_{u \in\Phi^{-1}((-\infty,\mu])} \Psi(u)\le 2 \biggl(\frac{p}{\min \{d,1 \}} \biggr)^{\frac {1}{p}}\mathcal{M} \bigl(2\mu^{\frac{1}{p}}+\mu^{\frac{\gamma _{1}}{p}}+ \mu^{\frac{\gamma_{2}}{p}} \bigr). $$
From (5.3) and the assumption (F5), we infer \(\frac{1}{\mu}\sup\Psi(u)<\frac{\Psi(\tilde{u}_{1})}{\Phi (\tilde{u}_{1})}\). As seen before, the functional \(I_{\lambda}\) is coercive for each \(\lambda>0\). Taking into account that \(\tilde{\Lambda}\subset (\frac{\Phi(\tilde{u}_{1})}{\Psi(\tilde{u}_{1})},\frac{\mu}{\sup_{\Phi(u)\le\mu}\Psi(u)} )\), Corollary 2.8 ensures that the problem (P) has at least distinct three solutions for each \(\lambda\in\tilde{\Lambda}\). □

Declarations

Acknowledgements

The authors would like to thank the referees for careful reading of the manuscript. The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827). The second author was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2014R1A1A2059536).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Institute of Mathematical Sciences, Ewha Womans University
(2)
Department of Mathematics Education, Sangmyung University

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