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 Caratheodory 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.
Highlights
1 Introduction In the present paper, we are concerned with multiple solutions for the nonlinear Neumann boundary value problem associated with p-Laplacian type
Based on [, ], Bonanno and Chinnì [ ] 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 [ ]: (AR) There exist positive constants M and θ such that θ > p and
Summary
First we show the existence of at least two weak solutions for (P) without assuming that f satisfies the (AR) condition. In Section , by using Zhong’s Ekeland variational principle, we state some critical point theorems for continuously differentiable functions with the Cerami condition.
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