Existence of three solutions for equations of p ( x ) $p(x)$ -Laplace type operators with nonlinear Neumann boundary conditions

Abstract

In this paper, we are concerned with nonlinear elliptic equations of the $p(x)$ -Laplace type operators $$\textstyle\begin{cases} -\operatorname{div}(a(x,\nabla u))+\vert u\vert ^{p(x)-2}u=\lambda f(x,u) &\mbox{in } \Omega, a(x,\nabla u)\frac{\partial u}{\partial n} = \lambda\theta g(x,u) &\mbox{on } \partial\Omega, \end{cases} $$ which are subject to nonlinear Neumann boundary conditions. Here the function $a(x,v)$ is of type $\vert v\vert ^{p(x)-2}v$ with a continuous function $p: \overline{\Omega} \to(1,\infty)$ and the functions $f, g$ satisfy a Caratheodory condition. The main purpose of this paper is to establish the existence of at least three weak solutions of the above problem by applying an abstract three critical points theorem which is inspired by the work of Ricceri (Nonlinear Anal. 74:7446-7454, 2011) Furthermore, we determine two intervals of λ’s precisely such that the first is where the given problem admits only the trivial solution, and the second is where the given problem has at least two nontrivial solutions as considering the positive principal eigenvalue for the $p(x)$ -Laplacian Neumann problems and an estimate of the Sobolev trace embedding’s constant.