INFINITELY MANY SOLUTIONS FOR A CLASS OF DEGENERATE ANISOTROPIC ELLIPTIC PROBLEMS WITH VARIABLE EXPONENT

Abstract

We study the nonlinear degenerate problem $-\sum_{i=1}^N \partial_{x_i} a_i \left(x,\partial_{x_i}u \right) = f(x,u)$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega \subset {\mathbb R}^N$ ($N \geq 3$) is a bounded domain with smooth boundary, $\sum_{i=1}^N \partial_{x_i} a_i \left(x,\partial_{x_i}u \right)$ is a $\overset{\rightarrow} p(\cdot)$ - Laplace type operator and the nonlinearity $f$ is $(P_+^+-1)$ - superlinear at infinity (with $\overset{\rightarrow} p(x) = (p_1(x), p_2(x), ... p_N(x))$ and $P_+^+ = \max_{i \in \{1,...,N\}} \left\{\sup_{x \in \Omega} p_i(x) \right\}$). By means of the symmetric mountain-pass theorem of Ambrosetti and Rabinowitz, we establish the existence of a sequence of weak solutions in appropriate anisotropic variable exponent Sobolev spaces.