Spectrum for anisotropic equations involving weights and variable exponents

Abstract

We study the problem $$ -\sum_{i=1}^{N}\Big[\partial_{x_{i}}\Big(|\partial_{x_{i}}u|^{p_{i}(x)-2} \partial_{x_{i}}u\Big) +|u|^{p_{i}(x)-2}u\Big]+|u|^{q(x)-2}u =\lambda g(x)|u|^{r(x)-2}u $$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ ($N\geq3$), with smooth boundary, $\lambda$ is a positive real number, the functions $p_{i}, q, r:\overline\Omega\to[2,\infty)$ are Lipschitz continuous, $g:\overline\Omega\to[0,\infty)$ is measurable and these fulfill certain conditions. The main result of this paper establish the existence of two positive constants $\lambda_0$ and $\lambda_{1}$ with $0<\lambda_0\leq\lambda_{1}$ such that any $\lambda\in[\lambda_{1},\infty)$ is an eigenvalue, while any $\lambda\in(0,\lambda_0)$ is not an eigenvalue of our problem.

Keywords

• p(.)-Laplace operator
• anisotropic variable exponent Sobolev space
• critical point
• weak solution
• eigenvalue