Electron. J. Diff. Equ., Vol. 2014 (2014), No. 241, pp. 1-16.

Spectrum for anisotropic equations involving weights and variable exponents

Ionela-Loredana Stancut

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, 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 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.

Submitted June 26, 2014. Published November 18, 2014.
Math Subject Classifications: 35D30, 35J60, 58E05.
Key Words: p(.)-Laplace operator; anisotropic variable exponent Sobolev space; critical point; weak solution; eigenvalue.

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Ionela-Loredana Stancut
Department of Mathematics
University of Craiova
200585, Romania
email: stancutloredana@yahoo.com

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