Year: 2013
Author: Nguyen Thanh Chung
Journal of Partial Differential Equations, Vol. 26 (2013), Iss. 3 : pp. 217–225
Abstract
Using variational methods, we prove the existence of a nontrivial weak solution for the problem \begin{align*} \left\{ \begin{array}{ll} -\sum_{i=1}^N\partial_{x_i}\Big(|\partial_{x_i}u|^{p_i-2}\partial_{x_i}u\Big) =\lambda a(x)|u|^{q(x)-2}u+|u|^{p^\ast-2}u, & \text{ in } \Omega, \\ u = 0, & \text{ in } \partial\Omega, \end{array} \right. \end{align*} where $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) is a bounded domain with smooth boundary $\partial\Omega$, $2 \leq p_i < N$, $i = \overline{1, N}$, $q: \overline \Omega \to (1, p^\ast)$ is a continuous function, $p^\ast=\frac{N}{\sum_{i=1}^N \frac{1}{p_i}-1}$ is the critical exponent for this class of problem, and $\lambda$ is a parameter.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jpde.v26.n3.2
Journal of Partial Differential Equations, Vol. 26 (2013), Iss. 3 : pp. 217–225
Published online: 2013-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 9
Keywords: Anisotropic equation