Abstract
In this paper, we consider the existence of multiple solutions for the following $p(x)$-Laplacian equations with critical Sobolev growth conditions \[ \begin{cases} -div(|\nabla u|^{p(x)-2} \nabla u) + |u|^{p(x)-2} u = f(x,u) \; \textrm{in } \Omega, \\ u = 0 \; \textrm{on } \partial \Omega.\end{cases} \]
We show the existence of infinitely many pairs of solutions by applying the Fountain Theorem and the Dual Fountain Theorem respectively. We also present a variant of the concentration-compactness principle, which is of independent interest.
Citation
Yuan Liang. Xianbin Wu. Qihu Zhang. Chunshan Zhao. "MULTIPLE SOLUTIONS OF A $p(x)$-LAPLACIAN EQUATION INVOLVING CRITICAL NONLINEARITIES." Taiwanese J. Math. 17 (6) 2055 - 2082, 2013. https://doi.org/10.11650/tjm.17.2013.3074
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