Open Access
Translator Disclaimer
2013 MULTIPLE SOLUTIONS OF A $p(x)$-LAPLACIAN EQUATION INVOLVING CRITICAL NONLINEARITIES
Yuan Liang, Xianbin Wu, Qihu Zhang, Chunshan Zhao
Taiwanese J. Math. 17(6): 2055-2082 (2013). DOI: 10.11650/tjm.17.2013.3074

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

Download 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

Information

Published: 2013
First available in Project Euclid: 10 July 2017

zbMATH: 1287.35009
MathSciNet: MR3141874
Digital Object Identifier: 10.11650/tjm.17.2013.3074

Subjects:
Primary: 35J20 , 35J25 , 35J60

Keywords: $p(x)$-Laplacian , Critical exponent , Integral functional , variable exponent Sobolev space

Rights: Copyright © 2013 The Mathematical Society of the Republic of China

JOURNAL ARTICLE
28 PAGES


SHARE
Vol.17 • No. 6 • 2013
Back to Top