On some multiple solutions for a p(x)-Laplacian equation with critical growth

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• 作者：Joã ; o Pablo P. da Silva ^{jpabloufpa@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Critical nonlinearity ; p(x)p(x)-Laplacian ; Variational methods ; Concentration compactness principle
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 April 2016
• 年：2016
• 卷：436
• 期：2
• 页码：782-795
• 全文大小：378 K

文摘

In this work we prove that, for each m∈N$m\in \mathbb{N}$, if λ is small, there are m   solutions for the equation −div(|∇u|^p(x)−2∇u)=f(x,u)+λ|u|^q(x)−2u$-\mathrm{div}({|\mathrm{\nabla }u|}^{p(x)-2}\mathrm{\nabla }u)=f(x,u)+\lambda {|u|}^{q(x)-2}u$, x∈Ω$x\in \mathrm{\Omega }$, in a bounded domain of R^N${\mathbb{R}}^N$, the nonlinearity is given by the critical growth in the context of variable exponents p^⁎(x)=Np(x)/(N−p(x))$p^{⁎}(x)=Np(x)/(N-p(x))$. The main tools used are the variational method and concentration compactness principle.