A new variational method for the \(p(x)\)-Laplacian equation. (English) Zbl 1115.35035
For a bounded region \(\Omega\subset\mathbb R^N\) and a function \(p\in C(\bar\Omega)\) such that \(p(x)\geq 2\) in \(\Omega\), the author studies the Dirichlet problem
\[
-\text{div}(| \nabla u(x)| ^{p(x)-2}\nabla u(x))=F_{u}(x,u(x)),\quad u| _{\partial\Omega}=0
\]
looking for weak solutions belonging to the generalized Orlicz-Sobolev Space \(W_{0}^{1,p(x)}(\Omega)\). Under suitable convexity assumptions on \(F\) he obtains the existence of a solution by applying a dual variational method.
Reviewer: Piero Montecchiari (Ancona)
MSC:
35J20 | Variational methods for second-order elliptic equations |
35F25 | Initial value problems for nonlinear first-order PDEs |
35J60 | Nonlinear elliptic equations |
47J30 | Variational methods involving nonlinear operators |
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