Existence of solutions for \(p(x)\)-Laplacian Dirichlet problem. (English) Zbl 1146.35353

Summary: The paper presents several sufficient conditions for the existence of solutions for the Dirichlet problem with the \(p(x)\)-Laplacian \[ -\text{div}(|\nabla u|^{p(x)-2}\nabla u)=f(x,u),\quad x\in\Omega,\quad u=0,\quad x\in\partial\Omega, \] where \(\Omega\subset\mathbb R^n\) is a bounded domain. In particular, a criterion for the existence of infinitely many pairs of solutions for the problem is obtained. The discussion is based on a theory of spaces \(L^{p(x)}(\Omega)\) and \(W_0^{1,p(x)}(\Omega)\).


35J60 Nonlinear elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
47J30 Variational methods involving nonlinear operators
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