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Existence and multiplicity of solutions for $p(x)$-Laplacian equations in $\Bbb R^N$. (English) Zbl 1134.35333
Summary: This paper investigates the existence and multiplicity of solutions for $p(x)$-Laplacian equations $-\mathrm{div}(|\nabla u|^{p(x)-2}\nabla u) + |u|^{p(x)-2}u = f(x,u)$ in $\Bbb R^N$, $u\in W^{1,p(x)}(\Bbb R^N)$ in the cases corresponding to “sublinear”, “superlinear” and “concave-convex nonlinearity” if $p=2$. They apply critical point theory in certain Sobolev spaces fitted to the problem.

##### MSC:
 35J60 Nonlinear elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 47J30 Variational methods (nonlinear operator equations) 58E05 Abstract critical point theory
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