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Perturbation for a $$p(x)$$-Laplacian equation involving oscillating nonlinearities in $$R^N$$. (English) Zbl 1152.35041
Summary: We study multiple solutions of the following problem in $$\mathbb{R}^N$$: $-\text{div}(|\nabla u|^{p(x)- 2}\nabla u)+ |u|^{p(x)- 2}u= f(x, u)+\lambda g(x,u),$ $u\in W^{1,p(x)}(\mathbb{R}^N),$ where the potential $$F(x, t)= \int^t_0 f(x, s)\,ds$$ has a suitable oscillating behavior in any neighborhood of the origin (respectively $$+\infty$$), and $$g$$ is a perturbation term. Our results are a generalization of the case of the constant exponent and bounded domain from [G. Anello and G. Cordaro, J. Differ. Equations 234, No. 1, 80-90 (2007; Zbl 1220.35030)] to the case of variable exponent and $$\mathbb{R}^N$$.

##### MSC:
 35J70 Degenerate elliptic equations 35J35 Variational methods for higher-order elliptic equations 35J60 Nonlinear elliptic equations
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##### References:
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