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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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