Arbitrarily many solutions for a perturbed \(p(x)\)-Laplacian equation involving oscillatory terms
This paper deals with a perturbed \(p(x)\)-Laplacian equation involving oscillatory terms: \[ \begin{cases} -\text{div}\left(|\nabla u|^{p(x)-2}\nabla u\right)+|u|^{p(x)-2}u=Q(x)[f(u)+\varepsilon g(u)]x\in\mathbb R^N,\\u\geq0,\\u(x)\to 0\text{ as }|x|\to\infty. \end{cases} \] By using variational methods and the non-smooth version of the symmetric criticality principle, we establish (a) the unperturbed problem \(P_0\) has infinitely many distinct solutions; (b) the number of distinct solutions for \(P_\varepsilon\) becomes greater whenever \(\varepsilon\) becomes smaller. Our results are a generalization of the case of the Laplacian from \textit{A. Kristály} [J. Differ. Equations 245, No. 12, 3849--3868 (2008; Zbl 1169.35025)] to the case of the \(p(x)\)-Laplacian.
scientific article; zbMATH DE number 5649904