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Perturbed elliptic equations with oscillatory nonlinearities

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  • In this paper, arbitrarily many solutions, in particular arbitrarily many nodal solutions, are proved to exist for perturbed elliptic equations of the form \begin{equation*}\label{} \left\{ \begin{array}{ll} \displaystyle -\Delta_p u+|u|^{p-2}u = Q(x)(f(u)+\varepsilon g(u)),\ \ \ x\in \mathbb R^N, \\ u\in W^{1,p}(\mathbb R^N), \end{array} \right. (P_\varepsilon) \end{equation*} where $\Delta_p$ is the $p$-Laplacian operator defined by $\Delta_p u=\text{div}(|\nabla u|^{p-2}\nabla u)$, $p>1$, $Q\in \mathcal{C}(\mathbb R^N,\mathbb R)$ is a positive function, $f\in\mathcal{C}(\mathbb R, \mathbb R)$ oscillates either near the origin or near the infinity, and $\epsilon$ is a real number. For $g$ it is only required that $g\in\mathcal{C}(\mathbb R, \mathbb R)$. Under appropriate assumptions on $Q$ and $f$ the following results which are special cases of more general ones are proved: the unperturbed problem $(P_0)$ has infinitely many nodal solutions, and for any $n\in\mathbb N$ the perturbed problem $(P_\varepsilon)$ has at least $n$ nodal solutions provided that $|\epsilon|$ is sufficiently small.
    Mathematics Subject Classification: 35J20, 35J65.


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