Existence and multiplicity of nontrivial solutions for \(p\)-Laplacian Schrödinger-Kirchhoff-type equations. (English) Zbl 1318.35021
Summary: In this paper, we are concerned with the following Schrödinger-Kirchhoff-type problem:
\[
\begin{cases} -(a+b\int_{\mathbb R^N}|\nabla u|^pdx)^{p-1}\Delta_pu+\lambda V(x)|u|^{p-2}u=f(x,u),\quad x\in\mathbb R^N, \\ u \in W^{1,p}(\mathbb R^N),\end{cases}\tag{P}
\]
where \(a>0\), \(b\geq 0\) are constants, \(\Delta_pu:=\mathrm{div}(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplacian operator with \(p\geq 2\), \(V(x)\) is the potential function satisfying some conditions which may not guarantee the compactness of the corresponding Sobolev embedding. By using the variational methods, we prove the existence and multiplicity of nontrivial solutions for problem (P).
MSC:
| 35J35 | Variational methods for higher-order elliptic equations |
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