Existence of nontrivial solutions for singular quasilinear elliptic equations on \(\mathbb R^N\). (English) Zbl 1343.35120
Summary: In this paper, we study the existence of weak solutions for the singular quasilinear elliptic problem
\[
\begin{cases} -\mathrm{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)+V(x)|u|^{p-2}u\\=h(x)|u|^{s-2}u\pm H(x)|u|^{r-2}u,\quad x\in\mathbb R^N,\\ u(x)\to 0,\quad \text{as }|x|\to\infty\end{cases}\eqno{(0.1)}
\]
where \(1<p<N\), \(-\infty <a<(N-p)/p\), \(1<s,r<p^\ast =pN/(N-p)\). Using an approach which strongly relies on the Nehari manifold and the fibering method in combination with related variational methods, we prove that problem (0.1) admits multiple solutions under appropriate assumptions on the weight functions \(V(x)\), \(h(x)\), and \(H(x)\).
MSC:
| 35J75 | Singular elliptic equations |
| 35J35 | Variational methods for higher-order elliptic equations |
| 35J62 | Quasilinear elliptic equations |
| 35D30 | Weak solutions to PDEs |
Keywords:
singular quasilinear elliptic equation; variational methods; Nehari manifold; fibering mapsReferences:
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