Positive solutions for singular \((p,2)\)-equations. (English) Zbl 1432.35101
The authors consider the following nonlinear, nonparametric singular Dirichlet problem:
\[
-\operatorname{div}(|\nabla u|^{p-2}\nabla u(z)) - \Delta u(z) = \mu(u(z)) + f(z, u(z)) \text{ in } \Omega, \ \
u|_{\partial \Omega} =0, \ p>2,\tag{P}
\]
where \(\Omega \subset \mathbb{R}^N\) is a bounded smooth domain, the function \(\mu(\cdot)\) is singular at
\(x = 0\), \(f(z, x)\) is a Carathéodory function and of \((p-1)\)-superlinear growth with respect to \(x\) near \(+\infty\) but the so called Ambrosetti-Rabinowitz condition may fail. Under some further conditions on \(f(z, x)\), the existence of a pair of positive smooth solutions is obtained for problem (P) by combining variational methods and some
truncation techniques.
Reviewer: Huansong Zhou (Wuhan)
MSC:
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 35J20 | Variational methods for second-order elliptic equations |
| 35J75 | Singular elliptic equations |
Keywords:
\(p\)-Laplacian; singular nonlinear term; Dirichlet problem; positive solutions; variational methodsReferences:
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