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Positive solutions for weighted singular \(p\)-Laplace equations via Nehari manifolds. (English) Zbl 1475.35158

Let \(\Omega \subseteq \mathbb{R}^N\) (\(N \geq 1\)) be a bounded domain with a Lipschitz boundary \(\partial \Omega\). The authors study a singular Dirichlet problem of the form \[ \begin{cases} -\operatorname{div}\left(\xi(x)|\nabla u|^{p-2}\nabla u\right)= a(x)u^{-\gamma}+\lambda u^{r-1} \quad\text{in } \Omega,\\ u|_{\partial \Omega}=0, \quad u\geq 0 ,\quad \lambda>0, \quad\gamma \in (0,1), \quad 1<p<r<p^\ast. \end{cases}\tag{\(P_\lambda\)} \] In this problem, \(s \to a(x) s^{-\gamma}\) is a singular term, and \(s \to \lambda s^{r-1}\) is a \((p-1)\)-superlinear perturbation. The weight \(\xi\in L^\infty(\Omega)\), \(\xi\ge 0\), is discontinuous in general and hence neither the global regularity theory nor the strong maximum principle can be used. Thus, the authors develop an approach based on the Nehari method. The main result of the paper establishes the existence of at least two positive bounded solutions of \((P_\lambda)\).

MSC:

35J62 Quasilinear elliptic equations
35J75 Singular elliptic equations
35J67 Boundary values of solutions to elliptic equations and elliptic systems
35J20 Variational methods for second-order elliptic equations
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