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Quasilinear equations with degenerate coerciveness and having multiple singularities. (English) Zbl 1307.35113

The purpose of the present paper is on giving sufficient conditions for having both a positive bounded solution and a distributional solution (Definition 1.1) of the following differential equation (for \(x\in \mathbb R^N\)) \[ \begin{aligned} -\text{div}\left(\frac{|\nabla u(x)|^{p-2}\nabla u(x)}{(u(x))^k}\mathbf{1}_{u^{-1}((0,\infty))}\right)+|u(x)|^{p-2}u(x)+\Lambda (x,u(x))\mathbf{1}_{u^{-1}((0,\infty))}|\nabla u(x)|^{p}\\ =f(x,u(x))+h(x),\end{aligned} \] these conditions are bearing on the positive bounded integrable function \(h\) and on \(f,\Lambda\) (which presents a singular behaviour at the origin), the real-valued Carathéodory on \(\mathbb R^N\times (0,\infty)\) and on \(\mathbb R^N\times\mathbb R\), respectively, that is, measurables w.r.t. the first component and continous w.r.t. the second one (Theorems 1.1 and 1.2).
\(\mathbf{1}_{u^{-1}((0,\infty))}\) stands for the indicator function on the open set \(u^{-1}((0,\infty))\) in \(\mathbb R^N\).

MSC:

35J62 Quasilinear elliptic equations
35D30 Weak solutions to PDEs
35J70 Degenerate elliptic equations
35J75 Singular elliptic equations
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