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Singular quasilinear equations with quadratic growth in the gradient without sign condition. (English) Zbl 1161.35013

Summary: Given a bounded domain \(\Omega\) in \(\mathbb R^N\), and a function \(a\in L^q(\Omega)\) with \(q>N/2\), we study the existence of a positive solution for the quasilinear problem
\[ \begin{aligned} & -\Delta w+g(x,w)|\nabla w|^2=a(x),\quad x\in\Omega,\\ & w\in H^1_0(\Omega),\end{aligned} \]
where \(g(x,s)\) is a Carathéodory function on \(\Omega\times(0,+\infty)\) which may have a singularity at \(s=0\) and may change of sign.

MSC:

35J60 Nonlinear elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35A35 Theoretical approximation in context of PDEs
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