Existence and nonexistence of solutions for singular quadratic quasilinear equations
type: Item
instance of: scholarly article
title: Existence and nonexistence of solutions for singular quadratic quasilinear equations
zbMATH Open document ID: 1173.35051
author: David Arcoya
author: José Carmona
author: Tommaso Leonori
author: Pedro J. Martínez-Aparicio
author: Luigi Orsina
author: Francesco Petitta
published in: Journal of Differential Equations
publication date: 2009-05-19T00:00:00Z
full work available at URL: http://hdl.handle.net/10316/10049
review text: The paper is concerned with quasilinear elliptic problems of the form \[ \begin{cases}-\text{div}(M(x,u)\nabla u)+g(x,u)|\nabla u|^2=f &\text{in }\Omega,\\ u=0 &\text{on }\partial\Omega. \end{cases}\tag{1} \] Here \(\Omega\subseteq\mathbb{R}^N\) is open and bounded, and \(N\geq3\). The coefficients are Caratheodory functions, and such that the principal part is uniformly elliptic with bounded coefficients. For the nonhomogeneous part \(f\) it is assumed that it lies in a suitable Lebesgue space, and that it is uniformly bounded from below by positive constants on compact subsets of \(\Omega\). The main interest lies in nonnegative functions \(g\) with a singularity in \(u=0\) that is uniform in \(x\). Suppose that \(h:(0,\infty)\to[0,\infty)\) is continuous, nonincreasing in a neighborhood of zero, and satisfies \[ \lim_{s\to0+}\int_s^1\sqrt{h(t)}\,dt<\infty. \] If \[ g(x,s)\leq h(s)\qquad\text{for a.e.\;}x\in\Omega,\;\forall s>0, \] then it is proved that (1) has a weak positive solution in \(H^1_0(\Omega)\). Conversely, a nonexistence result is given in the case that \(g\) grows faster in \(s\) than a function \(h\) whose square root is not integrable near \(0\). For the model problem \[ \begin{cases}-\Delta u+\frac{|\nabla u|^2}{u^\gamma}=f &\text{in }\Omega,\\ u=0 &\text{on }\partial\Omega. \end{cases}\tag{2} \] this amounts to the following assertion: Suppose that \(\gamma>0\). Then Eq.~(2) has a positive solution if and only if \(\gamma<2\). Existence of a solution to (1) is proved by applying classical results for quasilinear equations to a family of problems with truncated coefficients and then passing to the limit. The regularity of solutions to (1) is also considered. Moreover, the authors treat a general semilinear variant of (1).
reviewed by: Nils Ackermann
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 35D10
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 5555289
zbMATH Keywords: quasilinear elliptic equation
zbMATH Keywords: singular coefficient
zbMATH Keywords: positive solution
zbMATH Keywords: nonexistence
MaRDI profile type: Publication
OpenAlex ID: W2135620059
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DOI: 10.1016/J.JDE.2009.01.016
description: scientific article; zbMATH DE number 5555289