AIMS Mathematics, 2018, 3(1): 233-252. doi: 10.3934/Math.2018.1.233.

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Multiple finite-energy positive weak solutions to singular elliptic problems with a parameter

Facultad de Matematica, Astronomia y Fisica, Universidad Nacional de Cordoba, Ciudad Universitaria, 5000 Cordoba, Argentina

Consider the problem $-\Delta u=a\left( x\right) u^{-\alpha}+f\left(\lambda,x,u\right) $ in $\Omega,$ $u=0$ on $\partial\Omega,$ $u>0$ in$\Omega,$ \ where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with $C^{2}$ boundary, $0\leqa\in L^{\infty}\left( \Omega\right) ,$ $0<\alpha<3 and="" f="" left="" lambda="" x="" right="" is="" nonnegative="" and="" superlinear="" with="" subcritical="" growth="" at="" infty="" we="" prove="" that="" if="" f="" satisfies="" some="" additional="" conditions="" then="" for="" some="" lambda="">0 ,$ there are at least two weak solutions in$H_{0}^{1}\left( \Omega\right) \cap C\left( \overline{\Omega}\right) $ if$\lambda\in\left( 0,\Lambda\right)$, and there is no weak solution in$H_{0}^{1}\left( \Omega\right) \cap L^{\infty}\left( \Omega\right) $ if$\lambda>\Lambda.$ We also prove that, for each $\lambda\in\left[0,\Lambda\right] $, there exists a unique minimal weak solution $u_{\lambda}$ in $H_{0}^{1}\left( \Omega\right) \cap L^{\infty}\left( \Omega\right) $, which is strictly increasing in $\lambda.$
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Keywords singular elliptic problems; positive solutions; bifurcation problems; sub and supersolutions; fixed points; multiplicity theorems

Citation: Tomas Godoy, Alfredo Guerin. Multiple finite-energy positive weak solutions to singular elliptic problems with a parameter. AIMS Mathematics, 2018, 3(1): 233-252. doi: 10.3934/Math.2018.1.233


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This article has been cited by

  • 1. Tomás Godoy, Alfredo Guerin, Positive weak solutions of elliptic Dirichlet problems with singularities in both the dependent and the independent variables, Electronic Journal of Qualitative Theory of Differential Equations, 2019, 54, 1, 10.14232/ejqtde.2019.1.54
  • 2. T. Godoy, A. Guerin, Elliptic bifurcation problems that are singular in the dependent and in the independent variables, Complex Variables and Elliptic Equations, 2019, 1, 10.1080/17476933.2019.1664490

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