### AIMS Mathematics

2018, Issue 1: 233-252. doi: 10.3934/Math.2018.1.233
Research article

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

• Received: 20 March 2018 Accepted: 28 March 2018 Published: 04 April 2018
• Consider the problem $-\Delta u = a\left( x\right) u.{-\alpha}+f\left( \lambda, x, u\right)$ in $\Omega,$ $u = 0$ on $\partial\Omega,$ $u \gt 0$ in $\Omega,$ where $\Omega$ is a bounded domain in $\mathbb{R}.{n}$ with $C.{2}$ boundary, $0\leq a\in L.{\infty}\left( \Omega\right),$ $0 \lt \alpha \lt 3,$ 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 $\Lambda \gt 0,$ 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 $\lambda \gt \Lambda.$ 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 $\lambda.$

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

### Related Papers:

• Consider the problem $-\Delta u = a\left( x\right) u.{-\alpha}+f\left( \lambda, x, u\right)$ in $\Omega,$ $u = 0$ on $\partial\Omega,$ $u \gt 0$ in $\Omega,$ where $\Omega$ is a bounded domain in $\mathbb{R}.{n}$ with $C.{2}$ boundary, $0\leq a\in L.{\infty}\left( \Omega\right),$ $0 \lt \alpha \lt 3,$ 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 $\Lambda \gt 0,$ 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 $\lambda \gt \Lambda.$ 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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