New multiple positive solutions for elliptic equations with singularity and critical growth

In this note, the existence of multiple positive solutions is established for a semilinear elliptic equation $ -\Delta u=\frac{\lambda}{u^\gamma}+u^{2^*-1},~x\in\Omega,~u=0, x\in\partial\Omega$, where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ ($N\geq3$), $2^*=\frac{2N}{N-2}$, $\gamma\in(...

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Published in:Electronic journal of qualitative theory of differential equations 2019-01, Vol.2019 (20), p.1-14
Main Authors: Suo, Hong-Min, Lei, Chun-Yu, Liao, Jiafeng
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Language:English
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critical growth
positive solution
semilinear elliptic equations
singularity
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description In this note, the existence of multiple positive solutions is established for a semilinear elliptic equation $ -\Delta u=\frac{\lambda}{u^\gamma}+u^{2^*-1},~x\in\Omega,~u=0, x\in\partial\Omega$, where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ ($N\geq3$), $2^*=\frac{2N}{N-2}$, $\gamma\in(0,1)$ and $\lambda>0$ is a real parameter. We show by the variational methods and perturbation functional that the problem has at least two positive solutions $w_0(x)$ and $w_1(x)$ with $w_0(x)
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subjects critical growth
positive solution
semilinear elliptic equations
singularity
title New multiple positive solutions for elliptic equations with singularity and critical growth
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