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Singular elliptic equations with nonlinearities of subcritical and critical growth

Part of: Partial differential equations Elliptic equations and systems Representations of solutions

Published online by Cambridge University Press:  27 July 2022

Matheus F. Stapenhorst Open the ORCID record for Matheus F. Stapenhorst [Opens in a new window]
Matheus F. Stapenhorst*
Affiliation:
Departamento de Matemática, Rua Sérgio Buarque de Holanda, Universidade Estadual de Campinas, IMECC, 651 CEP 13083-859, Campinas, SP, Brazil (mstapenhorst@ime.unicamp.br)
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Abstract

In this paper, we consider the problem $-\Delta u =-u^{-\beta }\chi _{\{u>0\}} + f(u)$ in $\Omega$ with $u=0$ on $\partial \Omega$, where $0<\beta <1$ and $\Omega$ is a smooth bounded domain in $\mathbb {R}^{N}$, $N\geq 2$. We are able to solve this problem provided $f$ has subcritical growth and satisfy certain hypothesis. We also consider this problem with $f(s)=\lambda s+s^{\frac {N+2}{N-2}}$ and $N\geq 3$. In this case, we are able to obtain a solution for large values of $\lambda$. We replace the singular function $u^{-\beta }$ by a function $g_\epsilon (u)$ which pointwisely converges to $u^{-\beta }$ as $\epsilon \rightarrow 0$. The corresponding energy functional to the perturbed equation $-\Delta u + g_\epsilon (u) = f(u)$ has a critical point $u_\epsilon$ in $H_0^{1}(\Omega )$, which converges to a non-trivial non-negative solution of the original problem as $\epsilon \rightarrow 0$.

Keywords

singular problemvariational methodsa priori estimatescritical growth

MSC classification

Primary: 35A15: Variational methods
Secondary: 35B20: Perturbations 35B33: Critical exponents 35D30: Weak solutions 35J20: Variational methods for second-order elliptic equations 35J75: Singular elliptic equations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 3 , August 2022 , pp. 652 - 690
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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