Symmetry of large solutions for semilinear elliptic equations in a symmetric convex domain

Keqiang Li; Shangjiu Wang; Shaoyong Li; Keqiang Li; Shangjiu Wang; Shaoyong Li

doi:10.3934/math.2022607

AIMS Mathematics

2022, Volume 7, Issue 6: 10860-10866. doi: 10.3934/math.2022607

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Research article

Symmetry of large solutions for semilinear elliptic equations in a symmetric convex domain

1.
College of Digital Technology and Engineering, Ningbo University of Finance and Economics, Ningbo 315175, China
2.
School of Mathematics and Statistics, Shaoguan University, Shaoguan 512005, China
3.
School of Economics and Statistics, Guangzhou University, Guangzhou 510006, China

Received: 29 July 2021 Revised: 26 September 2021 Accepted: 08 October 2021 Published: 01 April 2022
MSC : 35A21, 35B06

In this paper, we consider the solutions of the boundary blow-up problem

$ \begin{eqnarray*} \begin{cases} \Delta u = \frac{1}{u^\gamma} +f(u) \ \ \ \ \mathrm{in}\ \ \ \Omega,\\ \ u>0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{in}\ \ \ \Omega, \\ \ u = +\infty \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{on} \ \ \partial\Omega, \end{cases} \end{eqnarray*} $

where $ \gamma > 0, \ \Omega $ is a bounded convex smooth domain and symmetric w.r.t. a direction. $ f $ is a locally Lipschitz continuous and non-decreasing function. We prove symmetry and monotonicity of solutions of the problem above by the moving planes method. A maximum principle in narrow domains plays an important role in proof of the main result.
- semilinear elliptic systems,
- moving plane method,
- maximum principle,
- symmetry of large solutions
Citation: Keqiang Li, Shangjiu Wang, Shaoyong Li. Symmetry of large solutions for semilinear elliptic equations in a symmetric convex domain[J]. AIMS Mathematics, 2022, 7(6): 10860-10866. doi: 10.3934/math.2022607

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Abstract

In this paper, we consider the solutions of the boundary blow-up problem

$ \begin{eqnarray*} \begin{cases} \Delta u = \frac{1}{u^\gamma} +f(u) \ \ \ \ \mathrm{in}\ \ \ \Omega,\\ \ u>0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{in}\ \ \ \Omega, \\ \ u = +\infty \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{on} \ \ \partial\Omega, \end{cases} \end{eqnarray*} $

where $ \gamma > 0, \ \Omega $ is a bounded convex smooth domain and symmetric w.r.t. a direction. $ f $ is a locally Lipschitz continuous and non-decreasing function. We prove symmetry and monotonicity of solutions of the problem above by the moving planes method. A maximum principle in narrow domains plays an important role in proof of the main result.

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