Existence of a positive radial solution for semilinear elliptic problem involving variable exponent

Changmu Chu; Shan Li; Hongmin Suo; Changmu Chu; Shan Li; Hongmin Suo

doi:10.3934/era.2023125

Electronic Research Archive

2023, Volume 31, Issue 5: 2472-2482. doi: 10.3934/era.2023125

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Research article

Existence of a positive radial solution for semilinear elliptic problem involving variable exponent

School of preparatory education, Guizhou Minzu University, Guizhou 550025, China

Received: 14 December 2022 Revised: 13 February 2023 Accepted: 15 February 2023 Published: 03 March 2023

This paper consider that the following semilinear elliptic equation

$ \begin{equation} \left\{ \begin{array}{ll} -\Delta u = u^{q(x)-1}, &\ \ {\mbox{in}}\ \ B_1,\\ u>0, &\ \ {\mbox{in}}\ \ B_1,\\ u = 0, &\ \ {\mbox{in}}\ \ \partial B_1, \end{array} \right. \end{equation} $

where $ B_1 $ is the unit ball in $ \mathbb{R}^N(N\geq 3) $, $ q(x) = q(|x|) $ is a continuous radial function satifying $ 2\leq q(x) < 2^* = \frac{2N}{N-2} $ and $ q(0) > 2 $. Using variational methods and a priori estimate, the existence of a positive radial solution for (0.1) is obtained.
- semilinear elliptic problem,
- variable exponent,
- mountain pass lamma,
- a priori estimate,
- positive radial solution
Citation: Changmu Chu, Shan Li, Hongmin Suo. Existence of a positive radial solution for semilinear elliptic problem involving variable exponent[J]. Electronic Research Archive, 2023, 31(5): 2472-2482. doi: 10.3934/era.2023125

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Abstract

This paper consider that the following semilinear elliptic equation

$ \begin{equation} \left\{ \begin{array}{ll} -\Delta u = u^{q(x)-1}, &\ \ {\mbox{in}}\ \ B_1,\\ u>0, &\ \ {\mbox{in}}\ \ B_1,\\ u = 0, &\ \ {\mbox{in}}\ \ \partial B_1, \end{array} \right. \end{equation} $

where $ B_1 $ is the unit ball in $ \mathbb{R}^N(N\geq 3) $, $ q(x) = q(|x|) $ is a continuous radial function satifying $ 2\leq q(x) < 2^* = \frac{2N}{N-2} $ and $ q(0) > 2 $. Using variational methods and a priori estimate, the existence of a positive radial solution for (0.1) is obtained.

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