Global multiplicity of solutions for a singular $ p $-Laplacian quasilinear Schrödinger equation

Siyu Chen; Yu Zheng; Jiazheng Zhou; Siyu Chen; Yu Zheng; Jiazheng Zhou

doi:10.3934/era.2026066

Electronic Research Archive

2026, Volume 34, Issue 3: 1448-1476. doi: 10.3934/era.2026066

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Global multiplicity of solutions for a singular $ p $-Laplacian quasilinear Schrödinger equation

1.
College of Data Science, Jiaxing University, Zhejiang 314001, China
2.
School of Mathematics and Statistics, Huizhou University, Guangdong 516007, China
3.
Departamento de Matemática, Universidade de Brasília, 70910-900 Brasília, Brazil

Received: 05 January 2026 Revised: 30 January 2026 Accepted: 05 February 2026 Published: 12 February 2026

We consider a class of $ p $-Laplace quasilinear Schrödinger Equations
$ \left \{ \begin{array}{c} -\Delta_p u-\frac{p}{2^{p-1}}u\Delta_p (u^2) = \lambda u^{-\gamma}+u^q\; \mbox{in}\; \Omega, \\ u > 0 \; \; \mbox{in} \; \; \Omega, \; \; \; u = 0 \; \; \mbox{on} \; \; \partial \Omega , \end{array}\right. $
where $ \Omega\subset\mathbb{R}^N $ is a bounded domain with regular boundary, $ 1 < p < \infty $, $ 0 < \gamma < 1 $, $ 2p - 1 < q\leq2\cdot p^*-1 $ for $ p\leq N $, $ 2p-1 < q < \infty $ for $ p > N $, where $ p^* = \frac{Np}{N-p} $ if $ 1 < p < N $, $ p^*\in(p, \infty) $ is arbitrarily large if $ p = N $, and $ p^* = \infty $ if $ p > N $. We establish global existence and multiplicity of positive solutions via a new strong comparison principle and a regularity result for weak solutions.
- global multiplicity,
- quasilinear Schrödinger equations,
- singular term
Citation: Siyu Chen, Yu Zheng, Jiazheng Zhou. Global multiplicity of solutions for a singular $ p $-Laplacian quasilinear Schrödinger equation[J]. Electronic Research Archive, 2026, 34(3): 1448-1476. doi: 10.3934/era.2026066

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Abstract

We consider a class of $ p $-Laplace quasilinear Schrödinger Equations

$ \left \{ \begin{array}{c} -\Delta_p u-\frac{p}{2^{p-1}}u\Delta_p (u^2) = \lambda u^{-\gamma}+u^q\; \mbox{in}\; \Omega, \\ u > 0 \; \; \mbox{in} \; \; \Omega, \; \; \; u = 0 \; \; \mbox{on} \; \; \partial \Omega , \end{array}\right. $

where $ \Omega\subset\mathbb{R}^N $ is a bounded domain with regular boundary, $ 1 < p < \infty $, $ 0 < \gamma < 1 $, $ 2p - 1 < q\leq2\cdot p^*-1 $ for $ p\leq N $, $ 2p-1 < q < \infty $ for $ p > N $, where $ p^* = \frac{Np}{N-p} $ if $ 1 < p < N $, $ p^*\in(p, \infty) $ is arbitrarily large if $ p = N $, and $ p^* = \infty $ if $ p > N $. We establish global existence and multiplicity of positive solutions via a new strong comparison principle and a regularity result for weak solutions.

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