Multiple solutions for quasi-linear elliptic equations with Berestycki-Lions type nonlinearity

Maomao Wu; Haidong Liu; Maomao Wu; Haidong Liu

doi:10.3934/cam.2024016

Communications in Analysis and Mechanics

2024, Volume 16, Issue 2: 334-344. doi: 10.3934/cam.2024016

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Multiple solutions for quasi-linear elliptic equations with Berestycki-Lions type nonlinearity

Maomao Wu ^1,2,
Haidong Liu ^{2
,
,}

1.
School of Mathematical Sciences, Zhejiang Normal University, Zhejiang 321004, P.R. China
2.
Institute of Mathematics, Jiaxing University, Zhejiang 314001, P.R. China

Received: 06 December 2023 Revised: 24 January 2024 Accepted: 26 March 2024 Published: 06 May 2024
35J20, 35J62

We studied the modified nonlinear Schrödinger equation

$ \begin{equation} -\Delta u-\frac12\Delta(u^2)u = g(u)+h(x), \quad u\in H^1({\mathbb{R}}^N), \end{equation} $

where $ N\geq3 $, $ g\in C({\mathbb{R}}, {\mathbb{R}}) $ is a nonlinear function of Berestycki-Lions type, and $ h\not\equiv 0 $ is a nonnegative function. When $ \|h\|_{L^2({\mathbb{R}}^N)} $ is suitably small, we proved that (0.1) possesses at least two positive solutions by variational approach, one of which is a ground state while the other is of mountain pass type.
- nonhomogeneous quasi-linear elliptic equation,
- ground state solution,
- mountain pass type solution,
- variational methods
Citation: Maomao Wu, Haidong Liu. Multiple solutions for quasi-linear elliptic equations with Berestycki-Lions type nonlinearity[J]. Communications in Analysis and Mechanics, 2024, 16(2): 334-344. doi: 10.3934/cam.2024016

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Abstract

We studied the modified nonlinear Schrödinger equation

$ \begin{equation} -\Delta u-\frac12\Delta(u^2)u = g(u)+h(x), \quad u\in H^1({\mathbb{R}}^N), \end{equation} $

where $ N\geq3 $, $ g\in C({\mathbb{R}}, {\mathbb{R}}) $ is a nonlinear function of Berestycki-Lions type, and $ h\not\equiv 0 $ is a nonnegative function. When $ \|h\|_{L^2({\mathbb{R}}^N)} $ is suitably small, we proved that (0.1) possesses at least two positive solutions by variational approach, one of which is a ground state while the other is of mountain pass type.

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