The existence results for a class of generalized quasilinear Schrödinger equation with nonlocal term

Die Hu; Peng Jin; Xianhua Tang; Die Hu; Peng Jin; Xianhua Tang

doi:10.3934/era.2022100

Electronic Research Archive

2022, Volume 30, Issue 5: 1973-1998. doi: 10.3934/era.2022100

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The existence results for a class of generalized quasilinear Schrödinger equation with nonlocal term

School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083, China

Received: 06 November 2021 Revised: 15 March 2022 Accepted: 21 March 2022 Published: 12 April 2022

In this paper, we discuss the generalized quasilinear Schrödinger equation with nonlocal term:

$ \begin{align} -\mathrm{div}(g^{2}(u)\nabla u)+g(u)g'(u)|\nabla u|^{2}+V(x)u = \left(|x|^{-\mu}\ast F(u)\right)f( u),\; \; x\in \mathbb{R}^{N}, \;\;\;\;\;\;\;\;({{\rm{P}}})\end{align} $

where $ N\geq 3 $, $ \mu\in(0, N) $, $ g\in \mathbb{C}^{1}(\mathbb{R}, \mathbb{R}^{+}) $, $ V\in \mathbb{C}^{1}(\mathbb{R}^N, \mathbb{R}) $ and $ f\in \mathbb{C}(\mathbb{R}, \mathbb{R}) $. Under some "Berestycki-Lions type conditions" on the nonlinearity $ f $ which are almost necessary, we prove that problem $ (\rm P) $ has a nontrivial solution $ \bar{u}\in H^{1}(\mathbb{R}^{N}) $ such that $ \bar{v} = G(\bar{u}) $ is a ground state solution of the following problem

$ \begin{align} - \Delta v+V(x)\frac{G^{-1}(v)}{g(G^{-1}(v))} = \left(|x|^{-\mu}\ast F(G^{-1}(v))\right)f( G^{-1}(v)),\; \; x\in \mathbb{R}^{N}, \;\;\;\;\;\;\;\;({{\rm{\bar P}}})\end{align} $

where $ G(t): = \int_{0}^{t} g(s) ds $. We also give a minimax characterization for the ground state solution $ \bar{v} $.
- quasilinear Schrödinger equation,
- nonlocal term,
- ground state solution,
- Berestycki-Lions conditions
Citation: Die Hu, Peng Jin, Xianhua Tang. The existence results for a class of generalized quasilinear Schrödinger equation with nonlocal term[J]. Electronic Research Archive, 2022, 30(5): 1973-1998. doi: 10.3934/era.2022100

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Abstract

In this paper, we discuss the generalized quasilinear Schrödinger equation with nonlocal term:

$ \begin{align} -\mathrm{div}(g^{2}(u)\nabla u)+g(u)g'(u)|\nabla u|^{2}+V(x)u = \left(|x|^{-\mu}\ast F(u)\right)f( u),\; \; x\in \mathbb{R}^{N}, \;\;\;\;\;\;\;\;({{\rm{P}}})\end{align} $

where $ N\geq 3 $, $ \mu\in(0, N) $, $ g\in \mathbb{C}^{1}(\mathbb{R}, \mathbb{R}^{+}) $, $ V\in \mathbb{C}^{1}(\mathbb{R}^N, \mathbb{R}) $ and $ f\in \mathbb{C}(\mathbb{R}, \mathbb{R}) $. Under some "Berestycki-Lions type conditions" on the nonlinearity $ f $ which are almost necessary, we prove that problem $ (\rm P) $ has a nontrivial solution $ \bar{u}\in H^{1}(\mathbb{R}^{N}) $ such that $ \bar{v} = G(\bar{u}) $ is a ground state solution of the following problem

$ \begin{align} - \Delta v+V(x)\frac{G^{-1}(v)}{g(G^{-1}(v))} = \left(|x|^{-\mu}\ast F(G^{-1}(v))\right)f( G^{-1}(v)),\; \; x\in \mathbb{R}^{N}, \;\;\;\;\;\;\;\;({{\rm{\bar P}}})\end{align} $

where $ G(t): = \int_{0}^{t} g(s) ds $. We also give a minimax characterization for the ground state solution $ \bar{v} $.

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