Solutions for Schrödinger equations with variable separated type nonlinear terms

Xia Su; Chunhua Deng; Xia Su; Chunhua Deng

doi:10.3934/math.20231557

AIMS Mathematics

2023, Volume 8, Issue 12: 30487-30500. doi: 10.3934/math.20231557

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Solutions for Schrödinger equations with variable separated type nonlinear terms

Xia Su ^,,
Chunhua Deng

Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Jiangsu 223003, China

Received: 17 July 2023 Revised: 09 October 2023 Accepted: 30 October 2023 Published: 09 November 2023
MSC : 35Q55, 47J30

In this paper, we consider the following semilinear Schrödinger equation:

$ \begin{eqnarray*} \left\{ \begin{array}{ll} -\Delta u+V(x)u = a(x)g(u)&{\mbox{for}}\; x\in \mathbb{R}^{N} ,\\ u(x)\rightarrow0&{\mbox{as}}\; |x|\rightarrow \infty , \end{array} \right. \end{eqnarray*} $

where $ a(x) > 0 $ for all $ \mathbb{R}^{N} $. Under some different superlinear conditions on $ g(u) $, we obtain the existence of solutions for the above problem. In order to regain the compactness of the Sobolev embedding, a competing condition between $ a(x) $ and $ V(x) $ is introduced.
- Schrödinger equations,
- compact embedding theorem,
- variable separated nonlinearities,
- superlinear
Citation: Xia Su, Chunhua Deng. Solutions for Schrödinger equations with variable separated type nonlinear terms[J]. AIMS Mathematics, 2023, 8(12): 30487-30500. doi: 10.3934/math.20231557

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Abstract

In this paper, we consider the following semilinear Schrödinger equation:

$ \begin{eqnarray*} \left\{ \begin{array}{ll} -\Delta u+V(x)u = a(x)g(u)&{\mbox{for}}\; x\in \mathbb{R}^{N} ,\\ u(x)\rightarrow0&{\mbox{as}}\; |x|\rightarrow \infty , \end{array} \right. \end{eqnarray*} $

where $ a(x) > 0 $ for all $ \mathbb{R}^{N} $. Under some different superlinear conditions on $ g(u) $, we obtain the existence of solutions for the above problem. In order to regain the compactness of the Sobolev embedding, a competing condition between $ a(x) $ and $ V(x) $ is introduced.

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