Existence and multiplicity of solutions for Schrödinger equations with sublinear nonlinearities

Ye Xue; Zhiqing Han; Ye Xue; Zhiqing Han

doi:10.3934/math.2021324

AIMS Mathematics

2021, Volume 6, Issue 6: 5479-5492. doi: 10.3934/math.2021324

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Research article

Existence and multiplicity of solutions for Schrödinger equations with sublinear nonlinearities

Ye Xue ,
Zhiqing Han ^,

School of Mathematical Sciences, Dalian University of Technology, 116024 Dalian, China

Received: 13 January 2021 Accepted: 09 March 2021 Published: 17 March 2021
MSC : 35A15, 35J60, 58E05

In the paper, we investigate a class of Schrödinger equations with sign-changing potentials $ V(x) $ and sublinear nonlinearities. We remove the coercive condition on $ V(x) $ usually required in the existing literature and also weaken the conditions on nonlinearities. By proving a Hardy-type inequality, extending the results in ^[1], and using it together with variational methods, we get at least one or infinitely many small energy solutions for the problem.
- Schrödinger equation,
- sublinear nonlinearity,
- Hardy-type inequality,
- variant fountain theorem,
- infinitely many solutions
Citation: Ye Xue, Zhiqing Han. Existence and multiplicity of solutions for Schrödinger equations with sublinear nonlinearities[J]. AIMS Mathematics, 2021, 6(6): 5479-5492. doi: 10.3934/math.2021324

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Abstract

In the paper, we investigate a class of Schrödinger equations with sign-changing potentials $ V(x) $ and sublinear nonlinearities. We remove the coercive condition on $ V(x) $ usually required in the existing literature and also weaken the conditions on nonlinearities. By proving a Hardy-type inequality, extending the results in ^[1], and using it together with variational methods, we get at least one or infinitely many small energy solutions for the problem.

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