Multiple solutions to the double phase problems involving concave-convex nonlinearities

Jae-Myoung Kim; Yun-Ho Kim; Jae-Myoung Kim; Yun-Ho Kim

doi:10.3934/math.2023254

AIMS Mathematics

2023, Volume 8, Issue 3: 5060-5079. doi: 10.3934/math.2023254

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Multiple solutions to the double phase problems involving concave-convex nonlinearities

Jae-Myoung Kim ¹,
Yun-Ho Kim ^{2
,
,}

1.
Department of Mathematics Education, Andong National University, Andong 36729, Korea
2.
Department of Mathematics Education, Sangmyung University, Seoul 03016, Korea

Received: 20 October 2022 Revised: 27 November 2022 Accepted: 28 November 2022 Published: 12 December 2022
MSC : 35B38, 35D30, 35J10, 35J20, 35J62

This paper is concerned with several existence results of multiple solutions for Schrödinger-type problems involving the double phase operator for the case of a combined effect of concave-convex nonlinearities. The first one is to discuss that our problem has infinitely many large energy solutions. Second, we obtain the existence of a sequence of infinitely many small energy solutions to the given problem. To establish such multiplicity results, we employ the fountain theorem and the dual fountain theorem as the primary tools, respectively. In particular we give the existence result of small energy solutions on a new class of nonlinear term.
- double phase problems,
- Musielak-Orlicz-Sobolev spaces,
- variational methods,
- multiple solutions
Citation: Jae-Myoung Kim, Yun-Ho Kim. Multiple solutions to the double phase problems involving concave-convex nonlinearities[J]. AIMS Mathematics, 2023, 8(3): 5060-5079. doi: 10.3934/math.2023254

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Abstract

This paper is concerned with several existence results of multiple solutions for Schrödinger-type problems involving the double phase operator for the case of a combined effect of concave-convex nonlinearities. The first one is to discuss that our problem has infinitely many large energy solutions. Second, we obtain the existence of a sequence of infinitely many small energy solutions to the given problem. To establish such multiplicity results, we employ the fountain theorem and the dual fountain theorem as the primary tools, respectively. In particular we give the existence result of small energy solutions on a new class of nonlinear term.

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