Multiple solutions to Kirchhoff-Schrödinger equations involving the $ p(\cdot) $-Laplace-type operator

Yun-Ho Kim; Yun-Ho Kim

doi:10.3934/math.2023477

AIMS Mathematics

2023, Volume 8, Issue 4: 9461-9482. doi: 10.3934/math.2023477

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Multiple solutions to Kirchhoff-Schrödinger equations involving the $ p(\cdot) $-Laplace-type operator

Yun-Ho Kim ^,

Department of Mathematics Education, Sangmyung University, Seoul 03016, Republic of Korea

Received: 30 November 2022 Revised: 03 February 2023 Accepted: 06 February 2023 Published: 17 February 2023
MSC : 35D30, 35J20, 35J60, 35J92, 47J30

This paper is devoted to deriving several multiplicity results of nontrivial weak solutions to Kirchhoff-Schrödinger equations involving the $ p(\cdot) $-Laplace-type operator. The aims of this paper are stated as follows. First, under some conditions on a nonlinear term, we show that our problem has a sequence of infinitely many large energy solutions. Second, we obtain the existence of a sequence of infinitely many small energy solutions to the problem on a new class of nonlinear term. The primary tools to obtain such multiplicity results are the fountain theorem and the dual fountain theorem, respectively.
- $ p(x) $-Laplace type,
- variable exponent Lebesgue-Sobolev spaces,
- weak solution,
- fountain theorem,
- dual fountain theorem
Citation: Yun-Ho Kim. Multiple solutions to Kirchhoff-Schrödinger equations involving the $ p(\cdot) $-Laplace-type operator[J]. AIMS Mathematics, 2023, 8(4): 9461-9482. doi: 10.3934/math.2023477

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Abstract

This paper is devoted to deriving several multiplicity results of nontrivial weak solutions to Kirchhoff-Schrödinger equations involving the $ p(\cdot) $-Laplace-type operator. The aims of this paper are stated as follows. First, under some conditions on a nonlinear term, we show that our problem has a sequence of infinitely many large energy solutions. Second, we obtain the existence of a sequence of infinitely many small energy solutions to the problem on a new class of nonlinear term. The primary tools to obtain such multiplicity results are the fountain theorem and the dual fountain theorem, respectively.

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