A multiplicity result for double phase problem in the whole space

Yanfeng Li; Haicheng Liu; Yanfeng Li; Haicheng Liu

doi:10.3934/math.2022963

AIMS Mathematics

2022, Volume 7, Issue 9: 17475-17485. doi: 10.3934/math.2022963

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Research article

A multiplicity result for double phase problem in the whole space

Yanfeng Li ¹,
Haicheng Liu ^{2
,
,}

1.
College of Science, Heilongjiang Bayi Agricultural University, Daqing, 163319, China
2.
College of Mathematical Sciences, Harbin Engineering University, Harbin, 150001, China

Received: 21 May 2022 Revised: 20 July 2022 Accepted: 25 July 2022 Published: 28 July 2022
MSC : 35D30, 35J20, 35J60

In the present paper, we discuss the solutions of the following double phase problem

$ -{\rm div}(|\nabla u|^{^{p-2}}\nabla u+ \mu(x) |\nabla u|^{^{q-2}}\nabla u)+ |u|^{^{p-2}}u+\mu(x)|u|^{^{q-2}}u = f(x, u), \;x\in \mathbb{R}^N, $

where $ N \geq2 $, $ 1 < p < q < N $ and $ 0\leq\mu\in C^{^{0, \alpha}}(\mathbb{R}^N), \; \alpha\in(0, 1] $. Based on the theory of the double phase Sobolev spaces $ W^{^{1, H}}(\mathbb{R}^N) $, we prove the existence of at least two non-trivial weak solutions.
- double phase operator,
- Musielak-Orlicz-Sobolev space,
- critical point
Citation: Yanfeng Li, Haicheng Liu. A multiplicity result for double phase problem in the whole space[J]. AIMS Mathematics, 2022, 7(9): 17475-17485. doi: 10.3934/math.2022963

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Abstract

In the present paper, we discuss the solutions of the following double phase problem

$ -{\rm div}(|\nabla u|^{^{p-2}}\nabla u+ \mu(x) |\nabla u|^{^{q-2}}\nabla u)+ |u|^{^{p-2}}u+\mu(x)|u|^{^{q-2}}u = f(x, u), \;x\in \mathbb{R}^N, $

where $ N \geq2 $, $ 1 < p < q < N $ and $ 0\leq\mu\in C^{^{0, \alpha}}(\mathbb{R}^N), \; \alpha\in(0, 1] $. Based on the theory of the double phase Sobolev spaces $ W^{^{1, H}}(\mathbb{R}^N) $, we prove the existence of at least two non-trivial weak solutions.

References

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