A multiplicity result for double phase problem in the whole space
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 $...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2022-07-01
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| Series: | AIMS Mathematics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2022963?viewType=HTML |
| Summary: | 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. |
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| ISSN: | 2473-6988 |