MULTIPLICITY OF SOLUTIONS FOR DOUBLE PHASE EQUATIONS WITH CONCAVE-CONVEX NONLINEARITIES

Woo Jin Joe; Seong Jin Kim; Yun-Ho Kim; Min Wook Oh

doi:10.11948/20210063

2021 Volume 11 Issue 6

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Woo Jin Joe, Seong Jin Kim, Yun-Ho Kim, Min Wook Oh. MULTIPLICITY OF SOLUTIONS FOR DOUBLE PHASE EQUATIONS WITH CONCAVE-CONVEX NONLINEARITIES[J]. Journal of Applied Analysis & Computation, 2021, 11(6): 2921-2946. doi: 10.11948/20210063

Citation:

Woo Jin Joe, Seong Jin Kim, Yun-Ho Kim, Min Wook Oh. MULTIPLICITY OF SOLUTIONS FOR DOUBLE PHASE EQUATIONS WITH CONCAVE-CONVEX NONLINEARITIES[J]. Journal of Applied Analysis & Computation, 2021, 11(6): 2921-2946. doi: 10.11948/20210063

MULTIPLICITY OF SOLUTIONS FOR DOUBLE PHASE EQUATIONS WITH CONCAVE-CONVEX NONLINEARITIES

1.
Department of Mathematics Education, Sangmyung University, Seoul 03016, Republic of Korea
2.
Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan 44919, Republic of Korea
3.
Department of Mathematics, Korea University, Seoul 02841, Republic of Korea

Corresponding author: Email: kyh1213@smu.ac.kr(Y.-H. Kim)

Abstract

Abstract

This paper is devoted to the study of the L^∞-bound of solutions to a double-phase problem with concave-convex nonlinearities by applying the De Giorgi's iteration method and the localization method. Employing this and a variant of Ekeland's variational principle, we provide the existence of at least two distinct nontrivial solutions belonging to L^∞-space when the convex term does not satisfy the Ambrosetti-Rabinowitz condition in general. In addition, our problem has a sequence of multiple small energy solutions whose L^∞-norms converge to zero. To achieve this result, we utilize the modified functional method and the dual fountain theorem as the main tools.
- Double phase equations /
- De Giorgi iteration /
- modified functional methods /
- dual fountain theorem
MSC: 35B45, 35J20, 35J62, 46E30

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References

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