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- Title
Existence and multiplicity of solutions for Schrödinger–Kirchhoff type problems involving the fractional p(⋅)-Laplacian in RN.
- Authors
Kim, In Hyoun; Kim, Yun-Ho; Park, Kisoeb
- Abstract
We are concerned with the following elliptic equations with variable exponents: M ([ u ] s , p (⋅ , ⋅)) L u (x) + V (x) | u | p (x) − 2 u = λ ρ (x) | u | r (x) − 2 u + h (x , u) in R N , where [ u ] s , p (⋅ , ⋅) : = ∫ R N ∫ R N | u (x) − u (y) | p (x , y) p (x , y) | x − y | N + s p (x , y) d x d y , the operator L is the fractional p (⋅) -Laplacian, p , r : R N → (1 , ∞) are continuous functions, M ∈ C (R +) is a Kirchhoff-type function, the potential function V : R N → (0 , ∞) is continuous, and h : R N × R → R satisfies a Carathéodory condition. Under suitable assumptions on h, the purpose of this paper is to show the existence of at least two non-trivial distinct solutions for the problem above for the case of a combined effect of concave–convex nonlinearities. To do this, we use the mountain pass theorem and variant of the Ekeland variational principle as the main tools.
- Subjects
MOUNTAIN pass theorem; LAPLACIAN operator; VARIATIONAL principles; ELLIPTIC equations; CONTINUOUS functions; MULTIPLICITY (Mathematics)
- Publication
Boundary Value Problems, 2020, Vol 2020, Issue 1, p1
- ISSN
1687-2762
- Publication type
Article
- DOI
10.1186/s13661-020-01419-z