Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p-Laplacian in RN

In this paper we investigate the existence of multiple solutions for the nonhomogeneous fractional p -Laplacian equations of Schrödinger–Kirchhoff type M ∫ ∫ R 2 N | u ( x ) - u ( y ) | p | x - y | N + p s d x d y ( - Δ ) p s u + V ( x ) | u | p - 2 u = f ( x , u ) + g ( x ) in R N , where ( - Δ ) p...

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Bibliographic Details
Published in:Calculus of variations and partial differential equations Vol. 54; no. 3; pp. 2785 - 2806
Main Authors: Pucci, Patrizia, Xiang, Mingqi, Zhang, Binlin
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2015
Subjects:
Analysis
Calculus of Variations and Optimal Control; Optimization
Control
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Systems Theory
Theoretical
35R11
35A15
47G20
35J60
ISSN:0944-2669, 1432-0835
Online Access:Get full text
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Summary:In this paper we investigate the existence of multiple solutions for the nonhomogeneous fractional p -Laplacian equations of Schrödinger–Kirchhoff type M ∫ ∫ R 2 N | u ( x ) - u ( y ) | p | x - y | N + p s d x d y ( - Δ ) p s u + V ( x ) | u | p - 2 u = f ( x , u ) + g ( x ) in R N , where ( - Δ ) p s is the fractional p -Laplacian operator, with 0 < s < 1 < p < ∞ and p s < N , the nonlinearity f : R N × R → R is a Carathéodory function and satisfies the Ambrosetti–Rabinowitz condition, V : R N → R + is a potential function and g : R N → R is a perturbation term. We first establish Batsch–Wang type compact embedding theorem for the fractional Sobolev spaces. Then multiplicity results are obtained by using the Ekeland variational principle and the Mountain Pass theorem.
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-015-0883-5