Multiplicity of positive solutions for the fractional Schrödinger-Poisson system with critical nonlocal term

Abstract

This paper deals with the following fractional Schrodinger-Poisson system: (-& UDelta;)su + u - K(x)f|u|2s*-3u = f ?(x)|u|q-2u,x & ISIN; Double-struck capital R3,(-& UDelta;)sf = K(x)|u|2s*-1,x & ISIN; Double-struck capital R3 with multiple competing potentials and a critical nonlocal term, where s & ISIN; (0, 1), q & ISIN; (1, 2) or q & ISIN; (4, 2s*), and 2s* = 6 3-2s is the fractional critical exponent. By combining the Nehari manifold analysis and the Ljusternik-Schnirelmann category theory, we establish how the coefficient K of the nonlocal critical nonlinearity affects the number of positive solutions. We propose a new relation between the number of positive solutions and the category of the global maximal set of K.This paper deals with the following fractional Schrodinger-Poisson system: (-& UDelta;)su + u - K(x)f|u|2s*-3u = f ?(x)|u|q-2u,x & ISIN; Double-struck capital R3,(-& UDelta;)sf = K(x)|u|2s*-1,x & ISIN; Double-struck capital R3 with multiple competing potentials and a critical nonlocal term, where s & ISIN; (0, 1), q & ISIN; (1, 2) or q & ISIN; (4, 2s*), and 2s* = 6 3-2s is the fractional critical exponent. By combining the Nehari manifold analysis and the Ljusternik-Schnirelmann category theory, we establish how the coefficient K of the nonlocal critical nonlinearity affects the number of positive solutions. We propose a new relation between the number of positive solutions and the category of the global maximal set of K.