Existence of positive solutions for a Schrödinger-Poisson system with critical growth. Issue 11 (17th August 2020) - Xiao, Lu; Wang, Jun
ABSTRACT: In this paper, we are concerned with the existence, multiplicity and concentration of positive solutions for the semilinear Schrödinger-Poisson system − ε 2 Δ u + a ( x ) u + λ φ ( x ) u = b ( x ) f ( u ) + c ( x ) | u | 4 u, x ∈ R 3, − ε 2 Δ φ = u 2, u ∈ H 1 ( R 3 ), x ∈ R 3, where ε > 0 (small) and λ are real parameters, a, b and c are continuous functions, and f is a continuous superlinear and subcritical nonlinearity. First, we prove that there are two families of semiclassical positive solutions for ε > 0 small. Then we show that these solutions are concentrating on some sets which is generated by the minimal points of a and maximal points of b and c . In addition, we investigate the relation between the number of positive solutions and the topology of the set of the global minima(or maxima) of the potentials by minimax theorems and the Ljusternik-Schnirelmann theory. Finally, we obtain some sufficient conditions for the nonexistence of positive ground state solutions.