Multiplicity and concentration behavior of positive solutions for a Schrödinger–Kirchhoff type problem via penalization method∗∗∗. (7th April 2014) - Figueiredo, Giovany M.; Santos, João R.
Abstract : In this paper we are concerned with questions of multiplicity and concentration behavior of positive solutions of the elliptic problem (P_)4cm { arrayrcl L_u=f(u) in ^3, [1.5mm] u>0 in ^3, [1.5mm] u H^1(^3), array . ( P ε ) ℒ ε u = f ( u ) in IR 3, u > 0 in IR 3, u ∈ H 1 ( IR 3 ), where ε is a small positive parameter, f : ℝ → ℝ is a continuous function, L_ ℒ ε is a nonlocal operator defined by L_u=M(1_^3|u|^2+1^3_^3V(x)u^2)[-^2Δ u + V(x)u ], ℒ ε u = M 1 ε ∫ IR 3 | ∇ u | 2 + 1 ε 3 ∫ IR 3 V ( x ) u 2 [ − ε 2 Δ u + V ( x ) u ], M : IR+ → IR+ and V : IR 3 → IR are continuous functions which verify some hypotheses.,