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 $$ (P_{\var})\hspace*{4cm} \left\{ \begin{array}{rcl} \mathcal{L}_{\var}u=f(u) \ \ \mbox{in} \ \ \R^{3}, \\[1.5mm] u>0 \ \ \mbox{in} \ \ \R^{3}, \\[1.5mm] u \in H^{1}(\R^3), \end{array} \right. $$ ( 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, $$ \mathcal{L}_{\var} $$ ℒ ε is a nonlocal operator defined by $$ \mathcal{L}_{\var}u=M\left(\dis\frac{1}{\var}\int_{\R^{3}}|\nabla u|^{2}+\frac{1}{\var^{3}}\dis\int_{\R^{3}}V(x)u^{2}\right)\left[-\var^{2}\Delta u + V(x)u \right], $$ 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.