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.
<x content-type="archive" xml:space="preserve">Abstract</x> In this paper we are concerned with questions of multiplicity and concentration behavior of positive solutions of the elliptic problem <alternatives> <![CDATA[ (P_)4cm { arrayrcl L_u=f(u) in ^3, [1.5mm] u>0 in ^3, [1.5mm] u H^1(^3), array . ]]> <inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2zjx9jp0" xmlns:xlink="http://www.w3.org/1999/xlink" /> ( P ε ) ℒ ε u = f ( u ) in IR 3 , u > 0 in IR 3 , u ∈ H 1 ( IR 3 ) , </alternatives> where ε is a small positive parameter, f : ℝ → ℝ is a continuous function, <alternatives> <![CDATA[ L_ ]]> <inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2zjx9jvr" xmlns:xlink="http://www.w3.org/1999/xlink" />ℒε</alternatives> is a nonlocal operator defined by <alternatives> <![CDATA[ L_u=M(1_^3|u|^2+1^3_^3V(x)u^2)[-^2Δ u + V(x)u ], ]]> <inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2zjx9jw9" xmlns:xlink="http://www.w3.org/1999/xlink" /> 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 ] , </alternatives> M : IR+ → IR+ and V : IR3 → IR are continuous functions which verify some hypotheses. ,