Skip to main content

Multiplicity and concentration behavior of positive solutions for a Schrödinger–Kirchhoff type problem via penalization method∗∗

  • Giovany M. Figueiredo (a1) and João R. Santos (a1)

In this paper we are concerned with questions of multiplicity and concentration behavior of positive solutions of the elliptic problem

$$ (P_{\var})\hspace*{4cm} \left\{ \begin{array}{rcl} \mathcal{L}_{\var}u=f(u) \ \ \mbox{in} \ \ \R^{3},\\[1.5mm] u>0 \ \ \mbox{in} \ \ \R^{3},\\[1.5mm] u \in H^{1}(\R^3), \end{array} \right. $$ (Pε)εu=f(u)inIR3,u>0inIR3,uH1(IR3),

where ε is a small positive parameter, f : ℝ → ℝ is a continuous function, $$ \mathcal{L}_{\var} $$ ε is a nonlocal operator defined by

$$ \mathcal{L}_{\var}u=M\left(\dis\frac{1}{\var}\int_{\R^{3}}|\nabla u|^{2}+\frac{1}{\var^{3}}\dis\int_{\R^{3}}V(x)u^{2}\right)\left[-\var^{2}\Delta u + V(x)u \right], $$ εu=M1εIR3|u|2+1ε3IR3V(x)u2[ε2Δu+V(x)u],

M : IR+ → IR+ and V : IR3 → IR are continuous functions which verify some hypotheses.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

ESAIM: Control, Optimisation and Calculus of Variations
  • ISSN: 1292-8119
  • EISSN: 1262-3377
  • URL: /core/journals/esaim-control-optimisation-and-calculus-of-variations
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 9 *
Loading metrics...

Abstract views

Total abstract views: 38 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 19th March 2018. This data will be updated every 24 hours.