Concentration of positive solutions for critical quasilinear Schrödinger equation with competing potentials

ABSTRACT

In this paper, we study a class of singularly perturbed critical quasilinear Schrödinger equation of the form $-{\epsilon }^2\mathrm{\Delta }u+V\left(x\right)u-{\epsilon }^2\left(\mathrm{\Delta }{u}^2\right)u=P\left(x\right)|u{|}^{p-2}u+Q\left(x\right)|u{|}^{2\cdot 2^{\ast }-2}u,\text{ in }{\mathbb{R}}^N.$ By using a change of variables and variational argument, we prove the existence of positive solutions and their concentration behavior.

Keywords:

• Quasilinear Schrödinger equation
• critical exponent
• concentration
• competing potential

AMS Subject Classifications:

• 35Q55
• 35J65

Disclosure statement

No potential conflict of interest was reported by the authors.