A Unified Approach to Singularly Perturbed Quasilinear Schrödinger Equations

Abstract

AbstractIn this paper we present a unified approach to investigate existence and concentration of positive solutions for the following class of quasilinear Schrödinger equations, $$-\varepsilon^2\Delta u+V(x)u\mp\varepsilon^{2+\gamma}u\Delta u^2=h(u),\ \ x\in \mathbb{R}^N, $$ - ε 2 Δ u + V ( x ) u ∓ ε 2 + γ u Δ u 2 = h ( u ) , x ∈ R N , where $$N\geqslant3, \varepsilon > 0, V(x)$$ N ⩾ 3 , ε > 0 , V ( x ) is a positive external potential,h is a real function with subcritical or critical growth. The problem is quite sensitive to the sign changing of the quasilinear term as well as to the presence of the parameter $$\gamma>0$$ γ > 0 . Nevertheless, by means of perturbation type techniques, we establish the existence of a positive solution $$u_{\varepsilon,\gamma}$$ u ε , γ concentrating, as $$\varepsilon\rightarrow 0$$ ε → 0 , around minima points of the potential.