On a defocusing quasilinear Schrödinger equation with singular term

Abstract

We consider a class of modified quasilinear Schrödinger equations $ -\Delta u+\frac{k}{2}u\Delta u^2+V(x)u = \lambda a(x) u^{-\alpha}+b(x)u^{\beta}\; \text{in}\; \mathbb{R}^N, $ where $ N\geq3 $, $ V $ is a suitable non-negative continuous potential; $ a, b $ are bounded mensurable functions, $ 0<\alpha<1<\beta\leq2^*-1 $ and $ k, \lambda\geq0 $ are two parameters. We establish global existence and local multiplicity results of positive solutions in $ H^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N) $ for the equation with appropriate classes of parameters $ \alpha, \beta $ and coefficients $ a(x), b(x) $.