Existence of Solutions to Quasilinear Schrödinger Equations Involving Critical Sobolev Exponent

Abstract

By using variational approaches, we study a class of quasilinear Schrödinger equations involving critical Sobolev exponents \[ -\Delta u + V(x)u + \frac{1}{2} \kappa [\Delta(u^2)]u = |u|^{p-2}u + |u|^{2^*-2}u, \quad x \in \mathbb{R}^N, \] where $V(x)$ is the potential function, $\kappa \gt 0$, $\max \{ (N+3)/(N-2),2 \} \lt p \lt 2^* := 2N/(N-2)$, $N \geq 4$. If $\kappa \in [0,\overline{\kappa})$ for some $\overline{\kappa} \gt 0$, we prove the existence of a positive solution $u(x)$ satisfying $\max_{x \in \mathbb{R}^N} |u(x)| \leq \sqrt{1/(2\kappa)}$.