A Unified Approach to Singularly Perturbed Quasilinear Schrodinger Equations

作者：Cassani, Daniele^*; Wang, Youjun; Zhang, Jianjun

来源：Milan Journal of Mathematics, 2020, 88(2): 507-534.

DOI：10.1007/s00032-020-00323-6

• Semiclassical states
• critical growth
• quasilinear Schrodinger equations
• variational methods

In this paper we present a unified approach to investigate existence and concentration of positive solutions for the following class of quasilinear Schrodinger equations, @@@ -epsilon(2)Delta u + V (x)u -/+ epsilon(2+gamma)u Delta u(2) = h(u), x is an element of R-N, @@@ where N >= 3, epsilon > 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. Nevertheless, by means of perturbation type techniques, we establish the existence of a positive solution u(epsilon,gamma) concentrating, as epsilon -> 0, around minima points of the potential.

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