Spikes of sign-changing solutions to the critical Schrödinger equations with trapping potentials

Abstract

ABSTRACTConsider the following Schrödinger equation: where , V(x) is a trapping potential, is a constant, is a small parameter and . By using the variational method, we obtain a least energy sign-changing solution to this equation for small enough. The concentration behavior of this sign-changing solution as is also studied. Furthermore, by combining the uniformly elliptic estimates and local energy estimates, we also obtain the location of the spikes as . To the best of our knowledge, this is the first attempt devoted to the spikes of sign-changing solutions to Schrödinger equations with Sobolev critical exponent and trapping potentials.