On a class of Kirchhoff type logarithmic Schrödinger equations involving the critical or supercritical Sobolev exponent

Abstract

In this paper, we study a class of Kirchhoff type logarithmic Schrödinger equations involving the critical or supercritical Sobolev exponent. Such problems cannot be studied by applying variational methods in a standard way, because the nonlinearities do not satisfy the Ambrosetti-Rabinowitz condition and change sign. Moreover, the appearance of the logarithmic term makes the associated energy functional lose differentiable in the sense of Gateaux. By analyzing the structure of the Nehari manifold and developing some analysis techniques, the above obstacles are overcome in subtle ways and several existence result are obtained. Furthermore, we investigate the regularity, the monotonicity, and the symmetric properties of the solutions via the iterative technique and the moving plane method.