一类具有临界指数的对数Schrodinger方程基态解的存在性和集中性

Existence and concentration of ground state solutions for a logarithmic Schrodinger equation with the critical exponent

本文研究一类具有竞争位势和临界指数的对数Schrodinger方程,它的能量泛函在其自然Sobolev空间中没有一阶连续导数.首先利用非光滑临界点理论,获得相关的自治对数Schrodinger方程基态解的存在性;然后,通过引入一个合适的基态能量函数,利用Nehari流形方法研究位势函数与对数Schrodinger方程基态解存在性和集中性的关系;最后,给出此类方程基态解不存在的一个充分条件.

We study a class of logarithmic Schrodinger equations involving competing for weight potentials and critical nonlinearities on an unbounded domain.The corresponding energy functional has no continuous first derivative in its natural Sobolev space.Firstly,using the non-smooth critical point theory,we show that the problem admits one ground state positive solution.Then,we relate the concentration of solutions to a global minimum set of a suitable ground energy function via a Nehari manifold method.Finally,we give a sufficient condition for the nonexistence of ground state solutions.