Ground state solutions for critical Schrödinger equations with Hardy potential

Abstract

In this paper, we investigate the following Schrödinger equation 0.1 −Δu−μ|x|2u=g(u)+|u|2*−2uinRN\\{0}, where N ⩾ 3, , is called the critical Sobolev exponent and g satisfies some appropriate subcritical conditions. For any , we prove that problem (0.1) has a positive radial ground state solution, which possesses exponential decaying property at infinity and blow-up property at origin. Moreover, for any sequence {μ n } ⊂ (0, +∞) satisfying μ n → 0+, the sequence of ground state solutions to problem (0.1) converges to a ground state solution of 0.2 −Δu=g(u)+|u|2*−2uinRN. when μ < 0, we prove that the mountain pass level of problem (0.1) in cannot be achieved. Further, we obtain a ground state radial solution of problem (0.1) whose energy is strictly greater than the mountain pass level in . Also, for any sequence {μ n } ⊂ (0, +∞) satisfying μ n → 0+, the sequence of ground state radial solutions to problem (0.1) converges to a ground state radial solution of the limiting problem as n → ∞.