Ground state solutions for critical Schrödinger equations with Hardy potential

Abstract

Abstract In this paper, we investigate the following Schrödinger equation 0.1 − Δ u − μ | x | 2 u = g ( u ) + | u | 2 * − 2 u i n R N \ { 0 } , where N ⩾ 3, μ < ( N − 2 ) 2 4 , 2 * ≔ 2 N N − 2 is called the critical Sobolev exponent and g satisfies some appropriate subcritical conditions. For any μ ∈ 0 , ( N − 2 ) 2 4 , 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 − Δ u = g ( u ) + | u | 2 * − 2 u i n R N . when μ < 0, we prove that the mountain pass level of problem (0.1) in H 1 ( R N ) 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 H 1 ( R N ) . 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 → ∞.