Existence and asymptotic behavior of ground state solutions for Schrodinger equations with Hardy potential and Berestycki-Lions type conditions

In this paper, we investigate the following Schrodinger equation { -Delta u = mu/vertical bar x vertical bar(2) u = g(u) in R-N \ {0}, u is an element of H-1 (R-N), where N >= 3, mu < (N-2)(2)/4, 1/vertical bar x vertical bar(2) called the Hardy potential (the inverse-square potential) and g satisfies 2 the Berestycki-Lions type condition. If 0 < mu < (N-2)(2)/4 , combining variational methods with analytical skills, we show that the above problem has a positive and radial ground state solution. At the same time, our results suggest that this solution together with its derivatives up to order 2 have exponential decay at infinity while this solution has the possibility of blow-up at the origin. Furthermore, we construct a family of ground state solutions which converges to a ground state solution of the limiting problem as mu -> 0(+). If < 0, we prove that the mountain pass level in H-1 (R-N) can not be achieved. Provided further assumption that the above problem in the radial space H-r(1) (R-N), we obtain the ground state solutions whose energy is strictly greater than the mountain pass level in H-1(R-N). We also construct a family of solutions which converges to the ground state solution of the limiting problem as mu -> 0(-). (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

In this paper, we investigate a specific Schrodinger equation and prove the existence of a positive radial ground state solution, showing that the solution decays exponentially with a possible blow-up at the origin. We also construct a family of solutions and validate their convergence.