Normalized solutions for Schrdinger equations with critical Sobolev exponent and mixed nonlinearities

In this paper, we consider the following nonlinear Schrodinger equations with mixed nonlinearities: {-Delta u = lambda u + mu vertical bar u vertical bar(q-2) u + vertical bar u(2*-2) u in R-N, u is an element of H-1(R-N), integral(RN) u(2) = a(2), where N >= 3, mu > 0, lambda is an element of R and 2 < q < 2* = 2N/N-2. We prove in this paper (1) Existence of solutions of mountain-pass type for N >= 3 and 2 < q < 2 + 4/N; (2) Existence and nonexistence of ground states for 2 + 4N <= q < 2* with mu > 0 large; (3) Precisely asymptotic behaviors of ground states and mountain-pass solutions as mu -> 0 and mu goes to its upper bound. Our studies answer some open questions proposed by Soave in [48]. (C) 2022 Elsevier Inc. All rights reserved.

Automated summary

In this paper, we study the nonlinear Schrodinger equations with mixed nonlinearities. We establish the existence of solutions and investigate the existence and nonexistence of ground states in different parameter ranges. Additionally, we provide precise asymptotic behaviors of the ground states and mountain-pass solutions as the parameters approach specific values.