Normalized solutions for nonlinear Schrodinger equations involving mass subcritical and supercritical exponents

fWe study nonlinear Schrodinger equation -Delta u + (lambda + V (y))u = u(p epsilon-1) in RN, with prescribed mass integral N-R u(2) = a, where lambda is a Lagrange multiplier, V (y) is a real-valued potential, a is an element of R+ is a constant, p pound = (p) over bar +/- epsilon and (p) over bar = 2 + 4/N is the L-2-critical exponent. Bartsch et al. (2021) [1] proved the existence of a solution with the assumption that V decays at infinity and is non-negative. In this paper, we prove that it is the number of the critical points of the potential V that affects the existence and the number of solutions for this problem. We also prove a local uniqueness for the solutions we construct. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.