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We study the existence of solutions (u, ₔ) H¹ (RN; R) R to \ - u + u = f (u) in RN \ with N 3 and prescribed L² norm, and the dynamics of the solutions to \ cases i ₜ + = f () \\ (, 0) = ₀ H¹ (RN; C) cases \ with ₀ close to u. Here, the nonlinear term f has mass-subcritical growth at the origin, mass-supercritical growth at infinity, and is more general than the sum of two powers. Under different assumptions, we prove the existence of a locally least-energy solution, the orbital stability of all such solutions, the existence of a second solution with higher energy, and the strong instability of such a solution.
Bieganowski et al. (Wed,) studied this question.