Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations

Abstract

Abstract. In this paper we study the existence and the instability of standing waves with prescribed L 2 -norm for a class of Schrödinger-Poisson-Slater equations in R 6). To obtain such solutions we look to critical points of the energy functionalon the constraints given byFor the values p ∈ ( 10 3 , 6) considered, the functional F is unbounded from below on S(c) and the existence of critical points is obtained by a mountain pass argument developed on S(c). We show that critical points exist provided that c > 0 is sufficiently small and that when c > 0 is not small a nonexistence result is expected. Concerning the dynamics we show for initial condition u 0 ∈ H 1 (R 3 ) of the associated Cauchy problem with u 0 2 2 = c that the mountain pass energy level γ(c) gives a threshold for global existence. Also the strong instability of standing waves at the mountain pass energy level is proved. Finally we draw a comparison between the Schrödinger-Poisson-Slater equation and the classical nonlinear Schrödinger equation.

Normalized solutions for the Chern–Simons–Schrödinger equation in R^2

“…The nonlocal term causes some mathematical difficulties that make the study of (1.1) more interesting. As we shall see, (1.1) is also different from the Schrödinger-Poisson equation (see [6,19,28]), which is another problem exhibiting the competition between local and nonlocal terms. We point out that (1.1) arises from seeking the standing wave solutions to the following nonlinear Schrödinger equations with the gauge field:…”

“…We would like to mention that the two methods used in [6] (see Theorem 1.2) and in [25] (see Lemma 2.9) seem difficult to be used here due to the existence of the Chern-Simons term in (1.1). Motivated by [5], after getting an equality related to m(c) (see Lemma 2.14), we succeed in proving the monotonicity property of m(c) by a scaling argument.…”

“…By using a mountain pass argument on [6] proved that for p ∈ ( , 6), there exists c 0 > 0 such that for any c ∈ (0, c 0 ), (1.7) admits an unbounded sequence of couple of weak solutions…”

“…Under certain assumptions on G(u), by using a minimax procedure inspired by [6], the authors of [27] proved that for any c > 0, there is at least a couple (u c , λ c ) ∈ H 1 (R N ) × R − of weak solution to the equation above with u c 2 = c.…”

“…[2,3,4,6,7,8,18,19,20,27,28,36]. In [18], Jeanjean considered the following semi-linear Schrödinger equation:…”