Sharp nonexistence results of prescribed L 2-norm solutions for some class of Schrödinger–Poisson and quasi-linear equations

Abstract

In this paper we study the existence of minimizers forwhere c > 0 is a given parameter. In the range p ∈ [3, 10 3 ] we explicit a threshold value of c > 0 separating existence and non-existence of minimizers. We also derive a non-existence result of critical points of F (u) restricted to S(c) when c > 0 is sufficiently small. Finally, as a byproduct of our approaches, we extend some results of [9] where a constrained minimization problem, associated to a quasilinear equation, is considered.

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

“…The key points of proving Theorem 1.1 are some established inequalities in Lemma 2.2-2.4. Here, we shall see the difference between (1.1) and the Schrödinger-Poisson equation (see [19]). In the mass-supercritical case p ∈ (4, +∞), the functional I(u) is no longer bounded from below on S(c) (Lemma 2.5), the minimization method on S r (c) used in [10] does not work.…”

“…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:…”

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

Multiple normalized solutions of Chern–Simons–Schrödinger system

“…[1,2,[6][7][8]13,17,20,22,26]. In [7,8,13], for p in some ranges and c > 0, by using the concentration-compactness method of Lions [18,19], the authors obtained the minimizers oñ…”

Existence and asymptotic behavior of high energy normalized solutions for the Kirchhoff type equations inR3

“…For fixed λ, the authors of [9][10][11] obtained weak solutions to (1.1) by looking for critical points of the C 1 functional Recently, normalized solutions to elliptic equations attract much attention of researchers, see e.g. [18][19][20][21][22][23][24][25][26]. In [23], Jeanjean considered the following semi-linear Schrödinger equation:…”