Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime

Bonheure, Denis; Casteras, Jean-Baptiste; Gou, Tianxiang; Jeanjean, Louis

doi:10.1090/tran/7769

Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime
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by Denis Bonheure, Jean-Baptiste Casteras, Tianxiang Gou and Louis Jeanjean PDF

Trans. Amer. Math. Soc. 372 (2019), 2167-2212 Request permission

Abstract:

In this paper, we study the existence of solutions to the mixed dispersion nonlinear Schrödinger equation \[ \gamma \Delta ^2 u -\Delta u + \alpha u=|u|^{2 \sigma } u, \qquad u \in H^2({\mathbb {R}}^N), \] under the constraint \[ \int _{{\mathbb {R}}^N}|u|^2 dx =c>0. \] We assume that $\gamma >0, N \geq 1, 4 \leq \sigma N < \frac {4N}{(N-4)^+}$, whereas the parameter $\alpha \in {\mathbb {R}}$ will appear as a Lagrange multiplier. Given $c \in {\mathbb {R}}^+$, we consider several questions including the existence of ground states and of positive solutions and the multiplicity of radial solutions. We also discuss the stability of the standing waves of the associated dispersive equation.

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