Multiple positive normalized solutions for nonlinear Schrödinger systems

Tianxiang Gou; Louis Jeanjean

doi:10.1088/1361-6544/aab0bf

paper

Multiple positive normalized solutions for nonlinear Schrödinger systems

arXiv:1705.09612 · doi:10.1088/1361-6544/aab0bf

Abstract

We consider the existence of multiple positive solutions to the nonlinear Schrödinger systems sets on $H^{1} (R^{N}) \times H^{1} (R^{N})$ , \[ \left\{ \begin{aligned} -Δu_1 &= λ_1 u_1 + μ_1 |u_1|^{p_1 -2}u_1 + βr_1 |u_1|^{r_1-2} u_1|u_2|^{r_2}, -Δu_2 &= λ_2 u_2 + μ_2 |u_2|^{p_2 -2}u_2 + βr_2 |u_1|^{r_1} |u_2|^{r_2 -2} u_2, \end{aligned} \right. \] under the constraint \[ \int_{\mathbb{R}^N}|u_1|^2 \, dx = a_1,\quad \int_{\mathbb{R}^N}|u_2|^2 \, dx = a_2. \] Here $a_{1}, a_{2} > 0$ are prescribed, $μ_{1}, μ_{2}, β > 0$ , and the frequencies $λ_{1}, λ_{2}$ are unknown and will appear as Lagrange multipliers. Two cases are studied, the first when $N \geq 1, 2 < p_{1}, p_{2} < 2 + \frac{4}{N}, r_{1}, r_{2} > 1, 2 + \frac{4}{N} < r_{1} + r_{2} < 2^{*}$ , the second when $N \geq 1, 2 + \frac{4}{N} < p_{1}, p_{2} < 2^{*}, r_{1}, r_{2} > 1, r_{1} + r_{2} < 2 + \frac{4}{N} .$ In both cases, assuming that $β > 0$ is sufficiently small, we prove the existence of two positive solutions. The first one is a local minimizer for which we establish the compactness of the minimizing sequences and also discuss the orbital stability of the associated standing waves. The second solution is obtained through a constrained mountain pass and a constrained linking respectively.

To appear in Nonlinearity

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Existence and Stability of standing waves for supercritical NLS with a Partial Confinement

Multiple positive normalized solutions for nonlinear Schrödinger systems

Abstract

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