## Least energy solutions to a cooperative system of Schrödinger equations with prescribed $$L^2$$-bounds: at least $$L^2$$-critical growth.(English)Zbl 07451515

Summary: We look for least energy solutions to the cooperative systems of coupled Schrödinger equations $\begin{cases} -\Delta u_i + \lambda_i u_i = \partial_iG(u)& \text{in } \mathbb{R}^N,\;N \ge 3,\\ u_i \in H^1(\mathbb{R}^N), &\qquad\qquad\qquad\quad i\in \{1,\ldots ,K\}\\ \int_{\mathbb{R}^N} |u_i|^2\,dx \le \rho_i^2 \end{cases}$ with $$G\ge 0$$, where $$\rho_i>0$$ is prescribed and $$(\lambda_i, u_i) \in\mathbb{R}\times H^1 (\mathbb{R}^N)$$ is to be determined, $$i\in \{1, \dots, K\}$$. Our approach is based on the minimization of the energy functional over a linear combination of the Nehari and Pohožaev constraints intersected with the product of the closed balls in $$L^2(\mathbb{R}^N)$$ of radii $$\rho_i$$, which allows to provide general growth assumptions about $$G$$ and to know in advance the sign of the corresponding Lagrange multipliers. We assume that $$G$$ has at least $$L^2$$-critical growth at 0 and admits Sobolev critical growth. The more assumptions we make about $$G$$, $$N$$, and $$K$$, the more can be said about the minimizers of the corresponding energy functional. In particular, if $$K=2$$, $$N\in \{3, 4\}$$, and $$G$$ satisfies further assumptions, then $$u=(u_1, u_2)$$ is normalized, i.e., $$\int_{\mathbb{R}^N} |u_i|^2\,dx=\rho_i^2$$ for $$i\in \{1, 2\}$$.

### MSC:

 35Q40 PDEs in connection with quantum mechanics 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q60 PDEs in connection with optics and electromagnetic theory 35J20 Variational methods for second-order elliptic equations 78A60 Lasers, masers, optical bistability, nonlinear optics

### Keywords:

Schrödinger equations; photonic crystal
Full Text:

### References:

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