Existence of normalized solutions for the coupled elliptic system with quadratic nonlinearity

Abstract

In the present paper, we study the existence of the normalized solutions for the following coupled elliptic system with quadratic nonlinearity −Δu−λ1u=μ1∣u∣u+βuvinRN,−Δv−λ2v=μ2∣v∣v+β2u2inRN,\left\{\begin{array}{ll}-\Delta u-{\lambda }_{1}u={\mu }_{1}| u| u+\beta uv\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ -\Delta v-{\lambda }_{2}v={\mu }_{2}| v| v+\frac{\beta }{2}{u}^{2}\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\end{array}\right. where u,vu,v satisfying the additional condition ∫RNu2dx=a1,∫RNv2dx=a2.\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{u}^{2}{\rm{d}}x={a}_{1},\hspace{1em}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{v}^{2}{\rm{d}}x={a}_{2}. On the one hand, we prove the existence of minimizer for the system with L2{L}^{2}-subcritical growth (N≤3N\le 3). On the other hand, we prove the existence results for different ranges of the coupling parameter β>0\beta \gt 0 with L2{L}^{2}-supercritical growth (N=5N=5). Our argument is based on the rearrangement techniques and the minimax construction.

Keywords

• normalized solutions
• quadratic nonlinearity
• variational methods
• 35j61
• 35j20
• 35q55
• 49j40