In this paper, we study the existence and concentration of normalized solutions to the supercritical nonlinear Schrödinger equation \[ \left\{\begin{array}{@{}ll} -\Delta u + V(x) u = \mu_q u + a \vert u \vert ^q u & {\rm in}\ \mathbb{R}^2,\\ \int_{\mathbb{R}^2} \vert u \vert ^2\,{\rm d}x =1, & \end{array} \right.\]where μq is the Lagrange multiplier. For ellipse-shaped potentials V(x), we show that for q > 2 close to 2, the equation admits an excited solution uq, and furthermore, we study the limiting behaviour of uq when q → 2+. Particularly, we describe precisely the blow-up formation of the excited state uq.

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关于具有椭圆形势的超临界非线性薛定谔方程

在本文中，我们研究了超临界非线性薛定谔方程归一化解的存在和集中\[ \left\{\begin{array}{@{}ll} -\Delta u + V(x) u = \mu_q u + a \vert u \vert ^qu & {\rm in}\ \mathbb{ R}^2,\\ \int_{\mathbb{R}^2} \vert u \vert ^2\,{\rm d}x =1, & \end{array} \right.\]在哪里μq是拉格朗日乘数。对于椭圆形势五(X)，我们证明对于q> 2 接近 2，方程允许激发解你q，此外，我们研究了你q什么时候q→ 2+. 特别是，我们精确地描述了激发态的爆炸形成你q.