Normalized solutions for nonhomogeneous Chern-Simons-Schrödinger equations with critical exponential growth
Chenlu Wei, Sitong Chen, Xinao Zhou
TL;DR
This work addresses the existence of normalized (constant $L^2$-norm) solutions to a planar Chern-Simons-Schrödinger equation with a nonhomogeneous perturbation $g$ and a nonlocal term, under a critical exponential nonlinearity $f(u)=(e^{u^2}-1)u$. The authors develop a refined variational framework on the $L^2$-constraint, proving the existence of two distinct solutions for small mass $c$: a negative-energy local minimizer and a positive-energy mountain-pass type solution. They overcome compactness challenges from the exponential growth and nonlocal interactions by deriving precise energy estimates, including a sharp bound $M(c)<m(c)+2\pi$, and by employing a Moser-type construction to build constrained paths. The results are new even in the absence of nonlocal terms and pave the way for handling broader constrained problems with nonhomogeneous perturbations. The techniques blend Pohozaev identities, sharp inequalities for the nonlocal term, and careful control of the exponential nonlinearity to obtain robust constrained critical points.
Abstract
This paper investigates the existence of normalized solutions for the following Chern-Simons-Schrödinger equation: \begin{equation*} \left\{ \begin{array}{ll} -Δu+λu+\left(\frac{h^{2}(\vert x\vert)}{\vert x\vert^{2}}+\int_{\vert x\vert}^{\infty}\frac{h(s)}{s}u^{2}(s)\mathrm{d}s\right)u =\left(e^{u^2}-1\right)u+g(x), & x\in \R^2, u\in H_r^1(\R^2),\ \int_{\R^2}u^2\mathrm{d}x=c, \end{array} \right. \end{equation*} where $c>0$, $λ\in \R$ acts as a Lagrange multiplier and $g\in \mathcal {C}(\mathbb{R}^2,[0,+\infty))$ satisfies suitable assumptions. In addition to the loss of compactness caused by the nonlinearity with critical exponential growth, the intricate interactions among it, the nonlocal term, and the nonhomogeneous term significantly affect the geometric structure of the constrained functional, thereby making this research particularly challenging. By specifying explicit conditions on $c$, we subtly establish a structure of local minima of the constrained functional. Based on the structure, we employ new analytical techniques to prove the existence of two solutions: one being a local minimizer and one of mountain-pass type. Our results are entirely new, even for the Schrödinger equation that is when nonlocal terms are absent. We believe our methods may be adapted and modified to deal with more constrained problems with nonhomogeneous perturbation.