Normalized solutions for nonhomogeneous Chern-Simons-Schr\"odinger equations with critical exponential growth

Abstract

This paper investigates the existence of normalized solutions for the following Chern-Simons-Schr\"odinger equation: \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+\lambda u+\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 $c>0$, $\lambda\in \R$ 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 Schr\"odinger 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.