Normalized solutions for planar Schrödinger-Poisson system with critical exponential growth and nonlocal interaction

Chenlu Wei
Sitong Chen
Muhua Shu

Abstract. This paper focuses on the following planar Schrödinger-Poisson system with critical exponential growth and nonlocal interaction \[\begin{cases}-\Delta u+\lambda u+\mu(\log|\cdot|*u^2)u = \gamma \left( I_\alpha * |u|^q \right) |u|^{q-2} u+\left(e^{u^2}-1-u^2\right)u, & x\in \mathbb{R}^2, \\ \displaystyle \int_{\mathbb{R}^2}u^2\mathrm{d}x=c,\end{cases}\] where \(c\gt 0\), \(\mu,\gamma\gt 0\), \(\lambda \in \mathbb{R}\) appears as a Lagrange multiplier, \(\alpha \in (0,2)\), \(1+\frac{\alpha}{2} \leq q \lt +\infty\), \( I_\alpha:\mathbb{R}^2\to\mathbb{R}\) denotes the Riesz potential and \(1+\frac{\alpha}{2}\) is the lower critical exponent with respect to the Hardy-Littlewood-Sobolev inequality. Through delicate energy estimates, under explicit conditions on \(c\), we prove the existence of two normalized solutions: one is a local minimizer and the other is of mountain-pass type. The presence of the logarithmic kernel and the competition between the two nonlocal terms necessitates the development of new tools to address the loss of compactness caused by the critical exponential growth, for which the variational techniques developed for the local problem are no longer applicable. Our work not only generalizes the special case \(\gamma=0\), but also provides an analytical approach that is applicable to more \(L^2\)-constrained problems with competing nonlocal terms modelling long-range attraction in particle physics.

Keywords: normalized solution, logarithmic convolution potential, nonlocal interaction, critical exponential growth, Trudinger-Moser inequality.

Mathematics Subject Classification: 35J20, 35J62, 35Q55.

Full text (pdf)