Multiple normalized solutions for the planar Schrödinger–Poisson system with critical exponential growth

Abstract

AbstractThe paper deals with the existence of normalized solutions for the following Schrödinger–Poisson system with $$L^2$$ L 2 -constraint: $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\lambda u+\mu \left( \log |\cdot |*u^2\right) u=\left( e^{u^2}-1-u^2\right) u, &{} x\in {\mathbb {R}}^2, \\ \int _{{\mathbb {R}}^2}u^2\textrm{d}x=c, \\ \end{array} \right. \end{aligned}$$ - Δ u + λ u + μ log | · | ∗ u 2 u = e u 2 - 1 - u 2 u , x ∈ R 2 , ∫ R 2 u 2 d x = c , where $$\mu >0$$ μ > 0 , $$\lambda \in {\mathbb {R}}$$ λ ∈ R will arise as a Lagrange multiplier and the nonlinearity enjoys critical exponential growth of Trudinger-Moser type. By specifying explicit conditions on the energy level c, we detect a geometry of local minimum and a minimax structure for the corresponding energy functional, and prove the existence of two solutions, one being a local minimizer and one of mountain-pass type. In particular, to catch a second solution of mountain-pass type, some sharp estimates of energy levels are proposed, suggesting a new threshold of compactness in the $$L^2$$ L 2 -constraint. Our study extends and complements the results of Cingolani–Jeanjean (SIAM J Math Anal 51(4): 3533-3568, 2019) dealing with the power nonlinearity $$a|u|^{p-2}u$$ a | u | p - 2 u in the case of $$a>0$$ a > 0 and $$p>4$$ p > 4 , which seems to be the first contribution in the context of normalized solutions. Our model presents some new difficulties due to the intricate interplay between a logarithmic convolution potential and a nonlinear term of critical exponential type and requires a novel analysis and the implementation of new ideas, especially in the compactness argument. We believe that our approach will open the door to the study of other $$L^2$$ L 2 -constrained problems with critical exponential growth, and the new underlying ideas are of future development and applicability.