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

The paper deals with the existence of normalized solutions for the following Schr & ouml;dinger-Poisson system with -constraint: { -Delta u+lambda u+mu(log||& lowast;u2)u=(e(u2-)1-u2)u,x is an element of R-2, integral R(2)u(2)dx=c, where mu>0,lambda is an element of 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 -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-2uin the case ofa>0andp>4, in the case of and , 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 -constrained problems with critical exponential growth, and the new underlying ideas are of future development and applicability.

Keywords

Critical exponential growth, Logarithmic convolution potential, Normalized solution, Planar Schrödinger–Poisson system, Trudinger–Moser inequality

Citation

MATHEMATISCHE ZEITSCHRIFT. 2024, vol. 306, issue 2, p. 1-32.
https://link.springer.com/article/10.1007/s00209-024-03432-9

Document type

Peer-reviewed

Document version

Published version

Language of document

en

Document licence

Creative Commons Attribution 4.0 International
http://creativecommons.org/licenses/by/4.0/