Axially symmetric solutions for the planar Schrödinger-Poisson system with critical exponential growth

By Sitong Chen, Xianhua Tang. Journal of Differential Equations, 2020

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Abstract This paper is concerned with the following planar Schrodinger-Poisson system { − Δ u + V ( x ) u + ϕ u = f ( x , u ) , x ∈ R 2 , Δ ϕ = u 2 , x ∈ R 2 , where V ∈ C ( R 2 , [ 0 , ∞ ) ) is axially symmetric and f ∈ C ( R 2 × R , R ) is of subcritical or critical exponential growth in the sense of Trudinger-Moser. We obtain the existence of a nontrivial solution or a ground state solution of Nehari-type and infinitely many solutions to the above system under weak assumptions on V and f. Our theorems extend the results of Cingolani and Weth [Ann. Inst. H. Poincare Anal. Non Lineaire, 33 (2016) 169-197] and of Du and Weth [Nonlinearity, 30 (2017) 3492-3515] and Chen and Tang [J. Differential Equations, 268 (2020) 945-976], where f ( x , u ) has polynomial growth on u. In particular, some new tricks and approaches are introduced to overcome the double difficulties resulting from the appearance of both the convolution ϕ 2 , u ( x ) with sign-changing and unbounded logarithmic integral kernel and the critical growth nonlinearity f ( x , u ) .

Subfields: Applied Mathematics

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