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Abstract

In this paper, we prove the existence of nontrivial solutions for the following planar quasilinear Schrödinger equation: $\begin{matrix} - Δ u + V (x) u - Δ (u^{2}) u = g (u), x \in R^{2}, \end{matrix}$ where $V \in C (R^{2}, [0, \infty))$ and $g \in C (R, R)$ is of subcritical exponential growth satisfying some mild conditions. In particular, by means of the Trudinger–Moser inequality, we give a different method from the one of the polynomial growth nonlinearities to prove the Brézis–Lieb split property when f has subcritical exponential growth. Our result extends and complements the one of Chen–Rădulescu–Tang–Zhang (Rev. Mat. Iberoam. 36 (2020) 1549–1570) dealing with the higher dimensions $N ⩾ 3$ to the dimension $N = 2$ .

Keywords

Quasilinear Schrödinger equation nontrivial solution exponential growth

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Combined effects in planar quasilinear Schrödinger equations with superlinear reaction

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