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AbstractThis paper deals with the following Choquard equation with a local nonlinear perturbation:$$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} - {\it\Delta} u+u=\left(I_{\alpha}*|u|^{\frac{\alpha}{2}+1}\right)|u|^{\frac{\alpha}{2}-1}u +f(u), & x\in \mathbb{R}^2; \\ u\in H^1(\mathbb{R}^2), \end{array} \right. \end{array}$$where α ∈ (0, 2), Iα : ℝ2 → ℝ is the Riesz potential and f ∈ 𝓒(ℝ, ℝ) is of critical exponential growth in the sense of Trudinger-Moser. The exponent $\begin{array}{} \displaystyle \frac{\alpha}{2}+1 \end{array}$ is critical with respect to the Hardy-Littlewood-Sobolev inequality. We obtain the existence of a nontrivial solution or a Nehari-type ground state solution for the above equation in the doubly critical case, i.e. the appearance of both the lower critical exponent $\begin{array}{} \displaystyle \frac{\alpha}{2}+1 \end{array}$ and the critical exponential growth of f(u).

This paper deals with the following Choquard equation with a local nonlinear perturbation: −normalΔu+u=()Iα∗false|ufalse|αN+1false|ufalse|αN−1u+λfalse|ufalse|p−2u,x∈double-struckRN;u∈H1false(double-struckRNfalse), where N ≥ 1, α∈(0,N), λ>0, 2λ∗ for some given number λ∗.

We consider the following two coupled nonlinear Schrödinger system: −normalΔu+u=f1false(x,ufalse)+λfalse(xfalse)v,x∈ℝ2,−normalΔv+v=f2false(x,vfalse)+λfalse(xfalse)u,x∈ℝ2, where the coupling parameter satisfies 0 < λ(x) ≤ λ0 < 1 and the reactions f1, f2 have critical exponential growth in the sense of Trudinger–Moser inequality. Using non‐Nehari manifold method together with the Lions's concentration compactness and the Trudinger‐Moser inequality, we show that the above system has a Nehari‐type ground state solution and a nontrivial solution. Our results improve and extend the previous results.