Abstract This paper deals with the following Choquard equation with a local nonlinear perturbation: −Δu+u=Iα∗|u|α2+1|u|α2−1u+f(u),x∈R2;u∈H1(R2), $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} - {\it\Delta} u+u=\left(I_{\alpha}*|u|^{\frac{\alpha}{2}+1}\right)|u|^{\frac{\alpha}{2}-1}u +f(u), & x\in \mathbb{R}^2; \\ u\in H^1(\mathbb{R}^2), \end{array} \right. \end{array}$$ 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 α2+1 $\begin{array}{} \displaystyle \frac{\alpha}{2}+1 \end{array}$ 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 α2+1 $\begin{array}{} \displaystyle \frac{\alpha}{2}+1 \end{array}$ and the critical exponential growth of f(u).

中文翻译：

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具有双临界指数的 Choquard 方程的 Nehari 型基态解

摘要 本文讨论了以下具有局部非线性扰动的 Choquard 方程： −Δu+u=Iα∗|u|α2+1|u|α2−1u+f(u),x∈R2;u∈H1(R2) , $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} - {\it\Delta} u+u=\left(I_{\alpha}*|u|^{\ frac{\alpha}{2}+1}\right)|u|^{\frac{\alpha}{2}-1}u +f(u), & x\in \mathbb{R}^2; \\ u\in H^1(\mathbb{R}^2), \end{array} \right。\end{array}$$ 其中 α ∈ (0, 2), Iα : ℝ2 → ℝ 是 Riesz 势，而 f ∈ 𝓒(ℝ, ℝ) 是 Trudinger-Moser 意义上的临界指数增长。指数 α2+1 $\begin{array}{} \displaystyle \frac{\alpha}{2}+1 \end{array}$ 是 Hardy-Littlewood-Sobolev 不等式的关键。我们在双重临界情况下获得上述方程的非平凡解或 Nehari 型基态解的存在性，即