Abstract
This 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).
Infinitely many solutions for quasilinear Schrödinger equation with critical exponential growth in RN
