Infinitely Many Solutions for Schrödinger-Choquard Equation with Critical Exponential Growth in $\mathbb {R}^{N}$

Author(s):  
Hongxue Song ◽  
Caisheng Chen
2020 ◽  
Vol 10 (1) ◽  
pp. 152-171
Author(s):  
Sitong Chen ◽  
Xianhua Tang ◽  
Jiuyang Wei

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).


2015 ◽  
Vol 15 (4) ◽  
Author(s):  
João Marcos do Ó ◽  
Abiel Costa Macedo

AbstractIn this paper we give a new Adams type inequality for the Sobolev space W(−Δ)where the nonlinearity is “superlinear” and has critical exponential growth at infinite.


2013 ◽  
Vol 13 (2) ◽  
Author(s):  
Nguyen Lam ◽  
Guozhen Lu

AbstractLet Ω be a bounded domain in ℝwhen f is of subcritical or critical exponential growth. This nonlinearity is motivated by the Moser-Trudinger inequality. In fact, we will prove the existence of a nontrivial nonnegative solution to (0.1) without the Ambrosetti-Rabinowitz (AR) condition. Earlier works in the literature on the existence of nontrivial solutions to N−Laplacian in ℝ


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