Existence and concentration of ground state solutions for a quasilinear Choquard equation with critical exponential growth in $\mathbb{R}^2$
首发时间:2024-05-31
Abstract:In this paper, we study the following perturbed quasilinear Choquard equation \begin{equation*} -\varepsilon^2\Delta u+V(x)u-\varepsilon^2\Delta (u^2)u=\varepsilon^{\mu-2}\big(\frac{1}{|x|^\mu}*F(u)\big)f(u),\quad x\in \ \mathbb{R}^2, \end{equation*} where $\varepsilon>0$ is a small parameter, $\frac{1}{|x|^\mu}$ with $0<\mu <2$ is the Riesz potential, $*$ is the convolution in $\mathbb{R}^2$, $V(x)\in C(\mathbb{R}^2, (0,+\infty))$, $F(u)$ is the primitive function of $f(u)$ and $f$ has critical exponential growth with respect to the Trudinger–Moser inequality. When $V$ verifies some assumptions, we apply variational methods and mountain pass theorem to obtain the existence and concentration behavior of positive ground state solutions for the above equation.
keywords: Quasilinear Choquard equation Critical exponential growth Trudinger–Moser inequality
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$\mathbb{R}^2$中带临界指数增长的拟线性Choquard方程基态解的存在性和集中性
摘要:本文我们研究下列带有扰动的拟线性Choquard方程 \begin{equation*} -\varepsilon^2\Delta u+V(x)u-\varepsilon^2\Delta (u^2)u=\varepsilon^{\mu-2}\big(\frac{1}{|x|^\mu}*F(u)\big)f(u),\quad x\in \ \mathbb{R}^2, \end{equation*} 其中$\varepsilon>0$是小的参数, $\frac{1}{|x|^\mu}$是里斯位势, $0<\mu <2$, $*$是$\mathbb{R}^2$中的卷积, $V(x)\in C(\mathbb{R}^2, (0,+\infty))$, $F(u)$是$f(u)$的原函数, $f$具有关于Trudinger–Moser不等式的临界指数增长. 在$V$满足一定的条件下, 我们运用变分法和山路定理, 得到了上述问题基态解的存在性和集中性.
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$\mathbb{R}^2$中带临界指数增长的拟线性Choquard方程基态解的存在性和集中性
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