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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.
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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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