论文标题
存在具有关键choquard非线性的一类准线性schrödinger方程的积极解决方案
Existence of positive solutions for a class of quasilinear Schrödinger equations with critical Choquard nonlinearity
论文作者
论文摘要
本文涉及以下sorilinearschrödingerChoquard方程的积极弱解决方案： \ begin {equation*} \ begin {array} {cc} \ displaystyle -div（g^2（u）\ nabla u） + g（u）g'（u）\ nabla u + a（x）u = k（x，x，u）\; \ text {in} \; \ mathbb {r}^n， \ end {array} \ end {equation*} 其中$ n \ geq 3 $，$ \ displayStyle k（x，u）：= h（x，u） +（i _ {\ vartheta}*| u |^|^{α\ cdot2^*_μm \ Mathbb {r}^+$是一个可区分的函数，$ g（0）= 1 $和$ g'（t）\ geq 0 $ for All $ t \ geq 0 $; $ h \ in C（\ mathbb {r}^n \ times \ times \ mathbb {r}，\ mathbb {r}）$和c（\ mathbb {r}^n，\ mathbb {r}）$ in c（\ mathbb {r}^n，\ mathbb {r}^n，\ mathbb {r}）$。我们使用在$ g $，$ h $和$ a $上的适当假设下的变量和变异方法的更改建立了积极解决方案的存在。
This article is concerned with the existence of positive weak solutions for the following quasilinear Schrödinger Choquard equation: \begin{equation*} \begin{array}{cc} \displaystyle -div(g^2(u)\nabla u) + g(u)g'(u)\nabla u + a(x) u = k(x, u) \;\text{in} \; \mathbb{R}^N, \end{array} \end{equation*} where $N \geq 3$, $\displaystyle k(x,u) := h(x,u) + (I_{\vartheta}*|u|^{α\cdot2^*_μ})|u|^{α\cdot2^*_μ-2}u$, $g : \mathbb{R} \to \mathbb{R}^+$ is a differentiable even function with $g(0) = 1$ and $g'(t) \geq 0$ for all $t \geq 0$; $h\in C( \mathbb{R}^N \times\mathbb{R}, \mathbb{R})$ and the potential $a \in C( \mathbb{R}^N, \mathbb{R})$. We establish the existence of a positive solution using the change of variable and variational methods under appropriate assumptions on $g$, $h$ and $a$.
