论文标题
$ \ mathbb {r}^n $中的quasilinear对数方程式具有指数增长
Quasilinear logarithmic Choquard equations with exponential growth in $\mathbb{R}^N$
论文作者
论文摘要
我们认为$ n $ laplacianschrödinger方程与高阶分数泊松方程式强烈结合。当Riesz电位$α$的顺序等于Euclidean dimension $ n $，因此它是对数时，系统原来等同于非局部Choquard类型方程。一方面，schrödinger能量的自然函数空间设置是sobolev限制空间$ w^{1，n}（\ mathbb {r}^n）$，其中最大非线性生长为指数类型。另一方面，为了使非局部能量定义良好并证明存在有限的能量解决方案，我们引入了Pohozaev-trudinger不平等的合适的$ log $ WEATER变体，该变体提供了适当的功能框架，我们使用变异方法。
We consider the $N$-Laplacian Schrödinger equation strongly coupled with higher order fractional Poisson's equations. When the order of the Riesz potential $α$ is equal to the Euclidean dimension $N$, and thus it is a logarithm, the system turns out to be equivalent to a nonlocal Choquard type equation. On the one hand, the natural function space setting in which the Schrödinger energy is well defined is the Sobolev limiting space $W^{1,N}(\mathbb{R}^N)$, where the maximal nonlinear growth is of exponential type. On the other hand, in order to have the nonlocal energy well defined and prove the existence of finite energy solutions, we introduce a suitable $log$-weighted variant of the Pohozaev-Trudinger inequality which provides a proper functional framework where we use variational methods.
