论文标题
存在针对双相问题的基态解决方案
Existence of ground state solutions for a Choquard double phase problem
论文作者
论文摘要
在本文中，我们研究了由双相操作员驱动的QuasilIrinear椭圆方程，该方程涉及该形式 \ begin {align*} - \ Mathcal {l} _ { u）} {| x-y |^μ} \，\ mathrm {d} y \ right）f（x，u） \ quad \ text {in} \ mathbb {r}^n， \ end {align*} 其中$ \ Mathcal {l} _ {p，q}^{a} $是双相操作员 \ begin {align*} \ Mathcal {l} _ { w^{1，\ Mathcal {h}}（\ Mathbb {r}^n）， \ end {align*} $0。 $f\colon\mathbb{R}^N\times\mathbb{R}\to\mathbb{R}$ is a continuous function that satisfies a subcritical growth. Based on the Hardy-Littlewood-Sobolev inequality, the Nehari manifold and variational tools, we prove the existence of ground state solutions of such problems under different assumptions on the data.
In this paper we study quasilinear elliptic equations driven by the double phase operator involving a Choquard term of the form \begin{align*} -\mathcal{L}_{p,q}^{a}(u) + |u|^{p-2}u+ a(x) |u|^{q-2}u = \left( \int_{\mathbb{R}^N} \frac{F(y, u)}{|x-y|^μ}\,\mathrm{d} y\right)f(x,u) \quad\text{in } \mathbb{R}^N, \end{align*} where $\mathcal{L}_{p,q}^{a}$ is the double phase operator given by \begin{align*} \mathcal{L}_{p,q}^{a}(u):= \operatorname{div}\big(|\nabla u|^{p-2}\nabla u + a(x) |\nabla u|^{q-2}\nabla u \big), \quad u\in W^{1,\mathcal{H}}(\mathbb{R}^N), \end{align*} $0<μ<N$, $1<p<N$, $p<q<p+ \frac{αp}{N}$, $0 \leq a(\cdot)\in C^{0,α}(\mathbb{R}^N)$ with $α\in (0,1]$ and $f\colon\mathbb{R}^N\times\mathbb{R}\to\mathbb{R}$ is a continuous function that satisfies a subcritical growth. Based on the Hardy-Littlewood-Sobolev inequality, the Nehari manifold and variational tools, we prove the existence of ground state solutions of such problems under different assumptions on the data.
