期刊论文

Ground State Solutions for Some Kirchhoff Type Problems with Critical Growth in ${{\mathbb{R}^3}}$

Shen, L.J.

中文

数学学报：中文版

0583-1431

2018

61

2

197-216

10.3969/j.issn.0583-1431.2018.02.002

国家自然科学基金资助项目（11371158） ***与创新团队发展计划资助项目（IRT13066）

本校为第一且通讯机构

本文主要考虑如下Kirchhoff问题$$\left\{ \begin{array}{l} - \left({a + b\int_{{\mathbb{R}^3}} {{{\left| {\nabla u} \right|}^2}dx} } \right)\Delta u + u = f\left({x,u} \right) + Q\left(x \right){\left| u \right|^4}u,\\u \in {H^1}\left({{\mathbb{R}^3}} \right),\end{array} \right.$$其中$a$,$b$是正的常数.我们证明了基态解,即上述问题的极小能量解的存在性.同时,如果假定$Q \equiv 1$,且$h\left(x \right)$满足一定的条件,可以证明下述问题$$\left\{ \begin{array}{l} - \left({a + b\int_{{\mathbb{R}^3}} {{{\left| {\nabla u} \right|}^2}dx} } \right)\Delta u + u = {\...

We are concerned with a class of Kirchhoff problem [Formula is presented] where a, b > 0 are constants. The existence of ground state solutions, i.e., nontrivial solutions with least possible energy of this Kirchhoff problem is obtained. Moreover, when Q ≡ 1, under suitable conditions on h(x), we prove the existence of ground state solutions for the the following Kirchhoff problem [Formula is presented]. © 2018,...