ABSTRACT

We are interested in the asymptotic behavior of ground states for a class of quasilinear elliptic equations in ${\mathbb{R}}^3$ when the nonlinear term has $H^1$-critical growth. In the previous result [Adachi et al. Asymptotic property of ground states for a class of quasilinear Schrödinger equation with $H^1$-critical growth. Calc Var Partial Differential Equations. 2019;58(3). Art. 88, 29 pp.], it was shown that, after a suitable scaling, the ground state converges to the Talenti function. However, the uniqueness of the limit of the full sequence was not obtained, which was essentially owning to the fact that the Talenti function does not belong to $L^2\left({\mathbb{R}}^3\right)$. In this paper, by constructing a refined test function and performing a detailed asymptotic analysis, we are able to obtain the uniqueness of asymptotic limit of ground states.

摘要

我们对一类拟线性椭圆方程的基态的渐近行为感兴趣${\mathbb{R}}^3$当非线性项有$H^1$- 临界增长。在之前的结果中 [Adachi et al. 一类拟线性薛定谔方程的基态渐近性质$H^1$- 临界增长。Calc Var 偏微分方程。2019；58(3)。艺术。88, 29 pp.]，结果表明，经过适当的缩放后，基态收敛到 Talenti 函数。但是没有得到全序列极限的唯一性，这本质上是由于Talenti函数不属于${\mathrm{大号}}^2\left({\mathbb{R}}^3\right)$. 在本文中，通过构造一个细化的测试函数并进行详细的渐近分析，我们能够得到基态渐近极限的唯一性。