Uniqueness of asymptotic limit of ground states for a class of quasilinear Schrödinger equation with H¹-critical growth in ℝ³

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.

Keywords:

• Quasilinear elliptic equation
• asymptotic behavior
• variational method

2010 Mathematics Subject Classifications:

• 35J62
• 35J20
• 35B40

Acknowledgments

The authors are supported by JSPS Grant-in-Aid for Scientific Research (C) (No. 18K03383, No. 18K03356, No. 18K03362). The first author is supported by JSPS-NSFC joint research project ‘Variational study of nonlinear PDEs’.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors are supported by JSPS Grant-in-Aid for Scientific Research (C) (No. 18K03383, No. 18K03356, No. 18K03362). The first author is supported by JSPS-NSFC joint research project ‘Variational study of nonlinear PDEs’.