Limiting profile of solutions for sublinear elliptic equations with shrinking self-focusing core

ABSTRACT

This paper is concerned with the following problem: $\{\begin{array}{ll}-\mathrm{\Delta }u+u=Q_{\epsilon }|u{|}^{p-2}u& \mathrm{in}{\mathbb{R}}^N,\\ u(x)\to 0,& \mathrm{as}|x|\to \mathrm{\infty },\end{array}$where $p\in (1,2)$, $Q_{\epsilon }\in L^{\mathrm{\infty }}({\mathbb{R}}^N)$ and the self-focusing core supp $Q_{\epsilon }^{+}$ shrinks to a point as $\epsilon \to 0$. We show that there is a least energy solution $u_{\epsilon }$ corresponding to $Q_{\epsilon }$ and then investigate the limiting profile of concentration for $u_{\epsilon }$ as $\epsilon \to 0$. The results are supplements to the works of [Ackermann and Szulkin. A concentration phenomenon for semilinear elliptic equations. Arch Ration Mech Anal. 2013;207(3):1075–1089; Fang and Wang. Limiting profile of solutions for Schrödinger equations with shrinking self-focusing core. Calc Var Partial Differential Equations. 2020;59(4):Paper No. 129, 18].

KEYWORDS:

• Sublinear elliptic equation
• self-focusing core
• limiting profile

2020 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35J60
• 35J20

Disclosure statement

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

Additional information

Funding

This work is partially supported by National Natural Science Foundation of China (Grant Nos. 12071266, 12101376, 12026218, 11801338) and Research Project Supported by Shanxi Scholarship Council of China (2020-005).