On existence and asymptotic behavior of solutions of elliptic equations with nearly critical exponent and singular coefficients

• Published in: Electronic journal of qualitative theory of differential equations Vol. 2021; no. 64; pp. 1 - 25
• Main Authors: Li, Shiyu, Wei, Gongming, Duan, Xueliang
• Format: Journal Article
• Language: English
• Published: University of Szeged 01.01.2021
• Subjects: asymptotic behavior; critical sobolev exponent; hardy exponents; singular coefficient
• ISSN: 1417-3875; 1417-3875
• DOI: 10.14232/ejqtde.2021.1.64

In this paper we study the existence and asymptotic behavior of solutions of $$-\Delta u=\mu\frac{u}{|x|^{2}}+|x|^{\alpha}u^{p(\alpha)-1-\varepsilon},\qquad u>0 \ \text{in}\ B_{R}(0)$$ with Dirichlet boundary condition. Here, $-2<\alpha<0$, $p(\alpha)=\frac{2(N+\alpha)}{N-2}$, $0<\varepsilon<p(\alpha)-1$ and $p(\alpha)-1-\varepsilon$ is a nearly critical exponent. We combine variational arguments with the moving plane method to prove the existence of a positive radial solution. Moreover, the asymptotic behaviour of the solutions, as $\varepsilon\to0$, is studied by using ODE techniques.