Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 114, pp. 1-17.

Title: Asymptotic behavior of positive radial solutions to elliptic equations 
       approaching critical growth
       
Authors: Rosa Pardo (Univ. Complutense de Madrid, Madrid, Spain)
         Arturo Sanjuan (Univ. Distrital Francisco Jose de Caldas, Bogota, Colombia)

Abstract:
 We study the asymptotic behavior of radially symmetric solutions
 to the subcritical semilinear elliptic problem
 $$\displaylines{
 -\Delta u = u^{\frac{N+2}{N-2}}/[\log(e+u)]^{\alpha}\quad \text{in }
 \Omega=B_R(0)\subset\mathbb{R}^N,\cr
 u>0,\quad  \text{in }  \Omega,\cr
 u=0,\quad  \text{on }  \partial \Omega,
 }$$
 as $\alpha\to 0^+$.
 Using asymptotic estimates, we prove that there exists an explicitly defined constant
 L(N,R)>0, only depending on N and R, such that
 $$
 \limsup_{\alpha\to0^+} \frac{\alpha u_\alpha (0)^2}
 {[\log(e+u_\alpha (0))]^{1+\frac{\alpha(N+2)}{2}}} 
 \leq L(N,R) 
 \le 2^*\liminf_{\alpha\to0^+}\frac{\alpha u_\alpha (0)^2}
 {[\log(e+u_\alpha (0))]^{\frac{\alpha(N-4)}2}}.
 $$

 
Submitted November 11, 2019. Published November 18, 2020.
Math Subject Classifications: 35B33, 35B45, 35B09, 35J60.
Key Words: A priori bounds; positive solutions; semilinear elliptic equations;
           Dirichlet boundary conditions; growth estimates; subcritical nonlinearites.