Asymptotic behavior of positive radial solutions to elliptic equations approaching critical growth

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 $$ \limsup_{\alpha\to0^+} \frac{\alpha u_\alpha (0)^2} {[\log(e+u_\alpha (0))]^{1+\frac{\alpha(N+2)}{2}}} \leq L(N,R) \le 2^*\liminf_{\alpha\to0^+}\frac{\alpha u_\alpha (0)^2} {[\log(e+u_\alpha (0))]^{\frac{\alpha(N-4)}2}}. $$

Keywords

• a priori bounds
• positive solutions
• semilinear elliptic equations
• dirichlet boundary conditions
• growth estimates
• subcritical nonlinearites