Asymptotic behavior of sign-changing radial solutions of a semilinear elliptic equation in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^2 $\end{document}</tex-math></inline-formula> when exponent approaches <inline-formula><tex-math id="M2">\begin{document}$ +\infty $\end{document}</tex-math></inline-formula>

Abstract

In this paper, we consider a semi-linear elliptic equation in R 2 with the nonlinear exponent approaching infinity. We study asymptotic behavior of sign-changing once radial solutions obtained by . Assuming up(0) > 0, we prove that a suitable rescaling of the positive part u + p converges to the unique regular solution of Liouville equation in R 2 , while a suitable rescaling of the negative part u − p converges to a solution of a singular Liouville equation in R 2 . We also obtain the asymptotic value of the L ∞ -norms of u − p and u + p . Moreover, we show that pup blow up at the origin and pup convergence to the fundamental solution of −∆ + 1 in R 2 (up to a multiplier).