Concentration profile, energy, and weak limits of radial solutions to semilinear elliptic equations with Trudinger-Moser critical nonlinearities

Abstract

We investigate the next Trudinger-Moser critical equations, \[ \begin{cases} -Δu=λue^{u^2+α|u|^β}&\text{ in }B,\\ u=0&\text{ on }\partial B, \end{cases} \] where $\alpha >0$, $(\lambda ,\beta )\in (0,\infty )\times (0,2)$ and $B\subset {\mathbb{R}}^2$ is the unit ball centered at the origin. We classify the asymptotic behavior of energy bounded sequences of radial solutions. Via the blow--up analysis and a scaling technique, we deduce the limit profile, energy, and several asymptotic formulas of concentrating solutions together with precise information of the weak limit. In particular, we obtain a new necessary condition on the amplitude of the weak limit at the concentration point. This gives a proof of the conjecture by Grossi-Mancini-Naimen-Pistoia in 2020 in the radial case. Moreover, in the case of $\beta \le 1$, we show that any sequence carries at most one bubble. This allows a new proof of the nonexistence of low energy nodal radial solutions for $(\lambda ,\beta )$ in a suitable range. Lastly, we discuss several counterparts of our classification result. Especially, we prove the existence of a sequence of solutions which carries multiple bubbles and weakly converges to a sign-changing solution.