On a Semilinear Elliptic Equation in R2 When the Exponent Approaches Infinity

We consider a semilinear elliptic equation in R2 with the nonlinear exponent approaching infinity. In contrast to the blow-up behavior of the corresponding problem in Rn with n ≥ 3, the L∞(R2) norms of the solutions to the equation in R2 remain bounded from below and above. After a careful study on the decay rates of several quantities, we prove that the normalized solutions will approach the fundamental solution of −Δ + 1 in R2. So as the exponent tends to infinity, the solutions to the problem look more and more like a peak.