ON A 2-DIMENSIONAL ELLIPTIC PROBLEM WITH LARGE EXPONENT IN NONLINEARITY

A semilinear elliptic equation on a bounded domain in R2 with large ex ponent in the nonlinear term is studied in this paper. We investigate positive solutions obtained by the variational method. It turns out th at the constrained minimizing problem Possesses nice asymptotic behavi or as the nonlinear exponent, serving as a parameter, gets large. We s hall Prove that c(p), the minimum of energy functional with the nonlin ear exponent equal to p, is like (8pie)1/2 p-1/2 as p tends to infinit y. Using this result, we shall prove that the variational solutions re main bounded uniformly in p. As p tends to infinity, the solutions dev elop one or two peaks. Precisely the solutions approach zero except at one or two points where they stay away from zero and bounded from abo ve. Then we consider the problem on a special class of domains. It tur ns out that the solutions then develop only one peak. For these domain s, the solutions enlarged by a suitable quantity behave like a Green's function of -DELTA. In this case we shall also prove that the peaks m ust appear at a critical point of the Robin function of the domain.