On a two-dimensional elliptic problem with large exponent in nonlinearity

A semilinear elliptic equation on a bounded domain in R2 with large exponent in the nonlinear term is studied in this paper. We investigate positive solutions obtained by the variational method. It turns put that the con- strained minimizing problem possesses nice asymptotic behavior as the nonlin- ear exponent, serving as a parameter, gets large. We shall prove that cp , the minimum of energy functional with the nonlinear exponent equal to p , is like (&Ke)lf2p~^2 as p tends to infinity. Using this result, we shall prove that the variational solutions remain bound- ed uniformly in p . As p tends to infinity, the solutions develop one or two peaks. Precisely the solutions approach zero except at one or two points where they stay away from zero and bounded from above. Then we consider the problem on a special class of domains. It turns out that the solutions then develop only one peak. For these domains, the solutions enlarged by a suitable quantity behave like a Green's function of -A. In this case we shall also prove that the peaks must appear at a critical point of the Robin function of the domain.