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

Abstract

A semilinear elliptic equation on a bounded domain in R 2 {R^2} 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 constrained minimizing problem possesses nice asymptotic behavior as the nonlinear exponent, serving as a parameter, gets large. We shall prove that c p {c_p} , the minimum of energy functional with the nonlinear exponent equal to p, is like ( 8 π e ) 1 / 2 p − 1 / 2 {(8\pi e)^{1/2}}{p^{ - 1/2}} as p tends to infinity. Using this result, we shall prove that the variational solutions remain bounded 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 − Δ - \Delta . In this case we shall also prove that the peaks must appear at a critical point of the Robin function of the domain.