Counting Peaks of Solutions to Some Quasilinear Elliptic Equations with Large Exponents

We consider the asymptotic behavior of certain solutions to a quasilinear problem with large exponent in the nonlinearity. Starting with the investigation of a Sobolev embedding, we get a sharp estimate for the embedding constant. Then we obtain a crucial L1-estimate for the N-Laplacian operators in RN. Using these estimates we prove that the solutions obtained by the standard variational method will develop a spiky pattern of peaks as the nonlinear exponent gets large, and we also have an upper bound depending on N only of the number of peaks. Stronger results for some special convex domains acid some special solutions are also achieved.