COUNTING PEAKS OF SOLUTIONS TO SOME QUASI-LINEAR ELLIPTIC-EQUATIONS WITH LARGE EXPONENTS

We consider the asymptotic behavior of certain solutions to a quasilin ear 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 L(1)-estimate for the N- Laplacian operators in R(N). Using these estimates we prove that the s olutions obtained by the standard variational method will develop a sp iky 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. Stron ger results for some special convex domains acid some special solution s are also achieved. (C) 1995 Academic Press, Inc.