Multi-peak Nodal Solutions for a Two-dimensional Elliptic Problem with Large Exponent in Weighted Nonlinearity

We consider the boundary value problem Delta u + vertical bar x vertical bar(2 alpha)vertical bar u vertical bar(p-1)u = 0, 1 < alpha not equal 0, in the unit ball B with the homogeneous Dirichlet boundary condition, when p is a large exponent. By a constructive way, we prove that for any positive integer m, there exists a multi-peak nodal solution up whose maxima and minima are located alternately near the origin and the other m points <(q(l))over tilde> = (lambda cos 2 pi(l-1)/m, lambda sin 2 pi(l-1)/m), l = 2, ..., m + 1, such that as p goes to +infinity, p vertical bar x vertical bar(2 alpha)vertical bar u(p)vertical bar(p-1)u(p) -> 8 pi e(1 + alpha)delta(0) + Sigma(m+1)(l=2) 8 pi e(-1)(l-1) delta(ql), where lambda is an element of(0, 1), m is an odd number with (1 + alpha)(m + 2)-1 > 0, or m is an even number. The same techniques lead also to a more general result on general domains.