We investigate the existence and multiplicity of weak solutions u 2 W 1;p 0 ðOÞ to the degenerate quasilinear Dirichlet boundary value problem
where z 2 R is a parameter. It is assumed that 1opo1; p=2; and O is a bounded domain in R N : The number l 1 stands for the first (smallest) eigenvalue of the positive p-Laplacian ÀD p ; where D p u divðjruj pÀ2 ruÞ: The eigenvalue l 1 being simple, let j 1 denote the eigenfunction associated with l 1 : Furthermore, f T 2 L 1 ðOÞ is a given function which is assumed to be L 2 -orthogonal to j 1 and f T c0 in O: We show the existence of a solution for problem (P) precisely when the parameter z satisfies z n 4 z4z n ; for some numbers À1oz n o0oz n o1 depending on f T : Otherwise, nonexistence occurs. Moreover, we show that problem (P) possesses at least two distinct solutions provided z # ozoz # and z=0; for some numbers z # and z # such that z n 4z # o0oz # 4z n : Finally, given any d > 0; we show that the set of all weak solutions to problem (P) is bounded in C 1 ð % O OÞ uniformly for jzj5d and for z ¼ 0 as well. Precise asymptotic behavior (blow-up) of every solution is given as z ! 0 (z=0) using the linearization of equation (P) about j 1 : # 2002 Elsevier Science (USA)