On the number and structure of solutions for a fredholm alternative with the p-Laplacian

[摘要] We investigate the existence and multiplicity of weak solutions u is an element of W-0(,1p) (Omega) to the degenerate quasilinear Dirichlet boundary value problem (P) - Delta(p)u = lambda(1) \u\(p-2)u + integral(T)(x) + zeta (.) phi(1) (x) in Omega; u = - on partial derivativeOmega where e R is a parameter. It is assumed that 1 < infinity, p not equal 2, and Omega is a bounded domain in R-N. The number l stands for the first (smallest) eigenvalue of the positive p-Laplacian -Delta(p), where Delta(p)u equivalent to div(vertical bar del u vertical bar(p-2)del u). The eigenvalue lambda(1) being simple, let phi(1) denote the eigenfunction associated with lambda(1). Furthermore, f(T) is an element of L-infinity(Omega) is a given kfunction which is assumed to be L-2-orthogonal to phi(1) and f(T) not equivalent to 0 in Omega. We show the existence of a solution for problem (P) precisely when the parameter zeta satisfies zeta(*) <= zeta <= zeta*, for some numbers -infinity < 0 < 0 < zeta(#) <= zeta*. Finally, given any delta > 0, we show that the set of all weak solutions to problem (P) is bounded in C-1(Q) uniformly for vertical bar zeta vertical bar >= delta and for zeta = 0 as well. Precise asymptotic behavior (blow-up) of every solution is given as zeta -> 0 (zeta not equal 0) using the linearization of equation (P) about phi(1.) (c) 2002 Elsexier Science (USA).

[发布日期] 2002-10-10

[关键词] nonlinear eigenvalue problem;Fredholm alternative;degenerate or singular quasilinear Dirichlet problem;p-Laplacian;global minimizer;second-order Taylor formula