The Fredholm alternative at the first eigenvalue for the one dimensional p-Laplacian

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:151

Article

del Pino, M ; Drábek, P ; Manásevich, R

关键词: one-dimensional p-Laplacian;    resonance;    Fredholm alternative;    Leray-Schauder degree;    upper and lower solutions;    refined asymptotics;   

DOI  :  10.1006/jdeq.1998.3506

来源: Elsevier

【 摘 要 】

In this work we study the range of the operator u bar right arrow (\u'\(p-2) u')' + lambda(1) \u\(p-2) u, u(0) = u(T) = 0, p > 1. We prove that all functions h is an element of C-1[0. T] satisfying integral(0)(T)h(t) sin(p)(pi(p)t/T) dt = 0 lie in the range, but that if p not equal 2 and h not equal 0 the solution set is bounded. Here sin(pi(p)t/T) is a first eigenfunction associated to lambda(1). We also show that in this case the associated energy functional u bar right arrow (1/p) integral(0)(T)\u'\(p) - (lambda(1)/p) integral(0)(T)\u\(p) + integral(0)(T) hu is unbounded from below if 1 < p < 2 and bounded from below (with a global minimizer) if p > 2, on W-0(t.p)(0, T) (lambda(1) corresponds precisely to the best constant in the L-p-Poincare inequality). Moreover, we show that unlike the linear case p = 2, for p not equal 2 the range contains a nonempty open set in L-infinity(0, T). (C) 1999 Academic Press.

【 授权许可】

Free