Journal
INDIANA UNIVERSITY MATHEMATICS JOURNAL
Volume 51, Issue 1, Pages 187-237Publisher
INDIANA UNIV MATH JOURNAL
DOI: 10.1512/iumj.2002.51.2156
Keywords
nonlinear eigenvalue problem; Fredholm alternative; degenerate or singular quasilinear Dirichlet problem; p-Laplacian; global minimizer; second-order Taylor formula
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We investigate the existence of a weak solution u is an element of W-0(1,p) (Omega) to the degenerate quasi-linear Dirichlet boundary value problem (p) -Delta(p)u = lambda(1)\u\(p-2)u + f(x) in Omega; u = 0 on partial derivativeOmega. It is assumed that 1 < p < infinity, p not equal 2, Omega is a bounded domain in R-N, f is an element of L-infinity (Omega) is a given function, and the number lambda(1) stands for the first (smallest) eigenvalue of the positive p-Laplacian -Delta(p), where Delta(p)u equivalent to div(\delu\(p-2)delu). The eigenvalue lambda(1) being simple, let phi(1) denote the eigenfunction associated with lambda(1). We show die existence of a solution for problem (P) when f satisfies the orthogonality condition integral(Omega) fphi(1) dx = 0, in which case the set of solutions is bounded in C-1((Omega) over bar) provided p not equal 2 and f not equivalent to 0 in Omega. A key role in our proofs is played by a second-order Taylor formula in its integral form (near phi(1)) which contains certain Gateaux derivatives and a positive semidefinite quadratic form. This quadratic form compensates for lack of coercivity in the energy functional corresponding to problem (P). When combined with well-known regularity results, its positive semidefiniteness renders a priori estimates from which existence and boundedness (for p not equal 2) follow.
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