On the Fredholm alternative for the p-Laplacian at the first eigenvalue
Peter Takáč · Indiana University Mathematics Journal · 2002
We investigate the existence of a weak solution u ∈ W 1,p 0 (Ω) to the degenerate quasi-linear Dirichlet boundary value problem (P) -Δ p u = λ 1 |u| p-2 u + f(x) in Ω; u = 0 on ∂Ω. It is assumed that 1 < p < ∞, p ¬= 2, Ω is a bounded domain in R N , f ∈ L∞ (Q) is a given function, and the number λ 1 stands for the first (smallest) eigenvalue of the positive p-Laplacian -Ap, where Δ p u ≡ div(|⊇u| p-2 ⊇u). The eigenvalue λ 1 being simple, let φ 1 denote the eigenfunction associated with λ 1 . We show the existence of a solution for problem (P) when f satisfies the orthogonality condition ⊃ Ω fφ 1 dx = 0, in which case the set of solutions is bounded in C 1 (Ω) provided p ¬= 2 and f ≢ 0 in Ω. A key role in our proofs is played by a second-order Taylor formula in its integral form (near φ 1 ) which contains certain Gâteaux 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 ¬= 2) follow.