Generalization of Fredholm alternative for nonlinear differential operators

On a nonresonance condition between the first and the second eigenvalues for the p‐Laplacian

“…In the quasilinear case p ≠ 2, (1.1) was investigated for N = 1 in [6] and for N ≥ 1 in [3]. In this latter work nonresonance is studied at the left of λ 1 .…”

Existence theorems for elliptic hemivariational inequalities involving the p‐Laplacian

“…In the smooth case, a generalization of the PS-condition was introduced by Cerami [4] and Bartolo et al [3] showed that this more general condition suffices to prove a deformation theorem and then using it to prove minimax theorems locating critical points of C 1 -energy functionals. In the context of the nonsmooth theory this was done by Kourogenis and Papageorgiou [13] who introduced the so-called "nonsmooth C-condition" which says the following: "A locally Lipschitz function f : X → R satisfies the "nonsmooth C-condition," if any sequence {x n } n≥1 ⊆ X along which { f (x n )} n≥1 is bounded and (1 + x n )m(x n ) → 0 as n → ∞ (here m(x n ) is as before), has a strongly convergent subsequence."…”

“…Recently, semilinear (i.e., p = 2) hemivariational inequalities were studied by Goeleven et al [10] and Gasiński and Papageorgiou [9]. Quasilinear problems with the p-Laplacian and a C 1 potential function were studied by Arcoya and Orsina [2], Boccardo et al [3], Costa and Magalhães [7], and Hachimi and Gossez [8].…”

Existence theorems for quasilinear multivalued boundary value problems in R N

“…Recall that x j j j j p 1 1 x H j j j j p for all x P W 1Yp 0 TY R N (see Boccardo et al [3]). Thus we have`v…”

“…In this paper we study quasilinear, second order di erential inclusions in R N with Dirichlet boundary conditions. Problems of this kind for scalar ordinary di erential equations were studied by Boccardo-Drabek-GiachettiKucera [3] and Pino-Elgueta-Manasevich [14], using degree theoretic techniques. Here in addition to the vectorial and multivalued character of the problem, we also propose a di erent approach based on the theory of multivalued operators of monotone type.…”