Strongly nonlinear multivalued elliptic equations on a bounded domain

A Berestycki–Lions type result and applications

“…A rich literature is available by now on problems with discontinuous nonlinearities, and we refer the reader to Alves, Bertone and Gonçalves [1], Alves and Bertone [2], Alves, Gonçalves and Santos [3], Ambrosetti and Turner [9], Ambrosetti, Calahorrano and Dobarro [10], Badiale and Tarantelo [11], Carl, Le and Motreanu [21], Clarke [26], Chang [19], Carl and Dietrich [22], Carl and S. Heikkila [23,24], Cerami [25], Hu, Kourogenis and Papageorgiou [40], Montreanu and Vargas [44], Radulescu [47] and their references. Several techniques have been developed or applied in their study, such as variational methods for nondifferentiable functionals, lower and upper solutions, global branching, fixed point theorem, and the theory of multivalued mappings.…”

Multiple solutions for a inclusion quasilinear problem with non-homogeneous boundary condition through Orlicz Sobolev spaces

“…are well-defined locally Lipschitz functionals. Moreover, due to Alves, Gonçalves and Santos [2], we have…”

“…Problems from the latter had been treated by several authors, such as Pucci and Serrin [24], Ricceri [26,27], Marano and Motreanu [20,22], Arcoya and Carmona [4], Bonanno [5,6], Gasiński and Papageorgiou [16], Kristály [19], Bonanno and Candito [7], Alves and Nascimento [3]. More recently, Alves, Gonçalves and Santos [2] established existence of nontrivial solutions for the problem −div(φ(|∇u|)∇u) − b(u)u ∈ λ∂F (x, u) in Ω, where λ > 0 is a parameter and φ : [0, +∞) → [0, +∞) is a C 1 -function satisfying (φ 1 ) lim s→0 + sφ(s) = 0 and lim s→+∞ sφ(s) = +∞, (φ 2 ) s → sφ(s) is increasing in [0, ∞), (φ 3 ) ℓ ≤ φ(t)t 2 Φ(t) ≤ m, t > 0, l, m > 0 and Φ(t) = |t| 0 sφ(s)ds, where b is a continous function and F , locally Lipschitz.…”

Multiple solutions for a problem with discontinuous nonlinearity

“…A rich literature is available by now on problems with discontinuous nonlinearities, and we refer the reader to Ambrosetti and Turner [2], Ambrosetti et al [5], Alves et al [6], Alves and Bertone [7], Alves et al [8], Badiale and Tarantelo [12], Carl et al [16], Clarke [17], Chang [18], Carl and Dietrich [21], Carl and Heikkila [22,23], Cerami [24], Hu et al [25], Montreanu and Vargas [27], Radulescu [28] and their references. Several techniques have been developed or applied in their study, such as variational methods for nondifferentiable functionals, lower and upper solutions, global branching, fixed point theorem, and the theory of multivalued mappings.…”