Multiplicity For Nonlinear Elliptic Boundary Value Problems Of P-Laplacian Type Without Ambrosetti-Rabinowitz Condition

In this paper , we study the existence of multiple solutions to the following nonlinear elliptic boundary value problem of p－Laplacian typewhere 1 < p <∞,Ω(?)R^N is a bounded smooth domain, (?)is the p－Laplacian of u , and f:Ω×R→R satisfies (?)uniformly with respect to x∈Ω, and l is not an eigenvalue of -Δ_p in W₀^1,p(Ω) and some other structure conditions.Under suitable assumptions on f(x,t) , we have proved that (*) has at least four nontrivitial solutions in W₀^1,p(Ω) by using nonsmooth mountain pass theorem under (C)_c condition. Our main result generalizes a result by N. S. Papageorgiou, E. M. Rocha and V. Staicu in [29] and a result by G. B. Li and H. S. Zhou in [23](see Theorem 1.3).The difference between our result and the result in [29] is that we assume that f(x, t) is of p－asymptoticly linear at t =∞hence f(x, t) does not satisfy the Ambrosetti-Rabinowitz condition , while [29] assumes that f(x,t) satisfies the Ambrosetti-Rabinowitz condition . The difference between our result and the result in [23] is that [23] also assumes that f(x, t) is of p－asymptoticly linear at t =∞, but is of p－super－linear at t = 0 ,i.e (?) uniformly with respect to x∈Ω, but we do not require that f(x, t) satisfies this condition.