Strict Convexity And The Existence Of Multiple Solutions To Nonlinear Elliptic Eigenvalue Problem In Divergence Form

In this paper, we study the existence of multiple weak solutions to the following nonlinear elliptic eigenvalue problem of an equation in divergence formwhereΩ(?) R~N is a bounded open domain,λ∈R~1 and N≥2, while the nonlinearities a :(?)×R~N→R~N and f : R~1→R~1 satisfy certain structural conditions. Especially, f is of (p- 1) - sublinear at infinity. Using a new Three Critical Points Theorem in [3], we get the existence of at least three solutions to (pλ) forλin some open intervalΛ(?)R~1, under weaker assumptions on A(x,ξ), where D_ξA(x,ξ) = a(x,ξ). The key point is that we only assume thatA(x,ξ+η/2)<1/2A(x,ξ)+1/2A(x,η),(?)x∈Ω,ξ,η∈R~N,ξ≠η(*)instead of the more restrict p-uniformly convexity assumption on A(x,ξ): There is a k > 0, such that for any x∈Ω,ξ,η∈R~N,A(x,ξ+η/2)<1/2A(x,ξ)+1/2A(x,η)-k|ξ-η|~pas [14] and other related works assumed.We also prove that if (*) holds, then A(x,ξ) is in fact strictly convex if a(x, is continuous inξ.