Existence Of Solutions Of Quasilinear Elliptic Equations With The Fucik Spectrum Of Resonance

In this paper,we firstly study the existence of the solutions for the Fucik type resonance problems of the quasilinear equation under the Landesman-Lazer condition.Set M(a,b)the set of solutions of the problem Define and setWe make the following assumptions.(G1)11 is a bounded domain with smooth boundary,â–³p denotes the p-Laplacian defined byâ–³p:=div(|â–½u|p-2â–½u).9(u):Râ†'R is a continuous function,h∈Lp'(Ω),where p'=p/p-1.(g2)One of the following happens:(i)The inequality hold for all v∈M(a,b)\{0),where v±:=max{0,±v}; (ii)The inequality hold for all v∈M(a,b)\{0}.By the Link Theorem,we can obtain the following result.Theorem 2.1 Assume that(G1),(g1)and(g2)hold.Then the problem (3)has at least a solution in W1,po(Ω)for every(a,b)∈(-∞,λ1)×(-∞,λ1)or [λκ,λκ+1)×[λκ,λκ+1).Then we consider the existenee of the solutions for the Fucik type resonance problems of the quasilinear equation under the nonquadratic conditions.We make the following assumption.(C1)Assume that g:Ω×â†'R is a Caratheodory function with subcritical growth:that is, for some a0,b0>0 and [resp.1<α<∞if1≤N≤p];(C2)Set G(x,t)=∫tog(x,s)ds and assume that uniformly for a.e.x∈Ω；(C3)Assume that G(x,t)is nonquadratic at infinity in the sense that uniformly for a.e.x∈Q.By the Link Theorem,we can obtain the following result. Theorem 3.1 Assume that(C1),(C2)and(C3)hold.Then the problem (4)has at least a solution in W1,po(Ω)for every(a,b)∈(-∞,λ1)×(-∞,λ1)or [λk,λk+1)×[λk,λk+1).