Existence Of Three Solutions For Elliptic Systems And Fourth-order Boundary Value Problem

In this paper, firstly, we consider the following quasilinear elliptic systems involving the (p, q)-Laplacian with Dirichlet boundary condition:where (?)is the p-Laplacian operator,λ∈[0, +∞),Ω(?) R^N(N≥1) is a non-empty bounded open set with a boundary (?)Ωof class C¹, p > N, q > N, F :Ω×R×R→R is a function such that F(·,s, t) is measurable inΩfor all(s,t)∈R×R and F(x,·, ?) is C¹ in R x R for a.e. x∈Ω, F_u denotes the partial derivative of F with respect to u. Fix x⁰∈Ωand pick r₁,r₂ with r₂ >r₁ >0, such that B(x⁰,r₂) (?)Ω, where B(x, r) stands for the open ball in R^N of radius r centered at x. PutThe main result is the following theorem:Theorem 1 Assume that there exist four positive constants c, d,γandβwithγ< p,β< q,(?)+(?)>(?), and a functionα∈L¹(Ω), such that where A = {(s,t)|(?)+(?)≤(?)}Then, there exist an open interval(?)(?)[0, +∞) and a positive real numberρsuch that, for eachλ∈(?),problem (ES) has at least three weak solutions whose norms in W₀^1,p×W₀^1,q are less thanρ.Corollary 1. Let g :Ω×R→R is a continuous function, and G is the real function given by G(x, t) = (?) g(x,ξ)dξfor each (x, t)∈Ω×R. Assume that there exist three positive constants c.d andγwithγ< p.d^Pσ₁^Ρ>(?), and a functionα6 L¹(Ω), such that(j'₃) G(x,t)≤α(x)(l + |t|^γ) for almost every x∈Ωand all t∈R.Then, there exist an open interval (?)(?) [0, +∞) and a positive real numberρsuch that,for eachλ∈(?), the problemhas at least three weak solutions whose norms in W₀^1,p are less thanρ.Next, we consider the following the fourth-order quasilinear boundary value problemwhereλ∈[0, +∞),Ω(?) R^N(N≥1) is a non-empty bounded open set with a boundary (?)Ωof class C¹, p > max{1,(?)},f :(?)×R→R is a continuous function. Fix x⁰∈Ωand pick r₁,r₂ with r₂ > r₁ > 0, such that B(x⁰,r₁) (?) B(x⁰, r₂) (?)Ω. Put The main result are the following theoremsTheorem 2 Assume that there exist three positive constants c, d and 7 withγ1 (Ω) such that(i₂) m(Ω) (?) F(x.t) < (?)^p (?)(x⁰,r₁)F(x, d)dx, where m(Ω) is the Lebesguemeasure ofΩThen, there exist an open interval (?)(?)[0, +∞) and a positive real number p such that, for eachλ∈(?), problem (NB) has at least three weak solutions whose norms in W^2,p(Ω)∩W₀^1,p(Ω) are less thanρ.Let N = 1,we fixΩ=]0,1[,p>l and consider a continuous function g: R→R. Moreover, put G(t) =(?)g(s)ds for all t∈R. We have the following result:Theorem 3 Assume that there exist four positive constants a, c, d andγwith(?)< 16d.γ< p, such that(i') g(s)≥0 for every s∈[-c,max{c, d}];Then, there exist an open interval (?)(?) [0, +∞) and a positive real numberρsuch that, for eachλ∈(?), problemhas at least three weak solutions whose norms in W^2,p(0,1)∩W₀^1,p (0,1)are less thanρ.