Existence And Multiplicity Of Solutions For A Class Of Fourth-order Elliptic Equations

This dissertation investigated a class of fourth-order elliptic equations.First, we considered the existence of solutions for a class of elliptic system involving fourth-order elliptic equationwhereΩis an open bounded domain in R^N(N≥3) with smooth boundary (?)Ω. HereΔ² denotes the biharmonic operator.In [10], under the well known (AR) condition, the author obtained the existence ofnontrivial solution for system (1) by the mountain pass theorem. (AR) condition usually plays a very important role in verifying that the corresponding functional has a Mountain-Pass geometry and showing a related (PS)_c sequence is bounded. In this paper, we will consider the existence of solutions for system (1) without (AR) condition.Second, under the weaker condition than those in [17], we considered the existence and multiplicity of solutions for the following fourth-order semilinear elliptic equation Also we consider the existence of positive solutions for problem (2) when f is superlinear at infinity. Then, by the same method, we obtain the existence and multiplicity of solutions for the following fourth-order quasilinear elliptic equationwhereΩis an open bounded domain in R^N with smooth boundary (?)Ω. c∈R, f:Ω×R→R is a Caratheodory function.The main results are the following theorems.Theorem 1 Now we state the assumptions imposed on F which is the potential in (1), whereâ–½_uF(x,u) =f(x,u) = (f₁(x,u),f₂(x,u)).(H₁) f :Ω×R²â†’R² is a Caratheodory function, that is, f(x,u) is measurable in x for each u = (y, z)∈R² and continuous in u= (y, z)∈R² for almost every x∈Ω, and there exist 2 < p < 2^* = 2N/(N-2),a₀,b₀>0 such that for all u= (y,z)∈R² and almost every x∈Ω,one has |f₁(x,u)|+|f₂(x,u)|≤a₀|u|^p-1+b₀.(H₂) There exist q > 2 and a₁ > 0 such that(H₃) There exist (N/2)(q-2)<μ≤q and b₁>0 such thatSuppose that -(1/2)(?)1) -(H₃) hold. Assume that there exist a₂,b₂>0 such thatuniformly for a.e. x∈Ω.where (?)= min{λ₁,λ₁²},(?)= max{λ₁,λ₁²},λ₁ is the first eigenvalue of -Δ. Then there exists at least one nontrivial solution for system (1).Theorem 2 Suppose that k≥0 and (H₁)-(H₃) hold. Assume that there exist a₃,b₃>0 such thatuniformly for a.e. x∈Ω. Then there exists at least one nontrivial solution for system (1). Corollary 1 Suppose that k> -1/2(?) and (H₁) - (H₃) hold. Assume that(?)(2F(x,u))/(|u|²)=0,(?)(2F(x,u))/(|u|²)=∞uniformly for a.e. x∈Ω. Then system (1) has at least one nontrivial solution.Theorem 3 For problem (2) suppose that f :Ω×R→R is a Caratheodory function and there exist C₀ > 0 and 2≤p <2N/(N - 4)for N≥5 (2≤p < +∞, for N≤4) such that |f(x,t)|≤C₀(|t|^p-1+ 1) for all t∈R and a.e. x∈Ω.Assume that there exist a positive measure subset E₁ ofΩand a functionβ∈L¹(Ω) such that F(x, t) - 1/2λ₁(λ₁ - c)t²â†’-∞as |t|→∞for a.e. x∈E₁, and F(x,t) - 1/2λ₁(λ₁ - c)t²â‰¤Î²(x) for all t∈R and a.e. x∈Ω, where F(x, t) = integral from n=1 to t f(x, s) ds.Then problem (2) has at least one solution in V = H²(Ω)∩H₀¹(Ω).Theorem 4 Suppose that the conditions of Theorem 3 hold. Assume that there exist an integer m≥1 andδ₁ > 0 such thatλ_m(λ_m - c)≤f(x,t)/t≤λ_m+1(λ_m+1 - c)for all 0 <|t|≤δ₁ and a.e. x G ft. Then problem (2) has at least two nonzero solutions in V.Theorem 5 In problem (2), suppose f satisfies:(H₄) f(x, t)≥0 for all t≥0, x∈(?) and f(x, t)≡0 for all t≤0, x∈(?);(H₅)(?) f(x,t)/t=∞a.e. on x∈(?);(H₆) There exists 2≤p< 2N/(N - 4) when N≥5; 2≤p < +∞when N≤4 suchthat (?) f(x,t)/(t^p-1)=0 a.e. on x∈(?);(H₇) (?) f(x,t)/t = a(x) a.e. on x∈(?), where a G L^∞(?) satisfies a(x)≤λ₁(λ₁ - c)for all x∈(?) and a(x) <λ₁(λ₁ - c) on someΩ' (?)Ωwith positive measure;(P) There existsθ≥1 such thatθG(x, t)≥G(x, st) for all x∈Ω, t∈R and s∈[0,1], where G(x,t) = f(x, t)t - 2F(x, t).Then problem (2) has at least one positive solution.Theorem 6 In problem (3), suppose that the following conditions hold:(H₈) There exist C₁>0 and 2≤p < 2N/(N - 4) for N≥5 (2≤p < +∞, for N≤4)such that |f(x,t)|≤C₁(|t|^p-1 + 1) for all t∈R and a.e. x∈Ω. (H₉) Let g₁,g₂∈C(R, R) and c <λ₁. Assume that g₁ is a continuous and nondecreasing function and cg2 is a continuous and nonincreasing function and there existα₁,α₂,β₁ andβ₂∈R such that 0 <α₁≤g₁(t)≤β₁, cα₂≤cg₂(t)≤cβ₂, whereα₁,β₂ satisfyingβ_/α₁≤1 if c≥0 andβ₂/α₁≥1 if c<0.(H₁₀) There exist a positive measure subset E₂ ofΩand a functionγ∈L¹(Ω) such that F(x, t)-(1/2)α₁λ₁(λ₁-c)t²â†’-∞as |t|→∞for a.e. x∈E₂, and F(x,t)-(1/2)α₁k_λ1-ct²â‰¤Î³(x) for all t∈R and a.e. x∈Ω, where F(x, t) = integral from n=0 to t f(x, s) ds.Then problem (3) has at least one solution in V.Theorem 7 Suppose that (H₈) - (H₁₀) hold with c≤0 andα₂/β₁≤1. Assume that there exists an integer i≥1 andδ₂ > 0 such thatβ₁λ_i(λ_i-c)≤f(x,t)/t≤α₁λ_i+1(λ_i+1-c)for all 0<|t|≤δ₂ and a.e. x∈Ω. Then problem (3) has at least two nonzero solutions in V.