The Property Of Solutions To Several Quasilinear Elliptic Systems

This paper is a comprehensive survey of the results to some quasilinear ellipticsystems. The main results are about existence, uniqueness, non-existence and multiplicity of positive solutions. Firstly, we recall the background and recent progress about quasilinear elliptic equation and systems. In chapter 2. we firstly introduce the results about radial solutions to the system of p-Laplace equation with homogeneous Dirichlet boundary condition, and then, we study existence, uniqueness, non-existence and multiplicity of positive solutions to quasilinear ellipticsystems with Dirichlet boundary condition. In chapter 3. we study the existence of positive solutions to the quasilinear elliptic systems with homogeneousNeumann boundary condition, and obtain the existence and multiplicity of positive solutions.In chapter 2, using some precise a priori estimates and topology degree theory, we study the radial solution of the homogeneous Dirichlet problem andwe have the following result:Theorem 1. For functionsφ,f and g, we have the following assumption:(i) For every constant c > 0 and z≥0, there exist a positive constant A(c) > 0. havingφ^-1(cz)≥A(c)φ^-1(z) hold, and if c→∞, then, A(c)→∞.(ii) f. g : [0,∞)²â†’R are continuous but non-increasing functions, and there exist constant M > 0, then, for every (u,v)∈[0.∞)². having f(u,v)≥-M/2, g(u,v)≥-M/2 hold.Leth^*(u,v) = max{f(u,v), g(u, v)}, h_*(u, v) = min{f(u,v), g(u,v)}.enough, then, systems (2) have positive solutions.In section 2, we study the following p-Laplacian systemsand obtain the existence of positive solutions to above systems. We have the following results:Theorem 2. Assume that f(x,u,v),f(x,u,v) satisfy (A₁)-(A₄). p = q,a = b. (A₁)f(x,u,v), g(x,u,v) : (?)×[0, +∞)×[0,+∞)→[0,+∞) are NonnegativeHolder continuous functions.(A₂) There exist nontrivial continuous function C(x)≥0 and constantsα₀,β₀ satisfying 0≤α₀≤P - 1, 0≤β₀≤q - 1, then, for all (x,u,v)∈(?)×[0,+∞)×[0,+∞).f(x,u,v)≥C(x)u^α₀v^p-1-α₀), g(x,u,v)≥C(x)u^q-1-β₀v^β₀,hold.(A₃) There exist non-negative continuous functions A₀(x),B₀(x),a_∞(x),B_∞(x). positive integer k, N, positive a_i,b_j,0≤a_i≤p - 1,0≤β_j≤q - 1(i = 1, 2,…, k, j = 1, 2,…, N), such that for (x,u,v)∈(?)×[0,+∞)×[0,+∞).(A₄) There exist positive constants C₁ and C₂, for any (u,v) satisfying 0≤C₁v≤u≤c₂v and any x∈(?),C₁^p-1g(x,u,v)≤f(x,u,v)≤C₂^p-1g(x,u,v).Ifλ(A₀(x),B₀(x)) < 1 <λ(A_∞(x),B_∞(x)). then, (3) has at least one positivesolution, andλ(A₀(x), B₀(x)) (λ(A_∞(x),B_∞(x))) is the unique eigenvalue with positive eigenfunction for the case A(x) = A₀(x), B(x) = B₀(x) ( A(x) = A_∞(x), B(x) = B_∞(x)); f(x,u, v) is monotonically in creaseing for u∈(0,∞). and monotone for v in (0,∞) (pickingσ(f) = 1, if f is increasing, andσ(f) = -1 if f is decreasing); g(x, u, v) is a monotonic function increasing for v∈(0,∞), and monotone in u (pickingσ(g) = 1 if g is increasing andσ(g) = -1 if g is decreasing).If for any(x,u,v)∈Ω×(0,∞)×(0,∞), t^1-pf(x,tu,t^σ(f)v),t^1-qg(x,t^σ(g)u,tv) are strictly increasing for t∈(0,∞), then, systems (3) has at most one positive solution.In section 3. we study the multiplicity of solutions to the quasilinear elliptic systems. Considerthe following systems:where p,q > 1.Δ_pu = div(|â–½u|^p-2â–½u),Ω(?) R^N is a bounded smooth region.λ,μare positive parameters,α,βare positive constants. a(x), b(x), c(x)∈(?) and may change sign.In the following, we introduce the existence and non-existence of positive solution for systems (4). When parametersλ,μare small enough and T⁺ is empty set, T^- is non-empty, then, systems (4) has a positive solution at least in T^-. When parametersλ,μare big enough, we introduce the relationship between different sets, and revealing the properties of T₊, T_-, T₀. therefore, systems (4) have two positive solution at least.We have the following results:In the following, we introduce in different cases for the existence and nonexistenceof solutions to the system (4). The main results are: If the parametersλ,μare all suitably small, the problems has at least one positive solution where the sign of∫_Ωφ^α+1ψ^β+1 plays a very important roll. And for suitably large pa- rameterλ,μ, the problem has at least two positive solutions. The results are the following:Ifλ<λ₁(a),μ<μ₁(b). then, systems(4) has at least, one positive solution;if∫_Ωφ^α+1ψ^β+1dx < 0. then, there existδ,σ> 0, such that forλ₁(a) <λ<λ₁(a) +δ,μ₁(b) <μ<μ₁(b) +σ,Systems (4) have at least two positive solutions:In Chapter 3, we introduce the existence of positive solutions for quasilinearelliptic systems with Neumann boundary condition. Consider the following problem:By eigenvalue theory and P. S. condition, we obtain the existence of positive solutions for above problem.Now, we give some conditions which make our results holds.(H₁) 1 < P <α+ 1, 1< q <β+1.(H₂) a(x),b(x) c(x) are smooth functions and change Symbol in (?).(H₃)∫_Ωa(x)dx < 0,∫_Ωb(x)dx < 0.(H′₃)∫_Ωa(x)dx > 0,∫_Ωb(x)dx > 0.(H″₃)∫_Ωa(x)dx = 0,∫_Ωb(x)dx = 0.(H₄)∫_Ωc(x)φ^α+1(?)^β+1dx < 0.(H₅) (N-p)/p(α+1) + (N-q)/q(β+1) 0,α+ 1 < p^*,β+1< q^*, where p^* = (N_p)/(N-p), q^* = (N_q)/(N-q) are critical index.Assume that (H₂), (H₃)((H′₃)), (H₅)hold, and p < p^*, q < q^*. then. Problem(5) has positive solutions; Assume that (H₁), (H₂), (H₃)((H′₃)), (H₄), (H₅)hold. then, there existλ^*,μ^*, whenλ^* >λ>λ₁(a),μ^* >μ>μ₁(b)(λ^* <λ<λ₁(a),μ^* <μ<μ₁(b)), problem (5) has at least one positive solution.Assume that (H₁), (H₂), (H₃)((H′₃)),(H₄),(H₅)hold. then, there existλ^*,μ^*, whenλ^* >λ>λ₁(a),μ^* >μ>μ₁(b)(λ^* <λ<λ₁(a),μ^* <μ<μ₁(b)). problem (5) has at least two positive solutions.