Qualitative Study Of Solutions To Some Quasilinear Elliptic Equations (Systems) With Subcritical Or Critical Exponents

In this paper, we apply variational methods to prove the existence and multiplicity of solutions to some quasilinear elliptic equations (systems) with subcritical or critical exponents. This thesis consists of four parts.In the introduction section, firstly, we review the histories of the devel-opment of variational methods; secondly, we state some main results about the semilinear or quasilinear elliptic problems; at the final of this Chapter, we give the main results of this thesis.In Chapter one, we investigate the existence and multiplicity of solu-tions to the following Neumann boundary problem of quasilinear elliptic equation involving critical Sobolev exponents: where Ω(?)RN is a bounded domain with a smooth boundary, parameters ε>0,1<p<N, p<q<p*, p*=Np/Np is the critical Sobolev exponent and functions V(χ), Q(χ), P(χ) satisfy the following conditions:(Al) Q(χ), P(χ) are continuous functions on Ω,, and Q(χ)>0, P(χ)≥0for all x∈Ω: (A2) V(x) is a continuous function on Ω satisfying V(x)≥0and V(x)≠0for all x G Ω.Due to the lack of compactness of the embedding W1,p(Ω)→Lp*(Ω,) and the coefficient V{x) which is allowed to be unbounded on boundary, these bring us some difficulties. In order to overcome these difficulties, we choose a suitable weighted Sobolev space and prove that functional satisfies (PS)。 condition under a suitable condition by the Lions concentration compactness principle, then by means of Mountain pass lemma, we obtain the existence and multiplicity of solutions of Problem (f). The main results are the following:Define Qm=maxQ(x), Qm=maxQ(x), Pm=maxP(χ), A=εs-p.Theorem1Suppose that (Al)(A2) hold and0G dΩ, H(0)>0, Qm Q(0). If functions Q(x), V(x) satisfy (A3) QM <2p/N-pQm.|Q(x)-Q(0)|=o(\x\a) for x→0, where1<a N/p-1(A4) fΩ∩B(0,δ)V.r1dχ<∞,1<r<N(p-1)/Np+2p-N-P2-1,1/r+1/r1=1. Then Problem (f) has at least one nontrivial solution for N≥2p, A>0.Theorem2Suppose that (Al)(A2) hold. If QM>2p/N-p and func-tions P(x), V(x) satisfy tions P(χ),V(χ)satisfy (A5) P(x)≠0for x∈Ω, V∈L1(Ω). Then there exists a constant λ*>0such that Problem (f) has at least one nontrivial solution for all0<λ <λ*.Theorem3Suppose that (Al)(A2) hold. If QM>2p/N-p and func-tions P(x), V(x) satisfy (A6) P(x)>0for x∈Ω;(A7) there exist x0∈Ω and S>0such that V(x)=0for x G B (x0, S) C Ω. Then there exists a constant.λ>0such that Problem(1) has at least one nontrivial solution fbr all λ>λ*..Theorem4Suppose that(A1)(A2)hold.If QM>p/N-pQm and fum tions P(χ),V(χ)satisfy(A6)P(χ)>0for χ∈Ω;(A7)there exist χ0∈Ω and δ>0such that V(χ)=0for χ∈B(χ0,δ)(?) Ω.Then for every integerm,there exists a constant Λm>0such that Problem(1)has at least m nontrivial solutions for all λ>Λm.In the first section of Chapter two,we study the fbllowing quasilinear elliptic problem with critical Hardy-Sobolev exponent: where aΩ∈C1and0∈aΩ,0<ι<p,p*(ι)=(?)is the critical Hardy-Sobolev exponent,α∈L∞(aΩ)is a nonnegative function,f satisfies the following conditions:(B1).f∈C(Ω×R,R)and there exists a function a(χ)∈L∞(Ω),a(χ)≤0such that(B2)lim(?)=0for all χ∈Ω；(B3)(?)-F(χ,τ)≥0；(B4).f∈C(Ω×R,R)satisfies.f(χ,-τ)=一.f(χ,τ),and there exists a function a(χ)∈L∞(Ω), a(χ)≤0such that(B5) lim f(χ,|τ|)/|τ|p*(t)-1=0for all χ∈Ω.Applying the Hardy-Sobolev inequality, Lions concentration compactness principle, Mountain pass lemma and strong maximum principle, we prove that Problem (2) has at least one positive solution. Furthermore, under certain conditions, the existence of two nontrivial solutions is also obtained. The main results are as the following:Theorem5Assume that (B1)(B2) and (B3) hold. If N≥p2and p≥2, then Problem (2) has at least one positive solution.Theorem6Assume that (B3)(B4) and (B5) hold. If N≥p2and p≥2, then Problem (2) has at least two solutions. In the second section of Chapter two, we discuss the following quasilinear elliptic problem with critical Sobolev exponent and critical Hardy-Sobolev exponent:When the parameter λ>0, by means of Mountain pass lemma and con-centration compactness principle, the existence of nonnegative solution is obtained. With regard to the parameter λ≤0, since the space W1'p(Ω) is not suitable for our problem, we have to introduce a suitable space, and apply Ekeland variational principle and concentration compactness princi-ple to prove the existence of local minimum for the corresponding energy functional of the problem, then obtain the existence of solutions of problem (3). The main results are as follows Theorem7Suppose thatλ>0,Q(χ),P(χ)are positive continuous funetions on Ω.If then there exists a constant λ*>0such that Problem (3) has at least one nontrivial nonnegative solution for0<λ<λ..Theorem8Suppose that c*=(?). If there exists a y∈aΩ Such that Qm=Q(y)and|Q(y)-Q(χ)|=o(|χ-y|)for χ→y,where1<σ<N/p1. Then Problem(3) has at least one nontrivial nonnegative solution for everyλ>0and N>2p一1.Theorem9Suppose that Q(χ),P(χ) satisfy the following conditions:(C1)Q(χ)is a positive continuous function on Ω；(C2)P(χ)is a continuous and changing sign functioon on Ω,satisfying (?) Then there exist a constant λ0>0and μ∈W1,p(Ω)such thatαλ=Jλ(μ) for all0≤λ<Aλ0.Furthermore,μis also a solution of Problem(3),where λ is the opposite number to the parameter of Problem(3).In Chapter three,we firstly investigate the existence of multiple solutions for a quasilinear elliptic systern involving sign-changing weight functions: where1<q<p,α,β>1and satisfy p<α+β<p*.In this section,we make some-assumptions on the weight functions.f,g and,h as the following(D1)f,9∈Lq*(Ω),where q*=(?),and either f±=max{±f,0)≠0org±=max{±g,0)≠0；(D2)h∈C(Ω)with||h||∞=1andh≥0. Secondly, we consider the existence of multiple solutions for singular quasilinear elliptic system with critical Sobolev-Hardy exponents and concave-convex terms:(5) where λ,μ>0,0≤ι<ι,0≤s,t<p,α>1,β>1satisfying α+β=p*(ι),ι=(Np/P)p is the best Hardy constant,functions.f,g and,h satisfy the following conditions:(E1)f(χ),g(χ)∈C(Ω),f(χ),g(χ)≥0and.f(χ),g(χ)≠0inΩ；(E2)There existβ0and ρ0>0such that B2ρ0(0)(?)Ω andf(χ),g(χ)≥β0for all χ∈B2ρ0(0)(?)Q；(E3)h∈C(Ω)with h(χ)≥0and h(χ)≠0inΩ；(E4)Vχ∈B2ρ0(0)(?)Ω,h(χ)>0.|h|∞=h(0)=maxχ∈Ωh(χ)and there exists aβ1>0Such that h(χ)=h(0)+o(|χ|β1)asχ→0,where ρ0is defined as in(E2)and β1satisfies β1<b(ι)p*(t)一N+p.The main features of these problemss are as follows.Firstly,the systems have the concave-convex nonlinear terms；secondly,the corresponding func—tionals of problems are not bounded below on the Sobolve space which we usually consider.In order to overcome these dimculties.we discuss the exis-tence of solutions of problems by a variational method involving the Nahari manifold which is the similar to a fibering method. The main resullts are the following:Theorem10Assume that(D1) and(D2)hold. Then Problem(4) has at least two nontrivial nonnegative solutions(υ0+,ν0+)and(υ0-,ν0-),such thatυ0=≥0,υ0±≥0in Ω and υ0±≠0if0<(|λ||f|Lq*)P/p-q+(|μ|||g||Lq*)p/p-q<C0,whereTheorem11Assume that(D1)and(D2)hold and.f≥0(f≤0).Then Problem(4)has at least two nontrivial nonnegative aolutions(υ0+,υ0+)and (υ0-,υ0-)such thatυ0±≥0,υ0±≥0inΩand μ0±≠0,υ0±≠0ifλ≤0(λ≥0) and satisfis0<|μ|<C0,whereTheorem12Assume that(E1)and(E3)hold.Then Problem(5)has at least one nontrivial nonnegative solution(υ0+,υ0+)for all0<(λ|f|∞)p/pq (μ|g|∞)p/pq<Λ1Theorem13Assume that(E1)-(E4)hold and p. Then there exists a constantΛ2>0such that Problem(1.1)has at least two nontrivial nonnegative solutions(υ0,υ0)and(υ0,υ0)for all0<(λ|f|∞)p/pq+(μ|g|∞)p/p-q<Λ2.