Existence And Multiplicity Of Solutions For Some Degenerate Elliptic Systems

In this paper,degenerate quasilinear elliptic systems are studied by the crit-ical Point theory.the variational methods and the implicit function theory.More precisely,we mainly consider existence and multiplicity of solutions of the follow-ing the degenerate quasilinear elliptic system where Ω is a bounded and connected subset of RN(N≥2),1<p,q<N,λ is a real parameter,Lu=—div(h1(x)|â–½u|p-2â–½u),Lu=—div(h2(x)|â–½u|q-2â–½v), and α≥0,β≥0satisfyina (α+1)/p+(β+1)/q=1.The coefficient functions a,d,b∈C(Ω)∩L∞(Ω),and F∈C1(Ω×R2,R),(Fu,Fv) denotes the gradient of F with respect to (u,v).The degeneracy of this system is considered in the sense that the measurable,nonnegative diffusion coefficients h1,h2,are allowed to vanish in Ω (as well as at the boundary (?)Ω) and/or to blow up in Ω.In Section2,first of all,if the nonlinearity F satisfies the coercive condition, that is uniformly with respect to x∈Ω,for λ<λ1sufficiently close to λ1,systems (0.3)has at least three nontrivial solutions by Ekeland’s variational principle and Mountain-Pass theorem in critical point theory. The second, if the nonlinearity F satisfies the non-quadraticity condition, that is uniformly with respect to x∈Ω, for λ∈(λ1,λ2), systems (0.3) has at least one nontrivial solutions by the Saddle Point Theorem in critical point theory. Where λ1is the principal eigenvalue of the following degenerate quasilinear elliptic system The associated normalized eigenfunction0satisfy each component is nonnegative, and we will determine in Sections2what it means λ2.In Section3, when p=q=2, α=β=0, using the standard spectral theory for compact, self-adjoint operator, we firstly consider the linear eigenvalue problem with weight A(x), we prove that the system (0.4) has the set of the eigenvalues λk, satisfying0<λ1<λ2≤…≤λk≤… such that λkâ†'∞as kâ†'∞. The second, the Sobolev space is resolved, and if the nonlinearity F satisfies the coercive condition (F1)(or (F2), or (F3), or (F4)). The assumptions concerning the coercive condition are the following.uniformly with respect to x∈Ω. (F2) uniformly with respect to x∈Ω.(F3) uniformly with respect to x∈Ω.(F4) uniformly with respect to x∈Ω. When p=q=2, α=β=0, some multiplicity results of solutions are obtained for the degenerate elliptic systems (0.1) which are near resonance at higher eigen-values by the classical saddle point theorem and a local saddle point theorem in critical point theory.