A Few Class Singular Quasilinear Elliptic Equations Existence Of Eigenvalue Problem Solutions

In this paper, we discuss three topics. The first is the solvability of the high order quasilinear elliptic equations; the second is the existence of the weighted singular quasilinear elliptic equations; and the third is the problem of singular elliptic equation with critical potential and indefinite weights.We first introduce in Chapter1the history, background, significance and the main content if this paper.In chapter2, we mainly study the high order quasilinear elliptic equation where Ω(?)RN(N≥1) is a bounded open connected set, α={α1…,αN). According to the critical point of function I(u) and mountain pass lemma, we obtain the existence of nontrivial weak solutions for the above equation.In chapter3, we study the high order quasilinear elliptic equation We utilize a new notation of Psuedo-eigenvalue to discuss the resonance problem with arbitrary eigenvalue. Then, an existence theorem is obtained under the Landesman-Lazer condition.In chapter4, we prove the existence of solutions in weighted Sobolev spaces Hp,p1(Ω,Г) for the singular quasilinear elliptic equation Qu=f(x,u), x∈Ω, u=0,x∈(?)Ω, The proofs rely on Galerkin-typetechniques, Brouwer fixed point theorem, a relationship between the quasilinearoperator Q and some linear uniformly elliptic operator L In chapter5, we study the eigenvalue problem in weighted Sobolev spacesWm,p(Ω;σ,σα) for the p-Laplace equationwhereΩ(?)RN(N≥1) is a bounded set,1<p <∞. Using the variation method, we obtain the existence of the eigenfunctions corresponding to the first eigenvalue.In chapter6, we study the singular quasilinear elliptic equation with critical potential and indefinite weightswhere Ω(?)RN is a bounded set, N=p,0∈Ω(?)Br(0), μ≥0,λ＞0. First we utilize the Hardy inequality and the Picone inequality to discuss the eigenvalue problem. Then using critical point theory and the properties of the first eigenvalue, we obtain the existence of nontrivial solutions.