A Study On Class Of Singular Elliptic Problems

Elliptic problems are one of the main problems in research for partial differ-ential equations and belong to the research areas of core mathematics. As early as1900, there are three problems related to elliptic problems in the famous23problems which proposed by D. Hilbert. Over the past century there has been considerable activity in the study of elliptic problems, the research of elliptic problems had rich results and formed huge theoretical system, the theory has a rich interplay with other subjects in geometry, hydromechanics, thermodynamics e.t.c. and promoted the development of these subjects. We discussed the ex-istence and multiplicity of weak solution and classical solution for problem (P) under various conditions.In this Ph.D. thesis we consider a family of singular elliptic problems, arising in the model of MEMS (micro electro mechanical system) where Ω(?)RN RN is a bounded domain, A>0,β>0,γ>0are real parameters, p is a non-negative real function in Lq(Q)(q>1) and not always zero in Ω.The main results we obtained are that there exists λ*>0, such that there exists a unique weak solution u∈H01(Ω) to (P) for all0<λ<λ*whenever0<β≤1,0<γ<1and p∈L2(Ω) as well as p(x)>C0>0a.e. in Ω.In the case of critical growth, i.e.β=2*-1,0<7<1, it follows from the Ekeland variational method and concentration compactness principle we also obtain that there exist λ*>0such that there exists two solutions vλ,ωλ∈H01(Ω) to (P) for all0<λ<λ*when p∈L2(Ω).If the perturbation term uβ is supercritical growth, i.e.β>2*-1, and or γ∈(0,1), p∈Lq(Ω)(q>N/2), or γ≥1, there exists positive function h∈C01(Ω) such that ph-γ∈L9(Ω)(q>N/2) we prove that there exists λ*>0, such that the problem (P) has at least a unique solution for all0<λ<λ*by the method of Moser iteration techniques and the Sup-Sub solution method.Finally, we research the classical solutions of the problem (P). If1<β<2*-1,0<γ<1, p∈C0α (Ω)(0<α<1), we show that there exists λ*>0, such that the problem (P) has two classical solutions u, u∈C2,α(Ω) D C1,v(Ω) for all0<λ<λ*via approximation scheme and blow-up arugment. Further, if0<γ<1,p∈C0α(Ω)(0<α<1), in view of the Moser iteration and the technique of truncation, we prove that there exists A*>0, such that the problem (P) has a classical solution u∈C2,α(Ω)∩C1,v(Ω) whenever β＜2*-1, where v∈(0,1+α-γ) when α<γ; v∈(0,1) when∈≥γ.