Some Problems to Solution of Singular Elliptic Equations

Abstract: In recent decades, singular equation as an important class of Differential Equations attract much attentions. This article is an overview which research the solutions of singular elliptic equations,By there parts we divided singular elliptic equations to semilinear elliptic equation、quasilinear elliptic equation、nonlinear elliptic equation.Firstly, we consider the solution of semilinear elliptic problem where f(x,u) be assumed to satisfy some conditions,Ω(?) RN is a bounded domain with smooth boundary (?)Ω, u∈H01(Ω), and f(u) satisfies:(1)f’(u)≤0:(2)f(u)>0,u>0;λu satisfies the conditions (1)-(3), thus (2.1) can be transformated to the following problem satisfy the following conditions. we definite a function J(u), there exists{um} is a P.S. sequence for J, Then u is a nontrivial solution of (2.2).Theorem 1 Supposing{um}∈H01(Ω) is a P.S. sequence for J, Then there exists u is a nontrivial solution of (2.2).In [2], f(x)=λu-q+up f(x) satisfies the conditions (1)-(3), then (2.1) can be transfor- mated to the following problem (2.3) weak solution. Thus We use variational method and sub-supper solution methods in this part. then there exists a constant c> 0, such that(1) For allλ∈(0, c), problem (2.3) has at least two weak solutions u, v, satisfyingu< v;(2) Forλ= c, problem (2.3) has at least one weak solution;(3) For allλ∈(c,+∞), problem (2.3) has no weak solution. is a weak solution of (2.3), is a classical solution of (2.3). And if k1>1, there k1 denotes the principal eigenvalue of -△with zero Derechletcondition, then(1)For p→1, cp→+∞;(2) For anyλ> 0, where, u1 denotes the unique solution of the following problem(3) For anyλ> 0, Moreover, there exists c*> 0,secondly, we consider the solution of quasilinear elliptic problem In [20], we considerSupposing V(x), f(x) satisfy some conditions.Theorem 4 Supposing 1< p< N,0<μ<μ*,0<β≤b(μ)p+p-N, V(x) satisfy (1)-(3), f(x, t) satisfy (Ⅰ)-(Ⅳ), then existsλ*> 0, such that, for anyλ∈(0,λ*), problem (3.2) has at least two positive solutions.finally, we consider the nonlinear elliptic problem.In this part, we consider the existence of positive solutions of singular nonlinear elliptic equations with Dirichlet boundary conditions. In [28], letΩbe a domain in Rn,ψis a nonnegative continuous function in (?)∞Ω, F is a Borel measurable function inΩ×(0,+∞), and F satisfies two conditions.Theorem 5 LetΩbe an arbitrary Dirichlet regular domain in Suppose that F is a Borel measurable function inΩ×(0,+∞), satisfying (1) and (2). Then, for every continuous functionψ∈(?)∞ΩsatisfyingThen, the problem (4.3)has at least one positive continuous solution u, such thatAssumingΩbe a bounded domain in In [29], we consider the singular elliptic problem where K(x) and V(x) are two given Holder continuous functions onΩ, and 0<γ< 1/N,0<α< 1 are two constants.Theorem 6 Let 0<γ<1/N,0<α< 1, Then the problem has at least one solution where 0<p< 1 be relevant to the exponentγ>0,0<p<1, MoreoverFurther, there is a family functions um(x) defined onΩ, such that for any initial function for some c tending uniformly to u1(x). And, the rate of convergence iswhere 0<k0< 1, which depends on the given initial function u0.
Key words: Singular elliptic equation; existence of solution; multiplicity; variational; sub-super solution;

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