Existence Of Solutions For A Class Of Singular Elliptic Equations

In this paper, firstly, we consider the following singular semilinear elliptic problemwhereΩ(?)R^N(N≥3) is a bounded domain with smooth boundary (?)Ω,γis a positive constant, g:Ω×R→R is a Carathe(?)odory function, that is, g(x, s) is measurable in x for every s∈R and continuous in s for a.e. x∈Ω.The proof is done by using the variational method and the least action principle.The main results are the following theorems.Theorem 1 Suppose g(x, s) satisfies the following conditions(g1) there exist two functions a∈L^2n/n+2(Ω) and b∈L^n/2(Ω) such that|g(x,s)|≤a(x)+b(x)|s|,for a.e. x∈Ωand every s∈R.(g2) there exists m∈R such thatfora.e. x∈ΩAssume that m<λ₁ then problem (1) has a solution in C((?))∩((?)W_loc^2,p(Ω)),whereλ₁ denotes the first eigenvalue of the operator -â–³with homogeneous Dirichlet condition. Theorem 2 Assume that g(x, s) satisfies (g1) in Theorem 1 and the following assumption(g3) there exist a positive measurable set E inΩand a function h∈L¹(Ω) such that(i)G(x,s)-1/2λ₁s²â†’-∞,as |s|→∞for a.e. x∈E, (ii)G(x,s)-1/2λ₁s²â‰¤h(x),for all s∈R and a.e.x∈Ω, where G(x,s)=integral from n=0 to s g(x,t)dt, then the conclusion of Theorem 1 also holds for problem (1).Next, we study the existence of solutions for the following singular elliptic equationwhereΩ(?)R^N(N≥1) is a bounded domain with C^2+α boundary for someα∈(0,1),λ,γand p are three positive constants, k is a nonnegative function inΩ.We prove the existence of the positive solution for problem (2) using the sub-supersolution method.The main results are the following theorems.Theorem 3 Suppose k∈C_loc^α(Ω)∩C((?)), k≥0 and k≠0. Assume that 0<γ<1 and 0λ∈C^2+α(Ω)∩C((?)) and u_λ^-γ∈L¹(Ω) for allλ>(?) and has no solution in C²(Ω)∩C((?))ifλ<(?). Moreover, problem (2) has a maximal solution v_λis increasing with respect toλfor allλ>(?).Theorem 4 Suppose k∈C_loc^α(Ω)∩C((?)) and k(x)>0 in (?).Ifγ≥1,problem (2) has no solution in C²(Ω)∩C((?)) for allλ>0 and p>0.At last, we consider a more general problem than problem (2) by the sub-supersolution method. We study the following problemwhereΩis a bounded domain in R^N(N≥1) with C^2+α boundary,λand p are two positive parameters, k is nonnegative function inΩ, f is a nonnegative and nonincreasing function in (0, +∞).The main results are the following theorems.Theorem 5 Suppose k∈C_loc^α(Ω)∩C((?)),k≥0 and k≠0,0loc^α(0,∞) satisfies the following assumptions(f1)(f2)then there exists (?)∈(0,∞) such that problem (3) has at least one solution u_λ∈C^2+α(Ω)∩C((?)) and f(u_λ)∈L¹(Ω) for allλ>(?) and has no solution in C²(Ω)∩C((?))ifλ<(?) Moreover, problem (3) has a maximal solution v_λis increasing with respect toλfor allλ>(?).Theorem 6 Suppose k∈C_loc^α(Ω)∩C((?)) and k(x)>0 in (?). If f∈C_loc^α(0,∞)satisfies the following condition(f3)problem (3) has no solution in C²(Ω)∩C((?)) for allλ>0 and p>0.