The Exact Boundary Behavior Of Solutions To Dirichlet Problems For Singular Nonlinear Elliptic Equations

In this paper we mainly study the exact asymptotic behavior near the bound-ary of classical solutions to a singular Dirichlet problem for nonlinear elliptic equation: where Ω is a bounded domain with smooth boundary in RN, λ,μ≥0, q ∈ (0,2],b, a ∈ Clocα(Ω)(0<α<1) is nonnegative and nontrivial in Ω, f∈ C([0,∞), [0,∞)), g∈C1((0,∞), [0,∞)), limtâ†'0+g(t)=∞, and there exists So>0, such that for any S ∈ (0, So), g’(s)< 0.For the problemWe first establish the local comparison principle of solutions, then by Kara-mata regular variation theory we study carefully the exact asymptotic behavior of the unique solution ψ at zero to the integral equation Moreover, by a perturbation method, when b satisfies structure condition (b2), and we construct proper comparison function, and reveal that all the solutions of problem (P1) have the same asymptotic behavior near the boundary For the problemBy constructing new local comparison principle, combining with the global estimates of the unique solution to the Dirichlet problem of Poisson equation and when a satisfies a proper condition, we reveal that the nonlinear term λa(x)f(u) does not affect the exact asymptotic behavior of any solution uλnear the bound-ary to problem (P2).For the problemFirst, when q=2, nonlinear transformation w=eμu-1 transforms problem (P3) into the equivalent problem (P2); when q ∈ (0,1), we construct a local comparison principle similar to (P1); and when q ∈ [1,2), by using an inequility, the problem (P3) can be changed into a new problem. We establish the local comparison principle of solutions near the boundary to problem (??) and (??), combining with the conclusion of chapter 3 and chapter 4, we reveal thatμ|(?)u|q+ σ does not affect the exact asymptotic behavior of any solution uμ near the boundary to problem (P3).In the sixth chapter, we give the result and proof of existence and non-existence of solutions to problem (P2).