Existence Of Solutions For A Nonlinear Elliptic Equation With Negative Exponents

In this thesis,we are interested in the solutions of nonlinear elliptic equation with negative exponents of the type (?) where p>1 and Ω C RN,V>1,is a bounded open set with smooth boundary(?)Ω.We first establish the existence of the solution:When k(x)∈L∞(Ω)is a non-negative function,and h(x)∈L1(Ω),h>0 almost everywhere in Ω,and g ∈ C1,g(u)is monotone decreasing and odd about u and there exists C>0 such that ∫Ωk(x)G(u)dx ≤ C,then there admits at least one solution u ∈ H01(Ω)to the equation(1)by the method of constrained manifolds.Then,we give a local description of the solution based on the existence of the solution:Let h be a positive function in Lolcα(Ω)∩ L1(Ω)with α>N/2 such that(?)ω(?)Ω(?)cω:h(x)cω>0,x∈ω.Let {un}(?)H01(Ω)∩Cloc0(Ω)be a solutions sequence of (?) and with the compatible condition n∈N sup∫Ωf(x)|un|1-pdx<+∞.we have a local uniform estimate for the solutions to the equation(2)by the iterative method.Namely,for any compact set K in Ω,there exists CK>0 such that x∈K maxun(x)≤ CK,for any n∈N.