Existence Of Radial Solutions To A Free Boundary Value Problems For Nonlinear Elliptic Equation

In this paper ,we considers the existence of a radial solution to the following free boundary problem for nonlinear elliptic equation:Where B is the unit ball in Rⁿ . μ is an unknown constant , M is a given positive constant. λ>0,ε>0,p>l,γ denotes the unit outer normal on (?)B .The following free boundary problem for nonlinear elliptic equation is a model which arises in the study of the equilibrium state of a plasma confined in a toroidal cavity:where Ω is a boundary domain in Rⁿ with a C¹ boundary . λ , μ , M are the same as the above paragraph . γ denotes the unit outer normal on (?)Ω .The method used to study the existence of the solutions to the problem (0.2) was variational method . Berestyki and Brezis who used the variational priciple first obtained the existence of the solutions of the problem (0.2) , if 1 < p < n/(n₂) = P^*. Gershon Wolansky further proved the existence of the solutions of (0.2) , if p^* ≤ p < 2^* =( (n+2)/(n-2)) Moreover the following result has been proved: when Ω is given by the unit ball B ∈ Rⁿ , (0.2) admits an unique solution for 0 < M < M^* , and is not solvable for M > M^* . For M = M^* , (0.2) admits a continuum of solution . Here M^* = ω_n (?)_o^ro s^n-1ψ^p*(s)ds . ω_n is the area of a unit sphere S^n-l ∈ Rⁿ . ψ(r) is a solution of the following initial value problem...