ON A NONLINEAR BOUNDARY VALUE PROBLEM WITH A MIXED BOUNDARY CONDITION: ASYMPTOTIC BEHAVIOR OF A SOLUTION

Abstract

We study the following nonlinear boundary value problem (-1)/x^y d/dx (x^y.|u^,(x)|^(p-2) u^,(x))+f(x,u(x))=F(x),0<x<1, (1) |lim┬(x→0_+ )⁡〖x^γ⁄p_(u^,(x)) 〗 | < +∞ ,|u^,(1)|^(p-2) u^,(1)+h.u(1)=g (2) where  > 0, p  2, h > 0, g are given constants, f, F are given functions. In this paper, we use the Galerkin and compactness method in appropriate Sobolev spaces with weight to prove the existence of a unique weak solution of the problem (1),(2). Afterwards, we also study the asymptotic behavior of the solution uh depending on h as h→0+. We also obtain that the function h  |u_h (1)| is nonincreasing on (0, +∞).