Weak solutions of quasilinear problems with nonlinear boundary condition

Abstract

The growing attention for the study of the p-Laplacian operator p in the last few decades is motivated by the fact that it arises in various applications. For instance, in Fluid Mechanics, the shear stress and the velocity gradient ∇pu of certain uids obey a relation of the form (x)= a(x)∇pu(x), where ∇pu= |∇u|p−2∇u. Here p? 1 is an arbitrary real number and the case p= 2 (respectively pi 2; p? 2) corresponds to a Newtonian (respectively pseudoplastic, dilatant) uid. The resulting equations of motion then involve div(a∇pu), which reduces to a pu = a div∇pu, provided that a is constant. The p-Laplacian also appears in the study of torsional creep (elastic for p = 2, plastic as p→∞, see [7]), ow through porous media (p = 2 , see [12]) or glacial sliding (p ∈ (1; 3 ], see [9]). Let ⊂RN be an unbounded domain with (possible noncompact) smooth boundary and n is the unit outward normal on . We consider the nonlinear elliptic boundary value problem: −div(a(x)|∇u|p−2∇u) = (1 + |x|) 1 |u|p−2u+ (1 + |x|) 2 |u|q−2u in ;