Homogenization of non-local nonlinear p-Laplacian equation with variable index and periodic structure

Abstract

This paper deals with the homogenization of a one-dimensional nonlinear non-local variable index p(x)-Laplacian operator Lɛ with a periodic structure and convolution kernel. By constructing a scale diffusive model and two corrector functions χ1 and χ2, as scale parameter ɛ → 0+, we first obtain that the limit operator L is a p-Laplacian operator with constant exponent and coefficients such that Lu=Rddx(|u′(x)|p−2u′(x)). Then, for a given function f∈Lq(R)(q=pp−1), we prove the asymptotic behavior of the solution uɛ(x) to the equation (Lɛ − I)uɛ(x) = f(x) such that uε(x)=u(x)+εχ1(xε)u′(x)+ε2χ2(xε)u″(x)+o(1)(ε→0+) in Lp(R), where u(x) is the solution of equation (L − I)u(x) = f(x).