Homogenization of a nonlinear elliptic problem with large nonlinear potential

Abstract

Homogenization is studied for a nonlinear elliptic boundary-value problem with a large nonlinear potential. More specifically we are interested in the asymptotic behavior of a sequence of p-Laplacians of the form $$-\text{div}(a(\frac{x}{\epsilon })|D{u}_{\epsilon }{|}^{p-2}D{u}_{\epsilon })+\frac{1}{\epsilon }V(\frac{x}{\epsilon })|u_{\epsilon }{|}^{p-2}{u}_{\epsilon }=f.$$ It is shown that, under a centring condition on the potential $V$, there exists a two-scale homogenized system with solution $(u,u_1)$ such that the sequence $u_{\epsilon }$ of solutions converges weakly to $u$ in $W^{1,p}$ and the gradients $D_x{u}_{\epsilon }$ two-scale converges weakly to $D_{x}u+D_y{u}_1$ in $L^p$, respectively. We characterize the limit system explicitly by means of two-scale convergence and a new convergence result.