Generalised vectorial 8-eigenvalue nonlinear problems for L∞ functionals

Let omega & nbsp; C= R-n, f E C1(RNxn) and g E C1(RN), where N, n E N. We study the minimisation problem of finding u E W01,ao(& OHM;; RN) that satisfies & nbsp;f(Du)L infinity(Q) {? = inf ?f (Dv)} : vE W1,& INFIN; 0 (Q; RN), IIg(v)IIL & INFIN;(Q) = 1 , L & INFIN;(Q)& nbsp;under natural assumptions on f, g. This includes the oo-eigenvalue problem as a special case. Herein we prove the existence of a minimiser uao with extra properties, derived as the limit of minimisers of approximating constrained Lp problems as p-+ oo. A central contribution and novelty of this work is that uao is shown to solve a divergence PDE with measure coefficients, whose leading term is a divergence counterpart equation of the non-divergence infinity-Laplacian. Our results are new even in the scalar case of the infinity-eigenvalue problem.(C) 2022 Published by Elsevier Ltd.