Abstract
We prove an existence (and regularity) result of weak solutions u∈W01,p(Ω)∩Wloc1,q(Ω), to a Dirichlet problem for a second order elliptic equation in divergence form, under general and p,q−growth conditions of the differential operator. This is a first attempt to extend to general growth the well known Leray-Lions existence theorem, which holds under the so-called natural growth conditions with q=p. We found a way to treat the general context with explicit dependence on (x,u), other than on the gradient variable ξ=Du; these aspects require particular attention due to the p,q-context, with some differences and new difficulties compared to the standard case p=q.
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