Nonlinear elliptic equations of order 2m and subdifferentials

Abstract

LetA be an operator of the calculus of variations of order 2m onW ^m,p(Ω) andj a normal convex integrand. Forf ∈L ^p(Ω), the equation

$$\mathcal{A}u + \partial j(x,u) \ni f, in \Omega , u - \phi \in W_0^{m,p} (\Omega ),$$

may have no strong solutions whenm>1, even ifj is independent ofx and φ=0. However, we obtain existence results whenj is everywhere finite and

$$\int_\Omega {j(x,\phi ) dx< + \infty ,} $$

by the study of the subdifferential of the function

$$\upsilon \mapsto \int_\Omega {j(x,\upsilon + \phi ) dx on W_0^{m,p} (\Omega ).} $$