Existence results for dirichlet problems in L¹ via minty's lemma

Abstract

In some recent papers, existence and qualitative properties of solutions for nonlinear elliptic equations with right hand side measures have been proved (see the references). This paper is concerned with the existence of solutions of elliptic problems as

where A(u) = −div(a(x,u, Δu)) is a Leray Lions operator defined from W^1,p ₀(ω) into its dual, ω is a bounded domain of R^N, N≥2,fεL¹(ω) and |F|εL^p'(ω). We will give an existence result for the so called entropy solutions, using as main tool an L¹ version of Minty's Lemma. This approach allows both to extend the known existence results (see [1]) to the case of operators defined by a function a which is not strictly monotone with respect to the gradient, and to avoid the technical result of almost everywhere convergence for gradients of suitable approximating solutions (see [2], [4], [5] and [11]) which was needed in order to prove existence.

Keywords:

• Elliptic equations
• measure data
• Minty's lemma

AMS:

• 35J65
• 35D05
• 35R05