Existence of renormalized solutions to nonlinear elliptic equations with two lower order terms and measure data

Abstract

Abstract. In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is is small enough.

Existence and Stability Results for Renormalized Solutions to Noncoercive Nonlinear Elliptic Equations with Measure Data

“…The counterpart of the existence result proved in the present paper can be found in [25], where in particular we prove the existence of a renormalized solution for the problem (1.1) in the case where µ is a Radon measure with bounded variation on ,…”

“…In the present paper and in [25], we prove the existence of renormalized solutions for the problems whose prototype is (1.1), where both the two lower terms −div(c(x)|u| γ ) and b(x)|∇u| λ appear and where 0 ≤ γ ≤ p − 1, 0 ≤ λ ≤ p − 1, |c| belongs to the Lorentz space L N p−1 ,r ( ), N p−1 ≤ r ≤ +∞, b belongs to Lorentz space L N,1 ( ) and µ is a Radon measure with bounded variation on . In both papers we do not make any coercivity assumption on the operator: we assume that the norm of one of the two coefficients is small when γ = λ = p − 1, while no smallness of such norms is required when γ or λ is less than p − 1.…”

“…Finally we explicitly remark that, as for the existence result proved in the present paper, the main difficulty in proving the existence result of [25] is to obtain some a priori estimate for |∇u| p−1 . Such a priori estimates are obtained by using a different method.…”

Nonlinear Elliptic Problems with L 1 Data: an Approach via Symmetrization Methods

“…We explicitely remark that similar results are in [8], [14], [24], [27] and in [6], [7], [9], [10], [17], [18], [20], [21], where operators with lower order terms are considered.…”

“…The method we used above works well when we consider operators with lower order terms or when the datum is a measure. About these subjects several results are available (see for example [7], [9], [10], [11], [12], [13], [16], [18], [20], [21], [25]). We refer to a forthcoming paper [6] for a thorough analysis of these questions.…”