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

Abstract

We consider nonlinear elliptic problems whose prototype iswith Ω bounded open subset of R N and p > 1. When f ∈ L 1 (Ω) several notions of solutions have been introduced; we refer to distributional solutions which can be obtained by an approximation procedure and point out that the question can be faced by a new method which uses symmetrization techniques. In this way we prove both a priori estimates and a continuity with respect to data result which allow us to deduce existence and uniqueness of the solution. Mathematics Subject Classification (2000). Primary 35J25; Secondary 35J60.

Gradient estimates via rearrangements for solutions of some Schrödinger equations

“…Thus, when f ∈ L 1 (Ω), we need another way to define the weak solution for the Dirichlet problem (1.1) and the Neumann problem (1.3). We also point out that (non-)linear elliptic problems with L 1 -data have attracted great interests for a long time, since they play important roles in partial differential equations (see, for example, [2,3,4,6,7,15,18,19,36] and the references therein).…”

“…We prove this proposition following the method used in [1,3,15]. We first assume that u and v are the solutions to the Dirichlet problem (1.1).…”

“…In this article, motivated by the work in [1,3,4,6,15], via applying the well known method for estimating the gradient of solutions to p-Laplace type elliptic boundary value problems and some estimates established by Shen [34,35] for the fundamental solution of Schrödinger equations, we obtain a sharp estimate for the decreasing rearrangement of the gradient of solutions to a class of Schrödinger equations with the Dirichlet or the Neumann boundary condition, under the weak assumption for the regularity on the boundary of domains. As applications, we further establish the gradient estimates of solutions to these Schrödinger equations in Lebesgue spaces and Lorentz spaces.…”

“…In this case, the generalized solutions for (1.1) and (1.3) can be defined by an approximating method (see Section 2 below for the details). We point out that the study for the theory of (non-)linear elliptic boundary value problems with L 1 -data has aroused the attention of many mathematicians for quite some time (see, for example, [1,2,3,4,6,7,18,19,36]). …”

An approach via symmetrization methods to nonlinear elliptic problems with a lower order term

“…Under these hypotheses it can be proved that there exists a weak solution u to problems of the type (1) with Φ = 0 (see [11], [12]); such a solution is found by a natural approximation method and is known as SOLA, that is, Solution Obtained as Limit of Approximation (see [15], [16] and [18]). Existence and uniqueness for SOLA to problem (1) has been obtained also in [2] with Φ = 0 and in [3] when the lower order term is of the type b(x) |∇u| p−1 (see also [8]). …”