The obstacle problem for non-coercive equations with lower order term and L^{1}-data

Jun Zheng

doi:10.1186/s13660-019-2157-9

Abstract

The aim of this paper is to study the obstacle problem associated with an elliptic operator having degenerate coercivity, a low order term, and L^{1}-data. We prove the existence of an entropy solution to the obstacle problem and show its continuous dependence on the L^{1}-data in W^{1,q}(varOmega ) with some q>1.

Highlights

1.1 Problem setting and main result Let Ω be a bounded domain in RN (N ≥ 2), 1 < p < +∞, and θ ≥ 0
Ψ ∈ W 1,p(Ω) ∩ L∞(Ω) and data f ∈ L1(Ω), the aim of this paper is to study the obstacle problem for nonlinear non-coercive elliptic equations with lower order term, governed by the operator
The classical methods used to prove the existence of a solution for elliptic equations, e.g., [14], cannot be applied even if b = 0 and the data f is regular

Summary

Introduction

1.1 Problem setting and main result Let Ω be a bounded domain in RN (N ≥ 2), 1 < p < +∞, and θ ≥ 0. Ψ ∈ W 1,p(Ω) ∩ L∞(Ω) and data f ∈ L1(Ω), the aim of this paper is to study the obstacle problem for nonlinear non-coercive elliptic equations with lower order term, governed by the operator. Where b > 0 is a constant, and a : Ω × RN → RN is a Carathéodory function, satisfying the following conditions: a(x, ξ ) · ξ ≥ α|ξ |p,. For almost every x in Ω and for every ξ , η, ζ in RN with ξ = η, where α, β, γ > 0 are constants, and j is a nonnegative function in Lp (Ω)

Zheng Journal of Inequalities and Applications

For r

Since exists tε such that

It follows

Using the fact that

Now assume that