Abstract
We consider a family of quasilinear second order elliptic differential operators which are not coercive and are defined by functions in Marcinkiewicz spaces. We prove the existence of a solution to the corresponding Dirichlet problem. The associated obstacle problem is also solved. Finally, we show higher integrability of a solution to the Dirichlet problem when the datum is more regular.
Highlights
Given a bounded domain Ω of RN, N ≥ 2, we considerA : (x, u, ξ) ∈ Ω × R × RN → RNCommunicated by A
By testing the problems with a suitable admissible test functions, we show that the sequence of solutions to the approximating problems is compact and its limit is a solution to the original problem (6)
We prove an existence result in the same spirit of [19]
Summary
In view of Sobolev embedding theorem in Lorentz spaces [2,21,31], by (2) and the assumptions on b, band φ, for each u ∈ W01,p(Ω). Our conditions allow us to consider operators with a singular coefficient in the lower-order term. The feature of Problem (6) is the lack of coercivity for the operator (4) and the singularity in the lower order term due to property of b and b. Page 3 of 20 83 embedding theorem the lower order term bu ∈ L p(Ω) and it has a norm comparable with the norm of |∇u|. It is well known that, if the operator in (4)-(5) is coercive, a solution to problem (6) exists.
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