Existence of Solutions of a Class of Nonlinear Singular Equations in Lorentz Spaces

Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 229)

Abstract

We consider the following nonlinear elliptic Dirichlet problem involving a Leray-Lions type differential operator \( \rm {-div}(\psi(x,u(x),\bigtriangledown u(x))) + a(x)u(x)=f(x), \quad \rm {in} \; \Omega, \;\; u \in W^{1,p}_{0}(\Omega)\) where \( \Omega \subset \mathbb{R}^{N} \) is a bounded domain with smooth boundary, \( 2 \leq p < N, a \in L^{\infty}_{\rm loc}(\Omega;)\mathbb{R}^{+}_{0}\; \rm{and}\; f \in L^{q,q_{1}(\Omega)} \) is a function in a Lorentz space. We show the existence of a solution \( u \in W^{1,p}_{0}(\Omega) \cap L ^{r,s}(\Omega) \) and an a priori estimate for the solution with respect to the Lorentz space norm of \( f \in L^{q,q_{1}}(\omega) \) Ω), for suitable values \( p,q,q_{1},r \; \rm{and}\; s \)

Keywords

Elliptic equations Leray–Lions operator Lorentz spaces existence of solutions. 

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Center for Research and Development in Mathematics and Applications Department of MathematicsUniversity of AveiroAveiroPortugal

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