Nonlinear elliptic unilateral problems with measure data in the anisotropic Sobolev space

Abstract

Abstract In this article, we consider a nonlinear elliptic unilateral equation whose model is − ∑ i = 1 N ∂ i σ i ( x , u , ∇ u ) + L ( x , u , ∇ u ) + N ( x , u , ∇ u ) = μ − div ϕ ( u ) in Ω . -\mathop{\sum }\limits_{i=1}^{N}{\partial }^{i}{\sigma }_{i}\left(x,u,\nabla u)+L\left(x,u,\nabla u)+N\left(x,u,\nabla u)=\mu -{\rm{div}}\phi \left(u)\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega . We prove the existence of entropy solutions for the aforementioned equation in the anisotropic Sobolev space, under the hypotheses, μ = f − div F \mu =f-{\rm{div}}F belongs to L 1 ( Ω ) + W − 1 , p ′ ( Ω ) {L}^{1}\left(\Omega )+{W}^{-1,{p}^{^{\prime} }}\left(\Omega ) . The nonlinear terms L ( x , s , ∇ u ) L\left(x,s,\nabla u) satisfy the sign and growth conditions, and N ( x , s , ∇ u ) N\left(x,s,\nabla u) verifies only the growth conditions.