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In this article, we consider a nonlinear elliptic unilateral equation whose model is −∑i=1N∂iσi(x,u,∇u)+L(x,u,∇u)+N(x,u,∇u)=μ−divϕ(u)inΩ.-\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−divF\mu =f-{\rm{div}}F belongs to L1(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. |
