L1 Existence and Uniqueness of Entropy Solutions to Nonlinear Multivalued Elliptic Equations with Homogeneous Neumann Boundary Condition and Variable Exponent

Abstract

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In this work, we study the following nonlinear homogeneous Neumann boundary value problem $\u03b2(u)\u2212diva(x,\u2207u) \u220b f in \u03a9, a(x,\u2207u)\u22c5\u03b7$ $=0$ on $\u2202\u03a9$, where $\u03a9$ is a smooth bounded open domain in $\u211c^N, N \u2265 3$ with smooth boundary $\u2202\u03a9$ and $\u03b7$ the outer unit normal vector on $\u2202\u03a9$. We prove the existence and uniqueness of an entropy solution for L\u00b9-data f. The functional setting involves Lebesgue and Sobolev spaces with variable exponent.<\/p>"