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Existence and uniqueness of entropy solutions to nonlinear elliptic problems with variable growth. (English) Zbl 1243.35056

The authors study the boundary value problem \[ \left\{ \begin{aligned} u-\text{div} (a(x,\nabla u))=f &\;\text{in}\;\Omega,\\ u=0 &\;\text{on}\;\Omega, \end{aligned} \right. \] where \(\Omega\) is a smooth bounded domain in \(R^N(N\geq 3)\) and \(\text{div} (a(x,\nabla u))\) is a \(p(x)\)-Laplace type operator. The main results presented are Theorems 3.2 and 4.3, obtained by using variational arguments, which establish the existence and uniqueness of weak energy solutions, for \(f\in L^\infty(\Omega)\), and entropy solutions, for \(f\in L^1(\Omega)\), to the above problem.
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MSC:

35J25 Boundary value problems for second-order elliptic equations
35J20 Variational methods for second-order elliptic equations
35D30 Weak solutions to PDEs
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
35J60 Nonlinear elliptic equations
35J70 Degenerate elliptic equations
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