Quasilinear elliptic equations with sub-natural growth aand nonlinar potentials
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Necessary and sufficient conditions for the existence of finite energy and weak solutions are given. Sharp global pointwise estimates of solutions are obtained as well. We also discuss the uniqueness and regularity properties of solutions. As a consequence, characterization of solvability of the equations with singular natural growth in the gradient terms is deduced. Our main tools are Wolff potential estimates, dyadic models, and related integral inequalities. Special nonlinear potentials of Wolff type ssociated with ``sublinear'' problems are constructed to obtain sharp bounds of solutions. We also treat equations with the fractional Laplacians. Our approach is applicable to more general quasilinear A-Laplace operators as well as the fully nonlinear k-Hessian operators.
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