Existence of solutions to the Dirichlet problem for implicit elliptic equationsis established by using Krasnoselskii–Schaefer type theorems owed to Burton–Kirk andGao–Li–Zhang. The nonlinearity of the equations splits into two terms: one term depending on the state, its gradient and the elliptic principal part is Lipschitz continuous,and the other one only depending on the state and its gradient has a superlinear growthand satisfies a sign condition. Correspondingly, the associated operator is a sum of acontraction with a completely continuous mapping. The solutions are found in a ballof a Lebesgue space of a sufficiently large radius established by the method of a prioribounds.
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