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Some recent results on singular p-Laplacian equations

Guarnotta Umberto, Livrea Roberto, Marano Salvatore A.

Demonstratio Mathematica

2022

Vol. 55, nr 1

416--428

A short account of some recent existence, multiplicity, and uniqueness results for singular p-Laplacian problems either in bounded domains or in the whole space is performed, with a special attention to the case of convective reactions. An extensive bibliography is also provided.

Optimality conditions for a class of optimal boundary control problems with quasilinear elliptic equations

Casas E., Dhamo V.

Control and Cybernetics

2011

Vol. 40, no 2

457-490

First- and second-order optimality conditions are established for the boundary optimal control of quasilinear elliptic equations with pointwise constraints on the control. The theory is developed for Neumann controls in polygonal domains of dimension two. For the derivation of second-order sufficient optimality conditions, which is the main goal of this paper, the regularity of the solutions to the state equation and its linearization is studied in detail. Moreover, a Pontryagin principle is proved. The main difficulty in the analysis of these problems is the nonmonotone character of the state equation.