A remark on the Leray–Lions condition

A New Topological Degree Theory for Perturbations of Demicontinuous Operators and Applications to Nonlinear Equations with Nonmonotone Nonlinearities

“…Consequently, Theorem 22 provides a new result concerning existence of weak solution for the nonlinear wave equation with lower order nonlinear part satisfying only continuity and sublinearity conditions. In conclusion, we like to mention that various examples of pseudomonotone type operators under Leray-Lion type growth conditions along with (83) can be found in the papers of Landes and Mustonen [26], Mustonen [27], and Mustonen and Tienari [28] and the references therein. Existence results for perturbations of maximal monotone operator by bounded demicontinuous operator of type ( + ) or bounded pseudomonotone can be found in the papers of Browder and Hess [16], Kenmochi [29], Kartsatos [30], Asfaw [31][32][33], and Le [34] and the references therein.…”

Controlling monotonicity of nonlinear operators

“…Controlling the monotonicity and the growth of operators like ∆ p or ∆ a p has a preeminent place in the theory of existence, regularity, and other properties of solutions to nonlinear PDE involving these operators. To give a flavour of a vast literature where such tools are applied we refer to [3,13,14,21,23,24,27,28,30,32]. We point out that in a study of regularity of minimizers to a related variational problem inf u∈u0+W 1,p loc |Du| p dx similar estimates are frequently employed as well, see e.g.…”