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Inspired by a biological model on genetic repression proposed by P. Jacob and J. Monod, we introduce a new class of delay equations with nonautonomous past and nonlinear delay operator. With the aid of some new techniques from functional analysis we prove that these equations, which cover the biological model, are well--posed.
The existence of nontrivial solutions for reversed variational inequalities involving $p$-Laplace operators is proved. The solutions are obtained as limits of solutions of suitable penalizing problems.
keywords: Reversed variational inequalities.
In this note we show that reversed variational inequalities cannot be studied in a general abstract framework as it happens for classical variational inequalities with Stampacchia’s Lemma. Indeed, we provide two different situations for reversed variational inequalities which are of the same type from an abstract point of view, but which behave quite differently.
We prove a stability result for damped nonlinear wave equations, when the damping changes sign and the nonlinear term satisfies a few natural assumptions.
We prove that a linear fractional operator with an asymptotically constant lower order term in the whole space admits eigenvalues.
We consider a parametric nonlinear Robin problem driven by the $p -$Laplacian plus an indefinite potential and a Carathéodory reaction which is $(p-1) -$ superlinear without satisfying the Ambrosetti - Rabinowitz condition. We prove a bifurcation-type result describing the dependence of the set of positive solutions on the parameter. We also prove the existence of nodal solutions. Our proofs use tools from critical point theory, Morse theory and suitable truncation techniques.
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