Nonlinear delay equations with nonautonomous past
Genni Fragnelli Dimitri Mugnai
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.
keywords: evolution family nonlinear delay equations genetic repression. local semigroup
Bounce on a p-Laplacian
Dimitri Mugnai
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.
Almost uniqueness result for reversed variational inequalities
Dimitri Mugnai
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.
keywords: Reversed variational inequalities uniqueness.
Stability of solutions for nonlinear wave equations with a positive--negative damping
Genni Fragnelli Dimitri Mugnai
We prove a stability result for damped nonlinear wave equations, when the damping changes sign and the nonlinear term satisfies a few natural assumptions.
keywords: positive-negative damping. Damped nonlinear wave equations
Robin problems for the p-Laplacian with gradient dependence
Genni Fragnelli Dimitri Mugnai Nikolaos S. Papageorgiou

We consider a nonlinear elliptic equation with Robin boundary condition driven by the p-Laplacian and with a reaction term which depends also on the gradient. By using a topological approach based on the Leray-Schauder alternative principle, we show the existence of a smooth solution.

keywords: Convection term nonlinear regularity Leray-Schauder alternative principle maximal monotone map compact embedding
A fractional eigenvalue problem in $\mathbb{R}^N$
Giacomo Bocerani Dimitri Mugnai
We prove that a linear fractional operator with an asymptotically constant lower order term in the whole space admits eigenvalues.
keywords: asymptotically linear problem. entire solution eigenvalue problem Fractional Laplacian
Positive and nodal solutions for parametric nonlinear Robin problems with indefinite potential
Genni Fragnelli Dimitri Mugnai Nikolaos S. Papageorgiou
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.
keywords: nonlinear regularity and maximum principle Picone identity nodal solutions superlinear reaction Indefinite potential critical groups. bifurcation-type theorem for positive solutions

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