Francesca Colasuonno

On the existence of stationary solutions for higher-order p-Kirchhoff problems

In this paper, we establish the existence of two nontrivial weak solutions of possibly degenerate nonlinear eigenvalue problems involving the p-polyharmonic Kirchhoff operator in bounded domains. The p-polyharmonic operators were recently introduced in [F. Colasuonno and P. Pucci, Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal.74 (2011) 5962–5974] for all orders L and independently, in the same volume of the journal, in [V. F. Lubyshev, Multiple solutions of an even-order nonlinear problem with convex-concave nonlinearity, Nonlinear Anal.74 (2011) 1345–1354] only for L even. In Sec. 3, the results are then extended to non-degeneratep(x)-polyharmonic Kirchhoff operators. The main tool of the paper is a three critical points theorem given in [F. Colasuonno, P. Pucci and Cs. Varga, Multiple solutions for an eigenvalue problem involving p-Laplacian type operators, Nonlinear Anal.75 (2012) 4496–4512]. Several useful properties of the underlying functional solution space , endowed with the natural norm arising from the variational structure of the problem, are also proved both in the homogeneous case p ≡ Const. and in the non-homogeneous case p = p(x). In the latter some sufficient conditions on the variable exponent p are given to prove the positivity of the infimum λ1 of the Rayleigh quotient for the p(x)-polyharmonic operator .

LIFESPAN ESTIMATES FOR SOLUTIONS OF POLYHARMONIC KIRCHHOFF SYSTEMS

In mathematical physics we increasingly encounter PDEs models connected with vibration problems for elastic bodies and deformation processes, as it happens in the Kirchhoff–Love theory for thin plates subjected to forces and moments. Recently Monneanu proved in Refs. 26 and 27 the existence of a solution of the nonlinear Kirchhoff–Love plate model. In this paper we treat several questions about non-continuation for maximal solutions of polyharmonic Kirchhoff systems, governed by time-dependent nonlinear dissipative and driving forces. In particular, we are interested in the strongly damped Kirchhoff–Love model, containing also an intrinsic dissipative term of Kelvin–Voigt type. Global non-existence and a priori estimates for the lifespan of maximal solutions are proved. Several applications are also presented in special subcases of the source term f and the nonlinear external damping Q.

A kinetic theory approach to the dynamics of crowd evacuation from bounded domains

A mathematical model of the evacuation of a crowd from bounded domains is derived by a hybrid approach with kinetic and macro-features. Interactions at the micro-scale, which modify the velocity direction, are modeled by using tools of game theory and are transferred to the dynamics of collective behaviors. The velocity modulus is assumed to depend on the local density. The modeling approach considers dynamics caused by interactions of pedestrians not only with all the other pedestrians, but also with the geometry of the domain, such as walls and exits. Interactions with the boundary of the domain are non-local and described by games. Numerical simulations are developed to study evacuation time depending on the size of the exit zone, on the initial distribution of the crowd and on a parameter which weighs the unconscious attraction of the stream and the search for less crowded walking directions.

Blow up at infinity of solutions of polyharmonic Kirchhoff systems

This article concerns the blow up at infinity of global solutions of strongly damped polyharmonic Kirchhoff systems, involving lower order terms, a time dependent nonlinear dissipative function Q and a driving force f, under homogeneous Dirichlet boundary conditions. Some applications are presented in special subcases of f and Q.

A p -Laplacian supercritical Neumann problem

For p > 2, we consider the quasilinear equation \begin{document}

Stability of eigenvalues for variable exponent problems

-\Delta_p u+|u|^{p-2}u=g(u)

On the Born–Infeld equation for electrostatic fields with a superposition of point charges

\end{document} in the unit ball B of \begin{document}

Three solutions for a neumann partial differential inclusion via nonsmooth Morse theory

\mathbb R^N

Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations

\end{document} , with homogeneous Neumann boundary conditions. The assumptions on g are very mild and allow the nonlinearity to be possibly supercritical in the sense of Sobolev embeddings. We prove the existence of a nonconstant, positive, radially nondecreasing solution via variational methods. In the case \begin{document}

Multiple solutions for an eigenvalue problem involving p-Laplacian type operators

g(u)=|u|^{q-2}u