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We prove the existence of multiple positive BV-solutions of the Neumann problem $$ \begin{cases} \displaystyle -\left(\frac{u'}{\sqrt{1+u'^2}}\right)'=a(x)f(u)\quad&\mbox{in }(0,1), u'(0)=u'(1)=0,& {cases} $$ where $a(x) > 0$ and $f$ belongs to a class of nonlinear functions whose prototype example is given by $f(u) = -��u + u^p$, for $��> 0$ and $p > 1$. In particular, $f(0)=0$ and $f$ has a unique positive zero, denoted by $u_0$. Solutions are distinguished by the number of intersections (in a generalized sense) with the constant solution $u = u_0$. We further prove that the solutions found have continuous energy and we also give sufficient conditions on the nonlinearity to get classical solutions. The analysis is performed using an approximation of the mean curvature operator and the shooting method., 28 pages, 1 figure

We investigate the existence and the multiplicity of solutions of the problem (Formula presented.) where Ω is a smooth, bounded domain of (Formula presented.), 1 < p < q < ∞, and the nonlinearity g behaves as uq − 1 at infinity. We use variational methods and find multiple solutions as minimax critical points of the associated energy functional. Under suitable assumptions on the nonlinearity, we cover also the resonant case.

In this paper we deal with the equation \[-\Delta_p u+|u|^{p-2}u=|u|^{q-2}u\] for $1p$, under Neumann boundary conditions in the unit ball of $\mathbb R^N$. We focus on the three positive, radial, and radially non-decreasing solutions, whose existence for $q$ large is proved in [13]. We detect the limit profile as $q\to\infty$ of the higher energy solution and show that, unlike the minimal energy one, it converges to the constant $1$. The proof requires several tools borrowed from the theory of minimization problems and accurate a priori estimates of the solutions, which are of independent interest., Comment: 14 pages, revised version

We consider the $p$-Laplacian equation $-\Delta_p u=1$ for $1, Comment: 18 pages, 0 figures

We establish existence of positive non-decreasing radial solutions for a nonlocal nonlinear Neumann problem both in the ball and in the annulus. The nonlinearity that we consider is rather general, allowing for supercritical growth (in the sense of Sobolev embedding). The consequent lack of compactness can be overcome, by working in the cone of non-negative and non-decreasing radial functions. Within this cone, we establish some a priori estimates which allow, via a truncation argument, to use variational methods for proving existence of solutions. As a side result, we prove a strong maximum principle for nonlocal Neumann problems, which is of independent interest., Comment: 32 pages, 0 figures. In version v2, two typos in Lemma 3.6 have been fixed, with respect to the published version

For p > 2, we consider the quasilinear equation \begin{document}$-\Delta_p u+|u|^{p-2}u=g(u)$\end{document} in the unit ball B of \begin{document}$\mathbb R^N$\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}$g(u)=|u|^{q-2}u$\end{document} , we detect the asymptotic behavior of these solutions as \begin{document}$q\to \infty$\end{document} .

We study a partial differential inclusion, driven by the p-Laplacian operator, involving a p-superlinear nonsmooth potential, and subject to Neumann boundary condi tions. By means of nonsmooth critical point theory, we prove the existence of at least two constant sign solutions (one positive, the other negative). Then, by applying the nonsmooth Morse identity, we find a third non-zero solution.

In this paper, we study the static Born–Infeld equation -div(∇u1, SCOPUS: ar.j, info:eu-repo/semantics/published

We analyze existence, multiplicity and oscillatory behavior of positive radial solutions to a class of quasilinear equations governed by the Lorentz-Minkowski mean curvature operator. The equation is set in a ball or an annulus of $\mathbb R^N$, is subject to homogeneous Neumann boundary conditions, and involves a nonlinear term on which we do not impose any growth condition at infinity. The main tool that we use is the shooting method for ODEs., 13 pages, 4 figures

The composite plate problem is an eigenvalue optimization problem related to the fourth order operator $(-\Delta)^2$. In this paper we continue the study started in [10], focusing on symmetry and rigidity issues in the case of the hinged composite plate problem, a specific situation that allows us to exploit classical techniques like the moving plane method., Comment: 19 pages, 2 figures

Let $10 \mbox{ in } \Omega, \quad \partial_\nu u = 0 \mbox{ on } \partial\Omega. \] We suppose that $f(0)=f(1)=0$ and that $f$ is negative between the two zeros and positive after. In case $\Omega$ is a ball, we also require that $f$ grows less than the Sobolev-critical power at infinity. We prove a priori bounds of radial solutions, focusing in particular on solutions which start above 1. As an application, we use the shooting technique to get existence, multiplicity and oscillatory behavior (around 1) of non-constant radial solutions., Comment: 26 pages, 3 figures

For $10\text{ in }\Omega,\quad\partial_\nu u=0\text{ on }\partial\Omega, $$ where $\Omega\subset\mathbb R^N$ is either a ball or an annulus. The nonlinearity $f$ is possibly supercritical in the sense of Sobolev embeddings; in particular our assumptions allow to include the prototype nonlinearity $f(s)=-s^{p-1}+s^{q-1}$ for every $q>p$. We use the shooting method to get existence and multiplicity of non-constant radial solutions. With the same technique, we also detect the oscillatory behavior of the solutions around the constant solution $u\equiv1$. In particular, we prove a conjecture proposed in [D. Bonheure, B. Noris, T. Weth, {\it Ann. Inst. H. Poincar\'e Anal. Non Lin\'aire} vol. 29, pp. 573-588 (2012)], that is to say, if $p=2$ and $f'(1)>\lambda_{k+1}^{rad}$, there exists a radial solution of the problem having exactly $k$ intersections with $u\equiv1$ for a large class of nonlinearities., Comment: 22 pages, 4 figures

In this paper we deal with the composite plate problem, namely the following optimization eigenvalue problem $$ \inf_{\rho \in \mathrm{P}} \inf_{u \in \mathcal{W}\setminus\{0\}} \frac{\int_{\Omega}(\Delta u)^2}{\int_{\Omega} \rho u^2}, $$ where $\mathrm{P}$ is a class of admissible densities, $\mathcal{W}= H^{2}_{0}(\Omega)$ for Dirichlet boundary conditions and $\mathcal W= H^2(\Omega) \cap H^1_{0}(\Omega)$ for Navier boundary conditions. The associated Euler-Lagrange equation is a fourth-order elliptic PDE governed by the biharmonic operator $\Delta^2$. In the spirit of [10], we study qualitative properties of the optimal pairs $(u,\rho)$. In particular, we prove existence and regularity and we find the explicit expression of $\rho$. When $\Omega$ is a ball, we can also prove uniqueness of the optimal pair, as well as positivity of $u$ and radial symmetry of both $u$ and $\rho$., Comment: 26 pages

In this paper we establish the existence of an unbounded sequence of solutions for a class of quasilinear elliptic p(x)-polyharmonic Kirchhoff equations, including the new delicate degenerate case, not yet covered in the literature. The main tool is the symmetric mountain pass theorem of Ambrosetti and Rabinowitz. © 2011 Elsevier Ltd. All rights reserved.

We look for nonconstant, positive, radially nondecreasing solutions of the quasilinear equation $-\Delta_p u+u^{p-1}=f(u)$ with $p>2$, in the unit ball $B$ of $\mathbb R^N$, subject to homogeneous Neumann boundary conditions. The assumptions on the nonlinearity $f$ are very mild and allow it to be possibly supercritical in the sense of Sobolev embeddings. The main tools used are the truncation method and a mountain pass-type argument. In the pure power case, i.e., $f(u)=u^{q-1}$, we detect the limit profile of the solutions of the problems as $q\to\infty$., Comment: 9 pages, 2 figures, BRU-TO PDE's Conference

We study an eigenvalue problem in the framework of double phase variational integrals and we introduce a sequence of nonlinear eigenvalues by a minimax procedure. We establish a continuity result for the nonlinear eigenvalues with respect to the variations of the phases. Furthermore, we investigate the growth rate of this sequence and get a Weyl-type law consistent with the classical law for the $p$-Laplacian operator when the two phases agree., Comment: 42 pages, typos corrected, final version, to appear in Ann. Mat. Pura Appl

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. Fil: Agnelli, Juan Pablo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina Fil: Colasuonno, F.. Politecnico di Torino; Italia Fil: Knopoff, Damián Alejandro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina

In the framework of variable exponent Sobolev spaces, we prove that the variational eigenvalues defined by inf sup procedures of Rayleigh ratios for the Luxemburg norms are all stable under uniform convergence of the exponents., 10 pages

This paper proposes a systems theory approach to the modeling of onset and evolution of criminality in a territory, which aims at capturing the complexity features of social systems. Complexity is related to the fact that individuals have the ability to develop specific heterogeneously distributed strategies, which depend also on those expressed by the other individuals. The modeling is developed by methods of generalized kinetic theory where interactions and decisional processes are modeled by theoretical tools of stochastic game theory., 20 pages

This paper presents the analysis of a system of ordinary differential equations modeling a socio-political competition toward a possible onset of extreme conflicts. The dependent variable is a probability density distribution, while the equations are characterized by quadratic type nonlinearities. The model was derived within the framework of the kinetic theory for active particles, where interactions are modeled according to game-theoretical tools. Global existence and uniqueness of the solutions to the initial value problem related to the model are proved in a special case. The main tool used is the Banach–Caccioppoli fixed point theorem.

In this paper we establish the existence of two nontrivial weak solutions of some eigenvalue problems, involving a general elliptic operator in divergence form. First we study a one parameter problem under homogeneous Dirichlet boundary conditions in bounded domains, and then a two parameters problem under inhomogeneous nonlinear Robin boundary conditions in regular bounded domains. These results complete in several directions some recent papers, where a Ricceri type critical point theorem was used. The main argument of this paper is based on two recent theorems due to Arcoya and Carmona (2007) in [2] which we use in a slightly modified version, see Section 2. © 2011 Elsevier Ltd. All rights reserved.

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. © 2012 World Scientific Publishing Company.

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. © 2012 Copyright Taylor and Francis Group, LLC.

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 [Formula: see text] 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 [Formula: see text], 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 λ1of the Rayleigh quotient for the p(x)-polyharmonic operator [Formula: see text].

By a combination of variational and topological techniques in the presence of invariant cones, we detect a new type of positive axially symmetric solutions of the Dirichlet problem for the elliptic equation $$ -\Delta u + u = a(x)|u|^{p-2}u $$ in an annulus $A \subset \mathbb R^N$ ($N\ge3$). Here $p>2$ is allowed to be supercritical and $a(x)$ is an axially symmetric but possibly nonradial function with additional symmetry and monotonicity properties, which are shared by the solution $u$ we construct. In the case where $a$ equals a positive constant, we detect conditions, only depending on the exponent $p$ and on the inner radius of the annulus, that ensure that the solution is nonradial., Comment: Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 23 pages

We face a rigidity problem for the fractional $p$-Laplace operator to extend to this new framework some tools useful for the linear case. It is known that $(-\Delta)^s(1-|x|^{2})^s_+$ and $-\Delta_p(1-|x|^{\frac{p}{p-1}})$ are constant functions in $(-1,1)$ for fixed $p$ and $s$. We evaluated $(-\Delta_p)^s(1-|x|^{\frac{p}{p-1}})^s_+$ proving that it is not constant in $(-1,1)$ for some $p\in (1,+\infty)$ and $s\in (0,1)$. This conclusion is obtained numerically thanks to the use of very accurate Gaussian numerical quadrature formulas., Comment: 18 pages, 8 figures

This paper deals with existence and multiplicity of positive solutions for a quasilinear problem with Neumann boundary conditions. The problem is set in a ball and admits at least one constant non-zero solution; moreover, it involves a nonlinearity that can be supercritical in the sense of Sobolev embeddings. The main tools used are variational techniques and the shooting method for ODE's. These results are contained in A. Boscaggin, F. Colasuonno, B. Noris. Multiple positive solutions for a class of p-Laplacian Neumann problems without growth conditions. ESAIM Control Optim. Calc. Var., DOI: 10.1051/cocv/2016064 (2017; F. Colasuonno, B. Noris. A p-Laplacian supercritical Neumann problem. Discrete Contin. Dyn. Syst., 37 (2017) 3025-3057., Bruno Pini Mathematical Analysis Seminar, Seminars 2017