In this note we reduce the problem of geodesic connectedness in a wide class of Godel type spacetimes to the search of critical points of a functional naturally involved in the study of geodesics in standard static spacetimes. Then, by using some known accurate results on the latter, we improve previous results on the former. (C) 2011 Elsevier B.V. All rights reserved.

In this paper we state an abstract multiplicity theorem which generalizes the well known Pucci-Serrin result as it allows one to prove the existence of a third critical point for functionals which are smooth in a Banach space but satisfy a kind of Palais-Smale condition with respect to a weaker norm. This result applies for proving that, under suitable assumptions, the functional \[ J_\lambda(u) = \int_\Omega A(x,u)(|\nabla u|^p - \lambda |u|^p)dx + \int_\Omega G(x,u) dx \] admits at least three distinct critical points in the Banach space $W^{1,p}_0(\Omega) \cap L^\infty(\Omega)$ but if $\lambda$ is large enough.

In the recent paper [3] we prove the existence of normal geodesics connecting quite general submanifolds of a globally hyperbolic stationary spacetime. In this note we focus on timelike geodesics. In particular we extend the Avez–Seifert theorem to normal geodesics connecting two submanifolds.

The second order differential equation x'' = F(x')-grad(V) on a Lorentzian manifold describes, in particular, the dynamics of particles under the action of an electromagnetic field F and a conservative force -grad(V). We provide a first study on the extendability of its solutions, by imposing some natural assumptions

We analyze the extendability of the solutions to a certain second order differential equation on a Riemannian manifold (M,g), which is defined by a general class of forces (both prescribed on M or depending on the velocity). The results include the general time-dependent anholonomic case, and further refinements for autonomous systems or forces derived from a potential are obtained. These extend classical results for Lagrangian and Hamiltonian systems. Several examples show the optimality of the assumptions as well as the applicability of the results, including an application to relativistic pp-waves.

We study a nonlinear elliptic system of Lane--Emden type \[\left\{ \begin{array}{ll} -\Delta u\ =\ { m sgn}(v) |v|^{p-1} & ext{in $\R^{N}$,} \\ -\Delta v\ =\ - { m sgn}(u) |u|^{\frac1{p-1}} + f(u) & ext{in $\R^{N}$,} \\ u,\ v\ ightarrow 0 \quad ext{as}\quad \left\vert x ight\vert ightarrow +\infty,& \end{array} ight. \] which is equivalent to a fourth order elliptic equation. By using variational methods the existence of radial solutions of the given problem is proved. To this aim, new compact imbeddings are stated.

The aim of this paper is to review and complete the study of geodesics on Gödel type spacetimes initiated in [CS] and improved in [BCF]. In particular, we prove some new results on geodesic connectedness and geodesic completeness for these spacetimes.

We find multiple solutions for a nonlinear perturbed Schr\"odinger equation by means of the so--called Bolle's method.

The aim of this paper is investigating the existence of weak solutions of the quasilinear elliptic model problem \[ \left\{ \begin{array}{lr} - \divg (A(x,u)\, |\nabla u|^{p-2}\, \nabla u) \dfrac1p\, A_t(x,u)\, |\nabla u|^p\ =\ f(x,u) & \hbox{in $\Omega$,}\\ u\ = \ 0 & \hbox{on $\partial\Omega$,} \end{array} ight.\] where $\Omega \subset \R^N$ is a bounded domain, $N\ge 2$, $p > 1$, $A$ is a given function which admits partial derivative $A_t(x,t) = \frac{\partial A}{\partial t}(x,t)$ and $f$ is asymptotically $p$-linear at infinity. Under suitable hypotheses both at the origin and at infinity, and if $A(x,\cdot)$ is even while $f(x,\cdot)$ is odd, by using variational tools, a cohomological index theory and a related pseudo--index argument, we prove a multiplicity result if $p > N$ in the non--resonant case.

In this paper we prove the existence of multiple nontrivial solutions for the quasilinear equation in divergence form \[ - { m div} (a(x,u,\nabla u)) + A_t(x,u,\nabla u) = \lambda b(x,u) - g(x,u) \;\hbox{in $\Omega$,}\quad u = 0\; \hbox{on $\partial\Omega$,} \] in an open bounded domain $\Omega \subset \R^N$, where $A :\Omega imes \R imes \R^N o \R$ is a given Carathéodory function with partial derivatives $A_t(x,t,\xi) = \frac{\partial A}{\partial t}(x,t,\xi)$ and $a(x,t,\xi) = (\frac{\partial A}{\partial \xi_1}(x,t,\xi),\dots,\frac{\partial A}{\partial \xi_N}(x,t,\xi))$. It generalizes the $p$-Laplacian problem \[ - \Delta_p u = \lambda |u|^{p-2}u - g(x,u), \qquad u \in W^{1,p}_0(\Omega), \] but, in general, the corresponding functional is not well defined in all the space $W^{1,p}_0(\Omega)$. Anyway, under suitable assumptions and by using variational tools, we are able to prove that the number of solutions of $(P_\lambda)$ depends on the parameter $\lambda$ and, even in lack of symmetry, at least three nontrivial solutions exist if $\lambda$ is large enough.

The aim of this paper is investigating the multiplicity of weak solutions of the quasilinear elliptic equation \[ -\Delta_p u + V(x)|u|^{p-2}u\ =\ g(x, u), \quad x \in\R^N, \] where $1<p<+\infty$, $\Delta_p u= { m div}(|\nabla u|^{p-2}\nabla u)$, the nonlinearity $g$ behaves as $|u|^{p-2}u$ at infinity and $V$ is a potential satisfying the assumptions in \cite{bf}, so that a suitable embedding theorem for weighted Sobolev spaces holds. Both the non--resonant and the resonant case are analyzed.

In this paper we obtain an existence theorem for normal geodesics joining two given submanifolds in a globally hyperbolic stationary spacetime M. The proof is based on both variational and geometric arguments involving the causal structure of M, the completeness of suitable Finsler metrics associated to it and some basic properties of a submersion. By this interaction, unlike previous results on the topic, also non-spacelike submanifolds can be handled.

This paper deals with a generalization of the $p$-Laplacian type boundary value problem \[ \left\{\begin{array}{ll} - { m div} (p \bar A(x,u) |\nabla u|^{p-2}\nabla u) + \bar A_t(x,u) |\nabla u|^p = g(x,u) & \hbox{in $\Omega$,}\\ u = 0 & \hbox{on $\bdry{\Omega}$,} \end{array} ight. \] $\Omega$ being a bounded domain in $\R^N$. Under suitable assumptions and if $p > N$, the existence of a nontrivial solution can be proved by means of variational tools and a cohomological local splitting.

Starting from a new sum decomposition of $ W^{1,p}(\R^N)\cap W^{1,q}(\R^N)$ and using a variational approach, we investigate the existence of multiple weak solutions of a (p,q)-Laplacian equation on $\R^N$, for 1<q<p<N, with a sign-changing potential and a Carath\'eodory reaction term satisfying the celebrated Ambrosetti-Rabinowitz condition. Our assumptions are mild and different from those used in related papers and moreover our results improve or complement previous ones for the single p-Laplacian.

The aim of this paper is investigating the existence of solutions of the semilinear elliptic problem \[\left\{\begin{array}{ll} \displaystyle{-\Delta u\ =\ p(x, u) + \varepsilon g(x, u)} & \mbox{ in } \Omega,\\ \displaystyle{u=0} & \mbox{ on } \partial\Omega,\\ \end{array} ight.\] where $\Omega$ is an open bounded domain of $\R^N$, $\varepsilon\in\R$, $p$ is subcritical and asymptotically linear at infinity and $g$ is just a continuous function. Even when this problem has not a variational structure on $H^1_0(\Omega)$, suitable procedures and estimates allow us to prove that the number of distinct crtitical levels of the functional associated to the unperturbed problem is ``stable'' under small perturbations, in particular obtaining multiplicity results if $p$ is odd, both in the non-resonant and in the resonant case.

The aim of this paper is investigating the existence and the multiplicity of solutions of a quasilinear elliptic problem of p-Laplacian type on an open bounded domain of $R^N$ with smooth boundary and the nonlinearity g behaves as $u^{p-1}$ at infinity. The main tools of the proof are some abstract critical point theorems, but extended to Banach spaces, and two sequences of quasi-eigenvalues for the p-Laplacian operator. The research that led to the present paper was partially supported by a grant of the group GNAMPA of INdAM

Recently, classical results on completeness of trajectories of Hamiltonian systems obtained at the beginning of the seventies, have been revisited, improved and applied to Lorentzian Geometry. Our aim here is threefold: to give explicit proofs of some technicalities in the background of the specialists, to show that the introduced tools allow to obtain more results for the completeness of the trajectories, and to apply these results to the completeness of spacetimes that generalize classical plane and pp--waves.

In this paper we study a class of quasilinear elliptic systems of the type \[\left\{\begin{array}{ll} - \divg(a_1(x,\nabla u_1,\nabla u_2))\ =\ f_1(x,u_1,u_2) & ext{in } \Omega,\\ - \divg(a_2(x,\nabla u_1,\nabla u_2))\ =\ f_2(x,u_1,u_2) & ext{in } \Omega,\\ u_1 = u_2 = 0 & ext{on } \partial \Omega, \end{array} ight.\] with $\Omega$ bounded domain in $\R^N$. We assume that $A : \Omega imes \mathbb{R}^N imes \mathbb{R}^N ightarrow \mathbb{R}$, $F : \Omega imes \mathbb{R} imes \mathbb{R} ightarrow \mathbb{R}$ exist such that $a=(a_1,a_2)=\nabla A$ satisfies the so called Leray-Lions conditions and $f_1=\frac{\partial F}{\partial u_1}$, $f_2=\frac{\partial F}{\partial u_2}$ are Carathéodory functions with {\sl subcritical growth}. The approach relies on variational methods and, in particular, on a cohomological local splitting which allows one to prove the existence of a nontrivial solution.