Non-existence of one-signed solutions for variational equations with locally amplifying potentials

In questo lavoro si considerano equazioni differenziali ordinarie del secondo ordine di tipo singolare della forma

$$[g(t)G_p (u,u')]' - g(t)G_u (u, u') + g(t)f(t, u) = 0$$     

Non-existence of measurable solutions of certain functional equations via probabilistic approaches

Aequationes mathematicae - This paper deals with functional equations in the form of $$f(x) + g(y) = h(x,y)$$ where h is given and f and g are unknown. We will show that if h is a Borel measurable...     

Spurious solutions of Faddeev equations with central potentials

Within the limits of the hyperharmonics approach, spurious solutions of the Faddeev differential equations are studied for a system of three distinguishable particles interacting via central potentials. The criterion for the existence of spurious solutions is proved. A straightforward way of constructing explicitly spurious solutions is suggested, and the results are illustrated by examples.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 3, pp. 501–512, June, 1996.     

Asymptotic behavior of solutions of variational equations

In questo articolo si considerano equazioni differenziali ordinarie del secondo ordine della forma $$\{ A(u')u'\} ' + \delta (r)A(u')u' + f(r,u) = 0,$$ dove cioè la nonlinearità è presente sia nella variabile soluzioneu che nella sua derivata. Si forniscono proprietà di monotonia, oscillazione e un accurato studio del comportamento asintotico all’ infinito delle soluzioni, quandoA, δ,f hanno crescite asintotiche di tipo algebrico.     

Non-existence of positive solutions of some elliptic equations in positive-type domains

In this work we prove a non-existence result for a positive solution of a Dirichlet boundary value problem in a class of domains that are more general than star-shaped ones. Our result extends some earlier non-existence results for the star-shaped case. We also present an example of a domain which satisfies our conditions but not star-shaped essentially.     

Infinitely many positive solutions for nonlinear equations with non-symmetric potentials

We consider the following nonlinear Schrödinger equation $$\begin{aligned} \left\{ \begin{array}{l} \Delta u-(1+\delta V)u+f(u)=0 \ \ \hbox { in }\mathbb {R}^N,\\ u>0 \ \hbox {in} \ \mathbb {R}^N, u\in H^1(\mathbb {R}^N) \end{array} \right. \end{aligned}$$ where \(V\) is a continuous potential and \( f(u)\) is a nonlinearity satisfying some decay condition and some non-degeneracy condition, respectively. Using localized energy method, we prove that there exists a \(\delta _0\) such that for \(0<\delta <\delta _0\) , the above problem has infinitely many positive solutions. This generalizes and gives a new proof of the results by Cerami et al. (Comm. Pure Appl. Math. 66, 372–413, 2013). The new techniques allow us to establish the existence of infinitely many positive bound states for elliptic systems.     

Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem

This paper is devoted to study the existence of periodic solutions of the second-order equation x″=f(t,x), where f is a Carathéodory function, by combining some new properties of Green's function together with Krasnoselskii fixed point theorem on compression and expansion of cones. As applications, we get new existence results for equations with jumping nonlinearities as well as equations with a repulsive or attractive singularity. In this latter case, our results cover equations with weak singularities and are compared with some recent results by I. Rachunková, M. Tvrdý and I. Vrkoc?.     

Consistent solutions of linear equations in variational derivatives

In this paper, we consider the classical existence problem for consistent and uniformly consistent solutions of linear equations in variational derivatives. In particular, we generalize several results obtained by V. V. Zhikov and B. A. Shcherbakov concerning the existence of almost-periodic, consistent, and uniformly consistent solutions of ordinary differential equations.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 176–187.Original Russian Text Copyright © 2005 by A. I. Gerko.This revised version was published online in April 2005 with a corrected issue number.     

Infinitely many solutions of some nonlinear variational equations

The aim of this paper is investigating the existence of one or more critical points of a family of functionals which generalizes the model problem

Consistent solutions of linear equations in variational derivatives

In this paper, we consider the classical existence problem for consistent and uniformly consistent solutions of linear equations in variational derivatives. In particular, we generalize several results obtained by V. V. Zhikov and B. A. Shcherbakov concerning the existence of almost-periodic, consistent, and uniformly consistent solutions of ordinary differential equations.     

Multiple solutions for a class of semilinear elliptic equations with general potentials

In this paper, we study the existence of infinitely many nontrivial solutions for a class of semilinear elliptic equations −△u+a(x)u=g(x,u)$-△u+a\left(x\right)u=g(x,u)$ in a bounded smooth domain of R^N(N≥3)${\mathbb{R}}^N(N\ge 3)$ with the Dirichlet boundary value, where the primitive of the nonlinearity g$g$ is of superquadratic growth near infinity in u$u$ and the potential a$a$ is allowed to be sign-changing. Recent results in the literature are generalized and significantly improved.     

Global variational solutions to the compressible magnetohydrodynamic equations

We prove the existence of globally defined variational solutions to the compressible magnetohydrodynamic (MHD) equations with the coefficients depending on the temperature. As a by-product, we give a simple proof for the nonexistence of nontrivial weak time-periodic solutions by the entropy principle of Clausius–Duhem and a new Poincaré-type inequality.     

Variationally trivial Lagrangians and locally variational differential equations of arbitrary order

We continue our investigation of the Lagrangian formalism on jet bundle extensions using Fock space methods. We are able to provide the most general form of a variationally trivial Lagrangian of arbitrary order and we also give a generic expression for the most general locally variational differential equation. As anticipated in the literature, these expressions involve some special combinations of the highest order derivatives, called hyper-Jacobians.