We study the effect of the coefficient of the critical nonlinearity for the Neumann problem on existence of symmetric least energy solutions. As a by-product we obtain two inequalities in symmetric Sobolev spaces involving a weighted critical Lebesgue norm and the $H^1$ norm.

In this paper we study multiplicity results for the critical points of a functional via topological information which ensures multiplicity of critical points for a sequence of approximating functionals. The main statement is quite simple, and it seems it could be usefully compared with a large class of problems. In particular we mention some problems that can be studied in this framework.

For a scalar, first order ordinary differential equation which depends periodically on time, Massera's Theorem says that the existence of a bounded solution implies the existence of a periodic solution. Though the statement is false when periodicity is replaced by quasi-periodicity, solutions with some kind of recurrence are anyway expected when the equation is quasi-periodic in time. Indeed we first prove that the existence of a bounded solution implies the existence of a solution which is quasi-periodic in a weak sense. The partial differential equation, having our original equation as its equation of characteristics, plays a key role in the introduction of this notion of weak quasi-periodicity. Then we compare our approach with others already known in the literature. Finally, we give an explicit example of the weak case, and an extension to higher dimension for a special class of equations.

In this paper we consider the singularly perturbed Dirichlet problem (P$_{\varepsilon}$), when the potential $a_{\varepsilon}(x)$, as $\varepsilon$ goes to $0$, is concentrating round a point $x_0\in\Omega$. Under suitable growth assumptions on $f$, we prove that (P$_{\varepsilon}$) has at least three distinct solutions whatever $\Omega$ is and that at least one solution is not a one-peak solution.

We present two generalizations of the famous Pascal theorem to the case of algebraic curves of degree 3.

We examine some overdetermined elliptic system in a domain in $\mathbb R^3$ which contains an axis. Assuming that data functions belong to Sobolev spaces with weights equal to a power of the distance from the axis we prove existence of solutions in the corresponding kind of weighted Sobolev spaces.

Sufficient conditions are given for the existence of multiple positive solutions of a boundary value problem of the form $x''(t)+q(t)f(x(t))=0$, $t\in [0,1]$, $x(0)=0$ and $x(1)=\int_{\alpha}^{\beta}x(s)dg(s)$, where $0< \alpha < \beta < 1$. A weaker boundary value problem is used to get information on the corresponding integral operator. Then the results follow by applying the Krasnosel'skiĭ fixed point theorem on a suitable cone.

In this paper we study topological lower bounds on the number of zeros of closed $1$-forms without Morse type assumptions. We prove that one may always find a representing closed $1$-form having at most one zero. We introduce and study a generalization ${\rm cat}(X,\xi)$ of the notion of the Lusternik-Schnirelman category, depending on a topological space $X$ and a $1$-dimensional real cohomology class $\xi\in H^1(X;\mathbb R)$. We prove that any closed $1$-form $\omega$ in class $\xi$ has at least ${\rm cat}(X,\xi)$ zeros assuming that $\omega$ admits a gradient-like vector field with no homoclinic cycles. We show that the number ${\rm cat}(X,\xi)$ can be estimated from below in terms of the cup-products and higher Massey products.

This paper corrects some my statements made in [Lusternik–Schnirelman theory for closed $1$-forms, Comment. Math. Helv. 75 (2000), 156–170] and [Topology of closed $1$-forms and their critical points, Topology 40 (2001), 235–258].

We develop a nonstandard description of Retarded Functional Differential Equations which consist of a formally finite iteration of vectors. We present two applications where the new description gives explicit formulae. The classical approach in these cases only offers a method to construct the solution.