We define an index of Fuller type counting the periodic orbits of a local topological semiflow on ANR spaces avoiding smoothness assumptions and approximation processes.

In this paper we prove a symmetry result for positive solutions of the Dirichlet problem \begin{equation} \begin{cases} -\Delta u=f(u) & \hbox{in }D,\\ u=0 & \hbox{on }\partial D, \end{cases} \tag{0.1} \end{equation} when $f$ satisfies suitable assumptions and $D$ is a small symmetric perturbation of a domain $\Omega$ for which the Gidas-Ni-Nirenberg symmetry theorem applies. We consider both the case when $f$ has subcritical growth and $f(s)=s^{(N+2)/(N-2)}+\lambda s$, $N\ge3$, $\lambda$ suitable positive constant.

In this paper we establish the existence of multiple solutions for the semilinear elliptic problem \begin{equation}\alignedat 2 -\Delta u&=g(x,u) &\quad&\text{in } \Omega, \\ u&=0 &\quad&\text{on } \partial\Omega, \endalignedat \tag{1.1} \end{equation} where $\Omega \subset {\mathbb R}^N$ is a bounded domain with smooth boundary $\partial \Omega$, a function $g\colon\Omega\times{\mathbb R}\to {\mathbb R}$ is of class $C^1$ such that $g(x,0)=0$ and which is asymptotically linear at infinity. We considered both cases, resonant and nonresonant. We use critical groups to distinguish the critical points.

We consider a class of nonlinear problems of the form $$ Lu+g(x,u)=f, $$ where $L$ is an unbounded self-adjoint operator on a Hilbert space $H$ of $L^{2}(\Omega)$-functions, $\Omega\subset\mathbb{R}^{N}$ an arbitrary domain, and $g\colon \Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a "jumping nonlinearity" in the sense that the limits $$ \lim_{s\rightarrow-\infty} \frac{g(x,s)}{s}=a \quad\text{and}\quad \lim_{s\rightarrow\infty}\frac{g(x,s)}{s}=b $$ exist and "jump" over an eigenvalue of the operator $-L$. Under rather general conditions on the operator $L$ and for suitable $a< b$, we show that a solution to our problem exists for any $f\in H$. Applications are given to the beam equation, the wave equation, and elliptic equations in the whole space $\mathbb{R}^{N}$.

We consider a Riemannian manifold $(\mathcal M,g)$ and a codimension one distribution $\Delta\subset T\mathcal M$ on $\mathcal M$ which is the orthogonal of a unit vector field $Y$ on $\mathcal M$. We do not make any nonintegrability assumption on $\Delta$. The aim of the paper is to develop a Morse Theory for the sub-Riemannian action functional $E$ on the space of horizontal curves, i.e. everywhere tangent to the distribution $\Delta$. We consider the case of horizontal curves joining a smooth submanifold $\mathcal P$ of $\mathcal M$ and a fixed point $q\in\mathcal M$. Under the assumption that $\mathcal P$ is transversal to $\Delta$, it is known (see [P. Piccione and D. V. Tausk, Variational aspects of the geodesic problem is sub-Riemannian geometry, J. Geom. Phys. 39 (2001), 183–206]) that the set of such curves has the structure of an infinite dimensional Hilbert manifold and that the critical points of $E$ are the so called normal extremals (see [W. Liu and H. J. Sussmann, Shortest paths for sub-Riemannian metrics on rank–$2$ distribution, Mem. Amer. Math. Soc. 564 (1995)]). We compute the second variation of $E$ at its critical points, we define the notions of $\mathcal P$-Jacobi field, of $\mathcal P$-focal point and of exponential map and we prove a Morse Index Theorem. Finally, we prove the Morse relations for the critical points of $E$ under the assumption of completeness for $(\mathcal M,g)$.

The existence of a solution to the Dirichlet boundary value problem for nonlinear Poisson equations with the nonlocal nonlinear term $$ -\Delta u=f\bigg(u,\int (g\circ u)\bigg),\quad u\vert\partial U=0, $$ is proved by means of fixed point theorems for increasing compact operators.

We construct and variationally characterize by a min-max procedure involving the Yang index a new sequence of eigenvalues of the $p$-Laplacian, and use the structure provided by this sequence to show that the associated variational functional always has a nontrivial critical group. As an application we obtain nontrivial solutions for a class of $p$-superlinear problems.

As well known the problem of global continuation of solutions to semilinear parabolic equations is completely solved when the nonlinear term is subordinated to an $\alpha$-power of the main linear operator with $\alpha\in[0,1)$. In this paper we study three examples of critical problems in which the mentioned subordination takes place with $\alpha=1$, i.e. the nonlinearity has the same order of magnitude as the linear main part. We use specific techniques of proving global solvability that fit well the considered examples for which general abstract methods fail.

Via nonsmooth critical point theory we prove the existence of at least two solutions in $W^{1,p}_0(\Omega)$ for a jumping problem involving the Euler equation of multiple integrals of calculus of variations under natural growth conditions. Some new difficulties arise in comparison with the study of the semilinear and also the quasilinear case.

Let $M$ and $N$ be compact manifolds with $\partial M\neq\emptyset$. We show that when $1< p< \dim M$, there are two different obstructions to extending a map in $W^{1-1/p,p}(\partial M,N)$ to a map in $W^{1,p}(M,N)$. We characterize one of these obstructions which is topological in nature. We also give properties of the other obstruction. For some cases, we give a characterization of $f\in W^{1-1/p,p}(\partial M,N)$ which has an extension $F\in W^{1,p}(M,N)$.

In this paper we prove that $d$-boundaries $$ D_d=\{x:{\rm dist}( x,Z) =d\} $$ of a compact $Z \subset \mathbb{R}^{2}$ are closed absolutely continuous curves for $d$ greater than some constant depending on $Z$. It is also shown that $D_d$ is a trajectory of solution to the Cauchy Problem of a differential equation with a discontinuous right-hand side.

This paper presents a continuation theory for ${\mathcal U}_c^{\kappa}$ maps. The analysis is elementary and relies on properties of retractions and fixed point theory for self maps. Also we present a separate theory for a certain subclass of ${\mathcal U}_c^{\kappa}$ maps, namely the $PK$ maps.

It is shown that group endomorphisms are exact flows if and only if they are exact in the measure-theoretic sense and that all flows which are exact with respect to an invariant measure with full support are exact. It is also proved that all locally eventually dense (led) flows have uniformly positive entropy (u.p.e.).