In this paper, we consider the existence of positive and negative entire solutions of semilinear elliptic problem $$ -\Delta u + u = g(x,u), \quad u \in H^{1}({\mathbb R}^{N})\tag{P} $$ where $N \geq 2$ and $g\colon{\mathbb R}^{N} \times {\mathbb R }\to {\mathbb R}$ is a continuous function with superlinear growth and $g(x,0) = 0$ on ${\mathbb R}^{N} $.

We consider nonlinear eigenvalue problems of the form $$ A_0 y + B(y) y = \lambda y \tag{$*$} $$ in a real Hilbert space $\mathcal H$, where $A_0$ is a semi-bounded self-adjoint operator and, for every $y$ from a certain dense subspace $X$ of $\mathcal H$, $B(y)$ is a bounded symmetric linear operator. The left hand side is assumed to be the gradient of a functional $\psi \in C^1(x)$, and the associated linear problems $$ A_0 v + B(y) v = \mu v \tag{$**$} $$ are supposed to have discrete spectrum $(y \in X)$. We present a new topological method which permits, under appropriate assumptions, to construct solutions of ($*$) on a sphere $S_R := \{ y \in X \mid \|y\|_{\mathcal H} = R\}$ whose $\psi$-value is the $n$th Ljusternik-Schnirelman level of $\psi |_{S_R}$ and whose corresponding eigenvalue is the $n$-th eigenvalue of the associated linear problem ($**$), where $R > 0$ and $n \in \mathbb N$ are given. In applications, the eigenfunctions thus found share any geometric property enjoyed by an $n$-th eigenfunction of a linear problem of the form ($**$). We discuss applications to nonlinear Sturm-Liouville problems, to the nonlinear Hill's equation, to periodic solutions of second-order systems, and to elliptic partial differential equations with radial symmetry.

A concept of topological essentiality is used to prove several existence results for nonlinear boundary value control problems. Some examples to illustrate the obtained results are presented.

We consider a two-point boundary value problem $$ \dot x=f(t,x), \quad x(a)=g(x(b)). $$ We assume that in the extended space of the equation there exist an isolating segment, a set such that $f$ properly behaves on its boundary. We give a formula for the fixed point index of the composition of $g$ with the translation operator in a neighbourhood of the set of the initial points of solutions contained in the isolating segment. We apply that formula to results on existence of solutions of some planar boundary value problem associated to equations of the form $\dot z=\overline z^q+\ldots$ and $\dot z=e^{it}\overline z^q+\ldots$.

We present the computations of the secondary obstruction groups for the first stem of stable equivariant homotopy groups, used in the setting for the equivariant degree introduced by Ize et al., in the case of the same action of a compact Lie group on the domain and co-domain.

The main goal of this paper is to define an equivariant degree theory for orthogonal maps. We apply our degree to study of bifurcations and existence of solutions of equivariant nonlinear problems.

Based on Morse homology of Morse functions, we give a new proof of the Morse-Bott inequalities for functions with non-degenerate critical manifolds. A proof of the Morse inequalities for functions with isolated critical points as developed by Gromoll-Meyer is also presented with the same method.

Let $\mathcal M^*$ be a non-complete Riemannian manifold with bounded topological boundary and $V\colon \mathcal M \to \mathbb R$ a $C^2$ potential function subquadratic at infinity.

In this paper we look for curves $x\colon [0,T]\to\mathcal M$ having prescribed period $T$ or joining two fixed points of $\mathcal M$, satisfying the system $$ D_t (\dot x(t))=-\nabla_R V(x(t)), $$ where $D_t(\dot x(t))$ is the covariant derivative of $\dot x$ along the direction of $\dot x$ and $\nabla_R V$ the Riemannian gradient of $V$.

We assume that $V(x) \to -\infty$ if $d(x,\partial\mathcal M)\to 0$ and, in the periodic case, suitable hypotheses on the sectional curvature of $\mathcal M$ at infinity.

We use variational methods in addition with a penalization technique and Morse index estimates.