We study the existence and non-existence of ground states for the Schrödinger equations $-\Delta u -\lambda\sum_{i< j}u/|x_i-x_j|^2 = |u|^{2^*-2}u$, $x=(x_1,\ldots,x_m)\in {\mathbb{R}}^{mN}$, and $-\Delta u -\lambda u/|y|^2 = |u|^{2^*-2}u$, $x=(y,z)\in {\mathbb{R}}^N$. In both cases we assume $\lambda\ne 0$ and $\lambda< \overline\lambda$, where $\overline\lambda$ is the Hardy constant corresponding to the problem.

We consider locally Lipschitz and completely continuous maps $A\colon C\to C$ defined on a closed convex subset $C\subset X$ of a Banach space $X$. The main interest lies in the case when $C$ has empty interior. We establish Poincaré-Hopf type formulas relating fixed point index information about $A$ with homology Conley index information about the semiflow on $C$ induced by $-{\rm id}+A$. If $A$ is a gradient we also obtain results on the critical groups of isolated fixed points of $A$ in $C$.

In this paper, we establish existence and multiplicity of multi-bump type nodal solutions for the following class of problems $$ -\Delta u + (\lambda V(x)+ 1)u= f(u), \quad u> 0 \quad \text{in } {\mathbb{R}}^N, $$ where $N \geq 1$, $\lambda \in (0, \infty)$, $f$ is a continuous function with subcritical growth and $V\colon {\mathbb{R}}^N \rightarrow \mathbb{R} $ is a continuous function verifying some hypotheses.

By means of a minimax argument, we prove the existence of at least one heteroclinic solution to a scalar equation of the kind $\ddot x=a(t)V'(x)$, where $V$ is a double well potential, $0< l\le a(t)\le L$, $a(t)\to l$ as $|t|\to\infty$ and the ratio $L/l$ is suitably bounded from above.

In this paper, we are concerned with existence of positive solutions for systems of nonlinear Hammerstein integral equations, in which one nonlinear term is superlinear and the other is sublinear. The discussion is based on the product formula of fixed point index on product cone and fixed point index theory in cones. As applications, we consider existence of positive solutions for systems of second-order ordinary differential equations with different boundary conditions.

In this paper, we consider the nonlinear eigenvalue problems \begin{gather*} u''+\lambda h(t)f(u)=0, \quad 0< t< 1, \\ u(0)=0,\quad u(1)=\sum_{i=1}^{m-2}\alpha_i u(\eta_i), \end{gather*} where $m\geq 3$, $ \eta_i\in (0,1)$ and $\alpha_i> 0$ for $i=1,\ldots,m-2$, with $\sum_{i=1}^{m-2}\alpha_i\eta_i< 1$; $h\in C([0,1], [0,\infty))$ and $h(t)\ge 0$ for $t\in [0,1]$ and $h(t_0)> 0$ for $t_0\in [0,1]$; $f\in C([0,\infty),[0,\infty))$ and $f(s)> 0$ for $s> 0$, and $f_0=\infty$, where $f_0=\lim_{s\rightarrow 0^+}f(s)/s$. We investigate the global structure of positive solutions by using the nonlinear Krein-Rutman Theorem.

In this paper we will be concerned with the existence and non-existence of constrained minimizers in Sobolev spaces $D^{k,p}({\mathbb{R}}^N)$, where the constraint involves the critical Sobolev exponent. Minimizing sequences are not, in general, relatively compact for the embedding $D^{k,p}({\mathbb{R}}^N)\hookrightarrow L^{p^*} ({\mathbb{R}}^N,Q)$ when $Q$ is a non-negative, continuous, bounded function. However if $Q$ has certain symmetry properties then all minimizing sequences are relatively compact in the Sobolev space of appropriately symmetric functions. For $Q$ which does not have the required symmetry, we give a condition under which an equivalent norm in $D^{k,p}({\mathbb{R}}^N)$ exists so that all minimizing sequences are relatively compact. In fact we give an example of a $Q$ and an equivalent norm in $D^{k,p}({\mathbb{R}}^N)$ so that all minimizing sequences are relatively compact.

We consider a nonautonomous second order periodic system with an indefinite linear part. We assume that the potential function is superquadratic, but it may not satisfy the Ambrosetti-Rabinowitz condition. Using an existence result for $C^1$-functionals having a local linking at the origin, we show that the system has at least one nontrivial solution.

We improve and simplify the minimization method for solitary waves in two cases: firstly, when the surface tension is weak (that is, the Bond number is $< 1/3$) and the depth is finite, and secondly, when the depth is infinite. In a previous work on the first case, minimizers were shown to exist for a sequence tending to $0$ of values of the horizontal impulse. The main difficulty is that strict subadditivity in the concentration-compactness method is unsettled. Here we observe in both examples that strict subadditivity nevertheless holds for a set of horizontal impulses of positive measure and the related propagation speeds are estimated from above.

In this paper, we investigate the finiteness of the Reidemeister number $R(f)$ of a selfmap $f\colon M\to M$ on an infra-nilmanifold $M$. We show that the Reidemeister number of an Anosov diffeomorphism on an infra-nilmanifold is always finite. A manifold $M$ is said to have the $R_\infty$ property if $R(f)=\infty$ for every homeomorphism $f\colon M\to M$. We show that every non-orientable generalised Hantzsche-Wendt manifold has the $R_\infty$ property. For an orientable Hantzsche-Wendt manifold $M$, we formulate a criterion, in terms of an associated graph, for $M$ to have the $R_\infty$ property.

In this paper we introduce vector metric spaces and we give some properties of these spaces. Also we prove Baire theorem and Banach fixed point theorem on these spaces.

The purpose of this paper is to prove the following theorem: Let $H$ be a Hilbert space, let $C$ be a nonempty bounded closed convex subset of $H$ and let $T\colon C\rightarrow C$ be an asymptotically regular mapping. If $$ \liminf_{n\rightarrow \infty} \|T^n\|< \sqrt{2}, $$ then ${\rm Fix}\, T=\{x\in C:Tx=x\}$ is a retract of $C$.