We consider the semilinear Schrödinger equation $$ -\Delta_A u+V(x)u =Q(x)|u|^{2^{*}-2}u. $$ Assuming that $V$ changes sign, we establish the existence of a solution $u\ne 0$ in the Sobolev space $ H_{A,V^+}^{1}(\mathbb R^N)$. The solution is obtained by a min-max type argument based on a topological linking. We also establish certain regularity properties of solutions for a rather general class of equations involving the operator $-\Delta_A$.

The study of the relation of matter and the electromagnetic field is a classical, intriguing problem both from physical and mathematical point of view. This relation can be interpreted from two different standpoints which, following [M. Born and L. Infeld, Foundations of the new field theory, Proc. Roy. Soc. London Ser. A 144 (1934), 425–451], are called unitarian standpoint and dualistic standpoint.

In this paper we briefly describe two models which are related to the unitarian and the dualistic standpoint respectively. For each model it is possible to prove the existence of solitary waves which can be interpreted as matter particles.

For a sequence of values of the total horizontal impulse that converges to $0$, there are solitary waves that minimise the energy in a given neighbourhood of the origin in $W^{2,2}({\mathbb R})$. The problem arises in the framework of the classical Euler equation when a two-dimensional layer of water above an infinite horizontal bottom is considered, at the surface of which solitary waves propagate under the action of gravity and weak surface tension. The adjective "weak" refers to the Bond number, which is assumed to be sub-critical ($< 1/3$).

This extends previous results on the conditional energetic stability of solitary waves in the super-critical case, namely those by A. Mielke ([On the energetic stability of solitary water waves, Philos. Trans. Roy. Soc. London Ser. A 360 (2002), 2337–2358]) and by the author ([Existence and conditional energetic stability of capillary-gravity solitary water waves by minimisation, Arch. Rational Mech. Anal.]). Like in the latter, the method is based on direct minimisation and concentrated compactness, but without relying on "strict sub-additivity", which is still unsettled in the present case. Instead, a complete and careful analysis of minimising sequences is performed that allows us to reach a conclusion, based only on the non-existence of "vanishing" minimising sequences. However, in contrast with [Existence and conditional energetic stability of capillary-gravity solitary water waves by minimisation, Arch. Rational Mech. Anal.], we are unable to prove the existence of minimisers for all small values of the total horizontal impulse.

In fact more is needed to get stability, namely that every minimising sequence has a subsequence that converges to a global minimiser, after possible shifts in the horizontal direction. This will be obtained as a consequence of the analysis of minimising sequences. Then exactly the same argument as in [Existence and conditional energetic stability of capillary-gravity solitary water waves by minimisation, Arch. Rational Mech. Anal.] gives conditional energetic stability and is therefore not repeated.

Perturbed geodesics are trajectories of particles moving on a semi-Riemannian manifold in the presence of a potential. Our purpose here is to extend to perturbed geodesics on semi-Riemannian manifolds the well known Morse Index Theorem. When the metric is indefinite, the Morse index of the energy functional becomes infinite and hence, in order to obtain a meaningful statement, we substitute the Morse index by its relative form, given by the spectral flow of an associated family of index forms. We also introduce a new counting for conjugate points, which need not to be isolated in this context, and prove that our generalized Morse index equals the total number of conjugate points. Finally we study the relation with the Maslov index of the flow induced on the Lagrangian Grassmannian.

We prove existence theorems for first order boundary value problems on $(0,\infty)$, of the form $\dot{u}+F(\cdot,u)=f$, $Pu(0)=\xi$, where the function $F=F(t,u)$ has a $t$-independent limit $F^{\infty}(u)$ at infinity and $P$ is a given projection. The right-hand side $f$ is in $L^{p} ((0,\infty),{\mathbb R}^{N})$ and the solutions $u$ are sought in $W^{1,p}((0,\infty),{\mathbb R}^{N})$, so that they tend to $0$ at infinity. By using a degree for Fredholm mappings of index zero, we reduce the existence question to finding a priori bounds for the solutions. Nevertheless, when the right-hand side has exponential decay, our existence results are valid even when the governing operator is not Fredholm.

We consider the solvability of the Neumann problem for the equation $$ -\Delta u+\lambda u =0, \quad \frac {\partial u}{\partial \nu}=Q(x)|u|^{q-2}u $$ on $\partial \Omega$, where $Q$ is a positive and continuous coefficient on $\partial \Omega$, $\lambda$ is a parameter and $q= {2(N-1)}/{(N-2)}$ is a critical Sobolev exponent for the trace embedding of $H^1(\Omega)$ into $L^q(\partial \Omega)$. We investigate the joint effect of the mean curvature of $\partial \Omega$ and the shape of the graph of $Q$ on the existence of solutions. As a by product we establish a sharp Sobolev inequality for the trace embedding. In Section 6 we establish the existence of solutions when a parameter $\lambda$ interferes with the spectrum of $-\Delta$ with the Neumann boundary conditions. We apply a min-max principle based on the topological linking.

In this paper, we show the multiple existence of positive solutions of semilinear elliptic problems of the form $$ -\Delta u=\vert u\vert ^{2^{*}-2}u+f, \quad u\in H_{0}^{1}(\Omega), $$ where $\Omega\subset{\mathbb R}^{N}$ is a bounded domain, $2^{*}$ is the Sobolev critical exponent and $f\in L^{2}(\Omega)$.

Let $f\colon M\to {\mathbb R}$ be an analytic proper function defined in a neighbourhood of a closed "regular" (for instance semi-analytic or sub-analytic) set $P\subset f^{-1}(y)$. We show that the set of non-trivial trajectories of the equation $\dot x =\nabla f(x)$ attracted by $P$ has the same Čech-Alexander cohomology groups as $\Omega\cap\{f< y\}$, where $\Omega$ is an appropriately choosen neighbourhood of $P$. There are also given necessary conditions for existence of a trajectory joining two closed "regular" subsets of $M$.

In this paper we address the existence and multiplicity results for $$ \begin{cases} -\Delta_p u -\lambda |u|^{p-2} u = h (x,u) &\text{in }\Omega, \\ u = 0 &\text{on } \partial \Omega, \end{cases} $$ where $p> 1$, $\Delta_p u = {\rm div}(|\nabla u|^{p-2}\nabla u)$, $h$ is a bounded function and the spectral parameter $\lambda$ stays "near" the principal eigenvalue of the $p$-Laplacian.

We show how the bifurcation theory combined with certain asymptotic estimates yield desired results.

We define spectral sequences associated with Morse decompositions of a compact metric space. We prove the existence and uniqueness of such spectral sequences for continuous dynamical systems.