Let $X$ be a metric space, $\pi$ be a local semiflow on $X$, $k\in\mathbb N$, $E$ be a $k$-dimensional normed space and $\widetilde\pi$ be the semiflow generated by the equation $\dot y=Ly$, where $L\colon E\to E$ is a linear map whose all eigenvalues have positive real parts. We show in this paper that for every admissible isolated $\pi$-invariant set $S$ there is a well-defined isomorphism of degree $-k$ from the homology categorial Conley-Morse index of $(\pi\times\widetilde\pi,S\times\{0\})$ to the homology categorial Conley-Morse index of $(\pi,S)$ such that the family of these isomorphisms commutes with homology index sequences. In particular, given a partially ordered Morse decomposition $(M_i)_{i\in P}$ of $S$ there is an isomorphism of degree $-k$ from the homology index braid of $(M_i\times\{0\})_{i\in P}$ to the homology index braid of $(M_i)_{i\in P}$, so $C$-connection matrices of $(M_i\times\{0\})_{i\in P}$ are just $C$-connection matrices of $(M_i)_{i\in P}$ shifted by $k$ to the right.

We study multiplicity of solutions of the following elliptic problems in which critical and supercritical Sobolev exponents are involved: $$ \begin{alignat}{2} -\Delta u& =g(x, u)+\lambda h(x, u) &\quad& \text{in } \Omega \text{ and } u=0 \text{ on } \partial\Omega, \\ -{\rm div}(|\nabla u|^{p-2}\nabla u)&=g(x, u)+\lambda h(x, u) &\quad& \text{in } \Omega \text{ and } u=0 \text{ on } \partial\Omega, \end{alignat} $$ where $\Omega$ is a smooth bounded domain in ${\mathbb R}^N$, $p> 1$, $\lambda$ is a parameter, and $\lambda h(x, u)$ is regarded as a perturbation term of the problems. Except oddness with respect to $u$ in some cases, we do not assume any condition on $h$. For the first problem, we get a result on existence of three nontrivial solutions for $|\lambda|$ small in the case where $g$ is superlinear and $\limsup_{|t| \to\infty}g(x, t)/|t|^{2^*-1}$ is suitably small. We also prove that the first problem has $2k$ distinct solutions for $|\lambda|$ small when $g$ and $h$ are odd and there are $k$ eigenvalues between $\lim_{t\to0}g(x, t)/t$ and $\lim_{|t|\to\infty}g(x, t)/t$. For the second problem, we prove that it has more and more distinct solutions as $\lambda$ tends to $0$ assuming that $g$ and $h$ are odd and $g$ is superlinear and $\lim_{|t| \to\infty}g(x, t)/|t|^{p^*-1}=0$.

We use Conley index theory to solve the singular boundary value problem $\varepsilon^2D u_{xx} + f(u,\varepsilon u_x,x) = 0$ on an interval $[-1,1]$, where $u \in \mathbb R^n$ and $D$ is a diagonal matrix, with separated boundary conditions. Since we use topological methods the assumptions we need are weaker then the standard set of assumptions. The Conley index theory is used here not for detection of an invariant set, but for tracking certain cohomological information, which guarantees existence of a solution to the boundary value problem.

In this paper, we study existence and nonexistence of solutions for the Dirichlet problem associated with the equation $-\Delta u=g(x,u)+\mu$ where $\mu$ is a Radon measure. Existence and nonexistence of solutions strictly depend on the nonlinearity $g(x,u)$ and suitable growth restrictions are assumed on it. Our proofs are obtained by standard arguments from critical theory and in order to find solutions of the equation, suitable functionals are introduced by mean of approximation arguments and iterative schemes.

A question about the possibility of using min-max methods on the double natural constraint, in spite of its lack of regularity, has been raised in some recent papers. In this note we give an answer by topological arguments which show the equivalence between constrained and unconstrained min-max classes, avoiding in this way any regularity problem.

In the paper a fixed-point index for a class of the so-called Krasnoselskiĭ-type set-valued maps defined locally on arbitrary absolute neighbourhood retracts is presented. Various applications to the existence problems for constrained differential inclusions and equations are provided.

We prove the existence of a strong solution of a periodic-Dirichlet problem for the semilinear wave equation with irrational period and with nonlinearity satisfying some general growth conditions locally around $0$. We construct a new variational method, called a dual method, and using relations between critical points and critical values of the primal action and the dual action functionals we prove that the solution exists. The dual functional which we define is different from the ones known so far in that it depends on two dual variables.