A jumping problem for quasilinear elliptic equations is considered. A local saddle argument in the framework of nonsmooth critical point theory is applied.

This paper is concerned with the multiplicity of positive solutions of the Dirichlet problem $$ -\varepsilon ^{2}\Delta u+u=K( x) \vert u\vert ^{p-2}u \quad\text{in }\Omega, $$ where $\Omega $ is a smooth domain in $\mathbb{R}^{N}$ which is either bounded or has bounded complement (including the case $\Omega =\mathbb{R}^{N}$), $N\geq 3$, $K$ is continuous and $p$ is subcritical. It is known that critical points of $K$ give rise to multibump solutions of this type of problems. It is also known that, in general, the presence of symmetries has the effect of producing many additional solutions. So, we consider domains $\Omega $ which are invariant under the action of a group $G$ of orthogonal transformations of $\mathbb{R}^{N}$, we assume that $K$ is $G$-invariant, and study the combined effect of symmetries and the nonautonomous term $K$ on the number of positive solutions of this problem. We obtain multiplicity results which extend previous results of Benci and Cerami (1994), Cingolani and Lazzo (1997) and Qiao and Wang (1999).

Existence and uniqueness of large positive solutions are obtained for some semilinear elliptic Dirichlet problems in bounded smooth domains $\Omega$ with a large parameter $\lambda$. It is shown that the large positive solution has flat core. The distance of its flat core to the boundary $\partial \Omega$ is exactly measured as $\lambda \to \infty$.

We discuss the asymptotic behavior of solutions of the nonlinearly damped wave equation $$ u_{tt} +\delta \vert u_t\vert ^{m-1}u_t -\phi (x)\Delta u = \lambda u\vert u\vert ^{\beta -1}, \quad x \in \mathbb R^n, \ t \geq 0, $$ with the initial conditions $ u(x,0) = u_0 (x)$ and $u_t(x,0) = u_1 (x)$, in the case where $N \geq 3$, $ \delta > 0$ and $(\phi (x))^{-1} =g (x)$ is a positive function lying in $L^{p}(\mathbb R^n)\cap L^{\infty}(\mathbb R^n)$, for some $p$. We prove blow-up of solutions when the source term dominates over the damping, and the initial energy is assumed to be positive. We also discuss global existence energy decay of solutions.

In this paper we study the sweeping processes by convex sets depending on time and the solution. We do some application to a dry friction's problem.

In the paper an optimization problem with parameters is considered. Some sufficient conditions under which the solutions of the problem continuosly depend on parameters (in the weak or the strong topology of a Banach space) are proved. Moreover, some applications to the eigenvalue and boundary value problems for differential operators are given.

This paper presents a general and systematic discussion of various symbolic representations of iterated maps through subshifts. A unified model for all continuous maps on a metric space is given. It is shown that at most the second order representation is enough for a continuous map. By introducing distillations, partial representations of some general continuous maps are obtained. Finally, partitions and representations of a class of discontinuous maps and some examples are discussed.

Necessary and sufficient conditions for the existence of weak solutions to stochastic differential inclusions with convex right-hand sides are given. The main results of the paper deal with the weak compactness with respect to the convergence in distribution of solution sets to such inclusions.

Some continuous selection results for a class of nonconvex-valued maps are obtained. One of them contains Michael's theorem, in the case of a Hilbert codomain. Methods of nonsmooth analysis and $\Gamma$-convergence are used.

In this paper we present the proof of the existence of symbolic dynamics for third iterate of the Rössler map. We combine an abstract topological results based on the fixed point index and covering relations with computer assisted rigorous computations.

We present in this paper several existence theorems for nonlinear complementarity problems in Hilbert spaces. Our results are based on the concept of "exceptional family of elements" and on Leray-Schauder type altrenatives.