In this paper, we construct multipeak solutions for a singularly perturbed Dirichlet problem. Under the conditions that the distance function $d(x,\partial\Omega)$ has $k$ isolated compact connected critical sets $T_1,\ldots,T_k$ satisfying $d(x,\partial\Omega) =c_j=\text{const.}$, for all $x\in T_j$, $\min_{i\ne j}d(T_i,T_j)> 2\max_{1\le j\le k}d(T_j,\partial\Omega)$, and the critical group of each critical set $T_i$ is nontrivial, we construct a solution which has exactly one local maximum point in a small neighbourhood of $T_i$, $i=1,\ldots,k$.

We consider a nonlinear model for time-periodic oscillations of a suspension bridge. Under some additional restrictive assumptions we describe our model by a standard bifurcation scheme which allows us to use global bifurcation theorems and make some new conclusions.

We consider the existence of multiple solutions for problem (1.1) below with either $\lambda\neq \lambda$ or $\lambda=\lambda_1$, where $\lambda_k$, $k=1,2,\ldots$ are eigenvalues of $(-\Delta, H^1_0(\Omega))$. The local bifurcation from $\lambda=\lambda_k$ is also investigated.

In this paper we study global bifurcation phenomena for a class of nonlinear elliptic equations governed by the $h$-Hessian operator. The bifurcation phenomena considered provide new methods for establishing existence results concerning fully nonlinear elliptic equations. Applications to the theory of critical exponents and the geometry of $k$-convex functions are considered. In addition, a related problem of Liouville-Gelfand type is analyzed.

We study the existence and multiplicity of positive solutions of the boundary value problem $$ u''+sf(t,u)=0,\quad u(0)=u(1)=0, $$ where $s$ is a positive parameter and $f$ is a non-negative function. We indicate examples with more than two solutions.

Given $X$ a Jiang space we know that all Nielsen classes have the same index. Now let us consider $X$ a $G$-space where $G$ is a finite group which acts freely on $X$. In [P. Wong, Equivariant Nielsen numbers, Pacific J. Math. l59 (1993), 153–175], we do have the notion of $X$ to be an equivariant Jiang space and under this condition it is true that all equivariant Nielsen classes have the same index. We study the question if the weaker condition of $X$ being just a Jiang space is sufficient for all equivariant Nielsen classes to have the same index. We show a family of spaces where all equivariant Nielsen classes have the same index. In many cases the spaces of such a family are not equivariant Jiang spaces. Finally, we also show an example of one Jiang space together with equivariant maps where the equivariant Nielsen classes have different indices.

We consider an autonomous conservative second order system defined by a potential which admits a connected set $\Gamma$ of critical points at level zero. We prove the existence of multiple periodic solutions of large period which are located near $\Gamma$.

We present a multidimensional generalization of the Sharkovskiĭ Theorem concerning the appearance of periodic points for the self-maps on the real line.

A selection problem for convex-valued mappings is studied. Two general results, so called "sandwich" theorems, are proved