We study the existence of the ground state solution of the problem $$ \begin{cases} -\Delta u+V(x)u=f'(u) & x\in\mathbb R^N, \\ u(x)> 0, \end{cases} $$ under the assumption that $\lim_{x\to\infty}V(x)=0$.

Global existence of regular solutions to the Navier-Stokes equations describing the motion of a fluid in a cylindrical pipe with large inflow and outflow in shown. The global existence is proved under the following conditions: \begin{enumerate} \item small variations of velocity and pressure with respect to the variable along the pipe, \item inflow and outflow are very close to homogeneous and decay exponentially with time, \item the external force decays exponentially with time. \end{enumerate} Global existence is proved in two steps. First by the Leray-Schauder fixed point theorem we prove local existence with large existence time which is inversely proportional to the above smallness restrictions. Next the local solution is prolonged step by step.

The existence is proved for a solution without any restrictions on the magnitudes of inflow, outflow, external force and the initial velocity.

The topological index for maps of infinite-dimensional Finsler manifolds, condensing with respect to internal Kuratowski's measure of non-compactness, is constructed under the hypothesis that the manifold can be embedded into a certain Banach linear space as a neighbourhood retract so that the Finsler norm in tangent spaces and the restriction of the norm from enveloping space on the tangent spaces are equivalent. It is shown that the index is an internal topological characteristic, i.e. it does not depend on the choice of enveloping space, embedding, etc. The total index (Lefschetz number) and the Nielsen number are also introduced. The developed machinery is applied to investigation of functional-differential equations of neutral type on Riemannian manifolds. A certain existence and uniqueness theorem is proved. It is shown that the shift operator, acting in the manifold of $C^1$-curves, is condensing, its total index is calculated to be equal to the Euler characteristic of (compact) finite-dimensional Riemannian manifold where the equation is given. Some examples of calculating the Nielsen number are also considered.

The generalized variational principle of Herglotz defines the functional, whose extrema are sought, by a differential equation rather than by an integral. For such functionals the classical Noether theorems are not applicable. First and second Noether-type theorems which do apply to the generalized variational principle of Herglotz were formulated and proved. These theorems contain the classical first and second Noether theorems as special cases. We published the first Noether-type theorem previously in this journal. Here we prove the second Noether-type theorem and show that it reduces to the classical second Noether theorem when the Herglotz variational principle reduces to the classical variational principle.

Consider a reaction-diffusion equation $u_t=\triangle u+f(u)$ on a family of net-shaped thin domains $\Omega_\varepsilon$ converging to a one dimensional set as $\varepsilon\downarrow 0$. With suitable growth and dissipativeness conditions on $f$ these equations define global semiflows which have attractors $\mathcal{A}_\varepsilon$. In [Th. Elsken, A reaction-diffusion equation on a net-shaped thin domain, Studia Math. 165 (2004), 159–199] it has been shown that there is a limit problem which also defines a semiflow having an attractor $\mathcal{A}_0$, and the family of attractors is upper-semi-continuous at $\varepsilon=0$. Here we show that under a stronger dissipativeness condition the family of attractors $\mathcal{A}_\varepsilon$, $\varepsilon\ge 0$, is actually continuous at $\varepsilon=0$.

In this paper we present a result concerning the existence of two nonzero nonnegative solutions for the following Dirichlet problem involving the $p$-Laplacian $$ \begin{cases} -\Delta_p u=\lambda f(x,u) &\text{\rm in\ } \Omega,\\ u=0 &\text{\rm on\ } \partial \Omega, \end{cases} $$ using variational methods. In particular, we will determine an explicit real interval $\Lambda$ for which these solutions exist for every $\lambda\in \Lambda$. We also point out that our result improves and extends to higher dimension a recent multiplicity result for ordinary differential equations.

In this paper we study the solvability of Sturm-Liouville BVPs for Duffing equations by means of homotopy continuation methods. We propose a new kind of solvable conditions on the nonlinear function in the equation. This kind of conditions can be seen as some limiting cases of the well-known asymptotically positive linear conditions. The obtained results generalize and unify some previous results by S. Villegas, T. Ma and L. Sanchez, and Y. Dong, respectively.

Addendum to "Dependence on parameters for the Dirichlet problem with superlinear nonlinearities" (Topol. Methods Nonlinear Anal. 16 (2000), 145–160)