We prove existence of global regular solutions for the $3$D Navier-Stokes equations with (or without) Coriolis force for a class of initial data $u_0$ in the space ${{{\mathrm{FM}}}_{\sigma,\delta}}$, i.e., for functions whose Fourier image ${\widehat{u}}_0$ is a vector-valued Radon measure and that are supported in sum-closed frequency sets with distance $\delta$ from the origin. In our main result we establish an upper bound for admissible initial data in terms of the Reynolds number, uniform on the Coriolis parameter $\Omega$. In particular this means that this upper bound is linearly growing in $\delta$. This implies that we obtain global-in-time regular solutions for large (in norm) initial data $u_0$ which may not decay at space infinity, provided that the distance $\delta$ of the sum-closed frequency set from the origin is sufficiently large.

We study the following nonlinear Schrödinger equation: $$ {\varepsilon}^2\Delta u-V(x) u+|u|^{p-1}u=0 \ \hbox{in}\ \mathbb R^N,$$ where ${\varepsilon}>0$, $u,V:\mathbb R^N\to\mathbb R$, $p>1$. We prove that, under suitable conditions on the symmetry of $V$, the set of sign-changing solutions has a rich structure in the semiclassical limit ${\varepsilon}\to 0$: we construct multipeak solutions with an arbitrarily large number of positive and negative peaks which collapse to either a local minimum or a local maximum of $V.$ The proof relies on a local approach and is based on the finite-dimensional reduction, in the spirit of the arguments employed in [15].

In this paper, we study the existence and nonexistence of solutions for some systems of semilinear elliptic equations in an unbounded domain $\Omega$. The main objective of this paper is to discuss the relationship between the existence of solutions and the translation invariance of the domain $\Omega$. More precisely, we point out how the translation invariance of our domain plays a crucial role for the existence of our problem.

In this text we introduce new classes of Sobolev--Morrey spaces being adequate for the regularity theory of second-order parabolic boundary-value problems on Lipschitz domains of space dimension $n \ge 3$ with nonsmooth coefficients and mixed boundary conditions. We prove embedding and trace theorems as well as invariance properties of these spaces with respect to localization, Lipschitz transformation, and reflection. In the second part [11] of our presentation we show that the class of second-order parabolic systems with diagonal principal part generates isomorphisms between the above-mentioned Sobolev--Morrey spaces of solutions and right-hand sides.