Here we study the local behavior of the continua in the case of Neumann boundary conditions, pointing out some qualitative differences with the Dirichlet case. We also combine local and global behavior of the bifurcating sets to obtain existence of solutions and study of their sign for some related problems.

In this paper we investigate positive radial solutions of the following equation $$ \Delta_{p}u+K(r) u|u|^{\sigma-2}=0 $$ where $r=|x|$, $x \in {\mathbb R}^n$, $n> p> 1$, $\sigma =n p/(n-p)$ is the Sobolev critical exponent and $K(r)$ is a function strictly positive and bounded.

This paper can be seen as a completion of the work started in [M. Franca, Structure theorems for positive radial solutions of the generalized scalar curvature equation, when the curvature exhibits a finite number of oscillations], where structure theorems for positive solutions are obtained for potentials $K(r)$ making a finite number of oscillations. Just as in [M. Franca, Structure theorems for positive radial solutions of the generalized scalar curvature equation, when the curvature exhibits a finite number of oscillations], the starting point is to introduce a dynamical system using a Fowler transform. In [M. Franca, Structure theorems for positive radial solutions of the generalized scalar curvature equation, when the curvature exhibits a finite number of oscillations] the results are obtained using invariant manifold theory and a dynamical interpretation of the Pohozaev identity; but the restriction $2 n/(n+2) \le p\le 2$ is necessary in order to ensure local uniqueness of the trajectories of the system. In this paper we remove this restriction, repeating the proof using a modification of Ważewski's principle; we prove for the cases $p> 2$ and $1< p< 2 n/(n+2)$ results similar to the ones obtained in the case $ 2 n/(n+2) \le p\le 2$.

We also introduce a method to prove the existence of Ground States with fast decay for potentials $K(r)$ which oscillates indefinitely. This new tool also shed some light on the role played by regular and singular perturbations in this problem, see [M. Franca and R. A. Johnson, Ground states and singular ground states for quasilinear partial differential equations with critical exponent in the perturbative case, Adv. Nonlinear Studies].

This work deals with the analysis of eigenvalues, bifurcation and Hölder continuity of solutions to mixed problems like $$ \begin{cases} -{\rm div} (|x|^{-p\gamma} |\nabla u|^{p-2}\nabla u) = f_{\lambda}(x,u) , &u > 0\ \text{ in }\Omega ,\\ u = 0 &\text{ on }\Sigma_1,\\ |x|^{-p\gamma}|\nabla u|^{p-2}\dfrac{\partial u}{\partial \nu} = 0 &\text{ on } \Sigma_2, \end{cases} $$ involving some potentials related with the Caffarelli-Kohn-Nirenberg inequalities, and with different kind of functions $f_\lambda (x,u)$.

Although the pioneering studies of G. I. Barenblatt [On some unsteady motions of a liquid or a gas in a porous medium, Prikl. Mat. Mekh. 16 (1952), 67–68] and A. G. Aronson and L. A. Peletier [Large time behaviour of solutions of some porous medium equation in bounded domains, J. Differential Equations 39 (1981), 378–412] did result into a huge industry around the porous media equation, none further study analyzed the effect of combining fast, slow, and linear diffusion simultaneously, in a spatially heterogeneous porous medium. Actually, it might be this is the first work where such a problem has been addressed. Our main findings show how the heterogeneous model possesses two different regimes in the presence of a priori bounds. The minimal steady-state of the model exhibits a genuine fast diffusion behavior, whereas the remaining states are rather reminiscent of the purely slow diffusion model. The mathematical treatment of these heterogeneous problems should deserve a huge interest from the point of view of its applications in fluid dynamics and population evolution.

The main purpose of this paper is to study the existence of single-peaked solutions of the Neumann problem $$ \begin{cases} -\varepsilon^2 {\rm div} (J(x)\nabla u)+V(x)u=u^p & \text{in }\Omega, \\ \dfrac{\partial u}{\partial \nu}=0 & \text{on }\partial\Omega, \end{cases} $$ where $\Omega$ is a smooth bounded domain of $\mathbb R^N$, $N\ge 3$, $1< p< (N+2)/(N-2)$ and $J$ and $V$ are positive bounded scalar value potentials. We will show that, for the existence of concentrating solutions, one has to check if at least one between $J$ and $V$ is not constant on $\partial \Omega$. In this case the concentration point is determined by $J$ and $V$ only. In the other case the concentration point is determined by an interplay among the derivatives of $J$ and $V$ calculated on $\partial \Omega$ and the mean curvature $H$ of $\partial \Omega$.

We study an optimization problem for the feedback control system emerging as a regularized model for the motion of a viscoelastic fluid subject to the Jeffris-Oldroyd rheological relation. The approach includes systems governed by the classical Navier-Stokes equation as a particular case. Using the topological degree theory for condensing multimaps we prove the solvability of the approximating problem and demonstrate the convergence of approximate solutions to a solution of a regularized one. At last we show the existence of a solution minimizing a given convex, lower semicontinuous functional.

Global motion of magnetohydrodynamic fluid in a domain bounded by a free surface and under the external electrodynamic field is proved. The motion is such that velocity and magnetic field are small in $H^3$-space.

A class of examples showing that a measure-theoretical characterization of regular cocycles in terms of essential values is not valid in topological dynamics is constructed. An example that in topological dynamics for the case of non-abelian groups, the groups of essential values of cohomologous cocycles need not be conjugate is given. A class of base preserving equivariant isomorphisms of Rokhlin cocycle extensions of topologically transitive flows is described. In particular, the topological centralizer of Rokhlin cocycle extension of minimal rotation defined by an action of the group $\mathbb R^m$ is determined.

We improve Theorem 1.2 of the paper [Yu. S. Ilyashenko, Covering manifolds for analytic families of leaves of foliations by analytic curves, Topol. Methods Nolinear Anal. 11 (1998), 361–373].

We remedy an error and simplify an argument in the article mentioned in the title.