Nonlinear Schrödinger equations are usually investigated with the use of the variational methods that are limited to energy-subcritical dimensions. Here we present the approach based on the shooting method that can give the proof of existence of the ground states in critical and supercritical cases. We formulate the assumptions on the system that are sufficient for this method to work. As examples, we consider Schrödinger-Newton and Gross-Pitaevskii equations with harmonic potentials.

The existence of steady states in the microcanonical case for a system describing the interaction of gravitationally attracting particles with a self-similar pressure term is proved. The system generalizes the Smoluchowski-Poisson equation. The presented theory covers the case of the model with diffusion that obeys the Fermi-Dirac statistic.

We investigate stationary energy models in heterostructures consisting of continuity equations for all involved species, of a Poisson equation for the electrostatic potential and of an energy balance equation. The resulting strongly coupled system of elliptic differential equations has to be supplemented by mixed boundary conditions. If the boundary data are compatible with thermodynamic equilibrium then there exists a unique steady state. We prove that in a suitable neighbourhood of such a thermodynamic...

Most of the paper deals with the application of the moving plane method to different questions concerning stationary accumulations of isentropic gases. The first part compares the concepts of stationarity arising from the points of view of dynamics and the calculus of variations. Then certain stationary solutions are shown to be unstable. Finally, using the moving plane method, a short proof of the existence of energy-minimizing gas balls is given.

In questo articolo studiamo problemi di Dirichlet singolari, lineari e semilineari, della forma ${\left|x\right|}^2\mathrm{\Delta }u=f\left(u\right)$ in $\mathrm{\Omega }$, $u=0$ su $\partial \mathrm{\Omega }$, dove $\mathrm{\Omega }$ è un dominio in ${\mathbb{R}}^2$ e $f\left(u\right)=\lambda u$ o $f\left(u\right)=\lambda u+{\left|u\right|}^{p-2}u$ con $p>2$ (o nonlinearità più generali). In tali problemi bidimensionali emergono alcune difficoltà a causa della non validità della disuguaglianza di Hardy in ${\mathbb{R}}^2$ e a causa delle invarianze dell'equazione $-{\left|x\right|}^2\mathrm{\Delta }u=f\left(u\right)$. Pertanto opportune condizioni su $\lambda$ e $\mathrm{\Omega }$ sono necessarie al fine di garantire l'esistenza di una soluzione positiva. Per esempio, se ${\mathrm{\Gamma }}_0$ è una curva non costante...

In this paper we are concerned with the steady Boussinesq system with mixed boundary conditions. The boundary conditions for fluid may include Tresca slip, leak, one-sided leak, velocity, vorticity, pressure and stress conditions together and the conditions for temperature may include Dirichlet, Neumann and Robin conditions together. For the problem involving the static pressure and stress boundary conditions, it is proved that if the data of the problem are small enough, then there exists a solution...

We study steady flow of a compressible heat conducting viscous fluid in a bounded two-dimensional domain, described by the Navier-Stokes-Fourier system. We assume that the pressure is given by the constitutive equation $p(\rho ,\theta )\sim {\rho }^{\gamma }+\rho \theta$, where $\rho$ is the density and $\theta$ is the temperature. For $\gamma >2$, we prove existence of a weak solution to these equations without any assumption on the smallness of the data. The proof uses special approximation of the original problem, which guarantees the pointwise boundedness of the...

We prove the existence of solution in the class H²(Ω) × H¹(Ω) to the steady compressible Oseen system with slip boundary conditions in a two dimensional, convex domain with boundary of class $H^{5/2}$. The method is to regularize a weak solution obtained via the Galerkin method. The problem of regularization is reduced to the problem of solvability of a certain transport equation by application of the Helmholtz decomposition. The method works under an additional assumption on the geometry of the boundary....

We consider the steady plane flow of certain classes of viscoelastic fluids in exterior domains with a non-zero velocity prescribed at infinity. We study existence as well as asymptotic behaviour of solutions near infinity and show that for sufficiently small data the solution decays near infinity as fast as the fundamental solution to the Oseen problem.

Two species of animals are competing in the same environment. Under what conditions do they coexist peacefully? Or under what conditions does either one of the two species become extinct, that is, is either one of the two species excluded by the other? It is natural to say that they can coexist peacefully if their rates of reproduction and self-limitation are relatively larger than those of competition rates. In other words, they can survive if they interact strongly among themselves and weakly...

The existence of a one-parameter family of stationary solutions to a fragmentation equation with size diffusion is established. The proof combines a fixed point argument and compactness techniques.