In order to obtain solutions to the problem $$ \left\{ \begin{array}{c} -\Delta u=\dfrac{A+h(x)} {|x|^2}u+k(x)u^{2^*-1},\,\,x\in {{{\mathbb R}}^N}, \\ u>0 \hbox{ in }{{{\mathbb R}}^N}, \mbox{ and }u\in {{\mathcal D}^{1,2}}({{{\mathbb R}}^N}), \end{array} \right. $$ $h$ and $k$ must be chosen taking into account not only the size of some norm but the shape. Moreover, if $h(x)\equiv 0$, to reach a multiplicity of solutions, some hypotheses about the local behavior of $k$ close to the points of maximum are needed.

We consider the Cauchy problem for nonlinear Schrödinger equations on $\mathbb{R}^{n}$ with nonzero boundary condition at infinity, a situation which occurs in stability studies of dark solitons. We prove that the Schrödinger operator generates a group on Zhidkov spaces $X^{k}(\mathbb{R}^{n})$ for $k>n/2$, and that the Cauchy problem for NLS is locally well-posed on the same Zhidkov spaces. We justify the conservation of classical invariants which implies in some cases the global well-posedness of the Cauchy problem.

This note is concerned with the recent paper, "Non-topological N-vortex condensates for the self-dual Chern-Simons theory" by M. Nolasco. Making modifications to her arguments, we show that the existence of "nontopological" multivortex condensates actually follows with any prescribed vortex points.

The global-in-time existence of solutions for a system describing the interaction of gravitationally attracting particles that obey the Fermi--Dirac statistics is proved. Stationary solutions of that system are also studied.

First we prove certain pointwise bounds for the fundamental solutions of the perturbed linearized Navier-Stokes equation (Theorem 1.1). Next, utilizing a new framework with very little $L^p$ theory or Fourier analysis, we prove existence of global classical solutions for the full Navier-Stokes equation when the initial value has a small norm in a new function class of Kato type (Theorem 1.2). The smallness in this function class does not require smallness in $L^2$ norm. Furthermore we prove that a Leray-Hopf solution is regular if it lies in this class, which allows much more singular functions then before (Corollary 1). For instance this includes the well-known result in [25]. A further regularity condition (form boundedness) was given in Section 5. We also give a different proof about the $L^2$ decay of Leray-Hopf solutions and prove pointwise decay of solutions for the three-dimensional Navier-Stokes equations (Corollary 2, Theorem 1.2). Whether such a method exists was asked in a survey paper [2].

We investigate the phenomenon of instantaneous shrinking of the support of nonnegative solutions to the Cauchy problem in $\\mathbb R^n$ for $$ u_t=f(u) \Delta u - g(u), \quad \mbox{where } f(0)=0. $$ Among other results, it is shown by means of comparison and integral techniques that under some structural assumptions on $f$ and $g$, a necessary {\em and} sufficient condition on the growth of $g$ near zero for instantaneous shrinking to occur is $\int_0^1 \frac{ds}{g(s)} <\infty$.

We develop critical-point theory in Banach spaces in order to find critical points inside or outside of given convex subsets of the space. In addition to localizing critical points we also obtain results on their critical groups. The abstract theory can be applied to $p$-Laplacian equations in order to prove the existence of multiple sign-changing solutions.

We construct local low-regularity solutions of the generalized Dirac-Klein-Gordon equations as well as global solutions with small initial data.

In the first part of this paper a variational characterization of parts of the Fučík spectrum for the p-Laplacian in an interval is given. The proof uses a linking theorem on suitably constructed sets in $W^{1,p}(0,1)$. In the second part, a superlinear equation with Neumann boundary conditions on an interval is considered, where the nonlinearity intersects all but the first eigenvalues. It is proved that under certain conditions this equation is solvable for arbitrary forcing terms. The proof uses a comparison of the minimax levels of the functional associated to this equation with suitable minimax values related to the Fučík spectrum.