American Institute of Mathematical Sciences

$ -\text{div}( M(x)\nabla u)- a\frac{u}{|x|^2}=f \text{ in } \Omega, \qquad u=0 \text{ on } \partial \Omega$,

with respect to the summability of $f$ and the value of the parameter $a$. Here $\Omega$ is a bounded domain in $\mathbb{R}^N$ containing the origin.

$ -\Delta u =|u|^{\frac 4{N-2} -\varepsilon} u \quad \text{in } B $

where $B$ is the unit ball in $\R^N$, $N\ge 3$, under zero Dirichlet boundary conditions. We construct radial solutions with $k$ nodal regions which resemble a superposition of "bubbles'' of different signs and blow-up orders, concentrating around the origin. A dual phenomenon is described for the slightly supercritical problem

$ -\Delta u =|u|^{\frac 4{N-2} +\varepsilon} u \quad \text{in } \R^N \setminus B $

under Dirichlet and fast vanishing-at-infinity conditions.

$\frac{\partial u}{\partial t}= \sum$ⁿ_k,j,m=1$ a_{kjm}\frac{\partial^2u}{\partial x_k\partial x_j} (x_1,...,x_{m-1},x_m+h_{kjm},x_{m+1},...,x_n,t),$

assuming that the operator on the right-hand side of the equation is strongly elliptic and the coefficients $a_{kjm}$ and $h_{kjm}$ are real. We prove that this Cauchy problem has a unique solution (in the sense of distributions) and this solution is classical in ${\mathbb R}^n \times (0,+\infty),$ find its integral representation, and construct a differential parabolic equation with constant coefficients such that the difference between its classical bounded solution satisfying the same initial-value function and the investigated solution of the differential-difference equation tends to zero as $t\to\infty$.

$-\varepsilon^2\Delta v+v+\omega\phi v- \varepsilon^{\frac{p-1}{2}} v^p=0 \quad \text{ in}\ B_1,$
$-\Delta \phi=4\pi \omega v^2 \quad \text{in}\ B_1$,
$v,\ \phi>0 \ \text{in}\ B_1 \quad \text{and}\quad v=\phi=0 \quad \text{on}\ \partial B_1$,

where $B_1$ is the unit ball in $\mathbb{R}^3,\ \omega>0$ and $\ \frac{7}{3}$<$p\leq 5$ are constants, and $\varepsilon$>$0$ is a small parameter. Using the localized energy method, we prove that for every sufficiently large integer $N$, the system has a family of radial solutions $(v_\varepsilon, \phi_\varepsilon)$ such that $v_\varepsilon$ has $N$ sharp spheres concentrating on a sphere $\{|x|=r_N\}$ as $\varepsilon\to 0$.