American Institute of Mathematical Sciences

$x_1'(t)=h_1(t,x_1(t))(a_1(t)-a_{1 1}(t)x_1(t)-\frac{a_{1 3}(t)x_3(t)}{m(t)x_3(t)+x_1(t)})+D_1(t)(x_2(t)-x_1(t))+S_1(t),$

$x_2'(t)=h_2(t,x_2(t))(a_2(t)-a_{2 2}(t)x_2(t))+D_2(t)(x_1(t)-x_2(t))+S_2(t),$

$x_3'(t)=h_3(t,x_3(t))(-a_3(t)+\frac{a_{3 1}(t)x_1(t-\tau)}{m(t)x_3(t-\tau)+x_1(t-\tau)})+S_3(t).$

Some corresponding results are generalized or improved.

$ -\Delta_{2-\varepsilon} u=\lambdae^u$ in $B\quad u =0 $ on $ \partial B,$

for small $\varepsilon>0$. We show that if $k\ge 1$ is given and $\lambda>0$ is fixed and small, then there is a family of radial solutions exhibiting multiple blow-up as $\varepsilon\to 0$ in the form of a superposition of $k$ bubbles of different blow-up orders and shapes. Similar phenomena is found for the same problem involving the operator $\Delta_{N-\varepsilon}$ in $\mathbb R^N$, $N\ge 3$.

$\ddot x + a^+ x^+ - a^-$$ x^-$ $+ b(t)x^2+r(t,x)=0,$

where $a^+,a^-$ are two different positive numbers, $b(t)$ is a $2\pi$-periodic function, and the remaining term $r(t,x)$ is $2\pi$-periodic with respect to the time $t$ and dominated by the power $x^3$ in a neighborhood of the equilibrium $x=0$.

$-\Delta_p u = k_1(|x| )f(u,v),$ for $|x| > 1$ and $x \in \mathbb R^N, $

$-\Delta_p v = k_2(|x|)g(u,v),$ for $|x| > 1 $ and $x \in \mathbb R^N, $

$u(x) = v(x) =0,$ for $|x| =1, $

$u(x), v(x) \rightarrow 0 $ as $|x| \rightarrow +\infty,$

where $1 < p < N $ and $\Delta_p u=$ div $(|\nabla u|^{p-2}\nabla u )$ is the p-Laplacian operator. We consider nonlinearities that are either superlinear or sublinear.

$Lu=u'-(\nu_0+\nu_1||u(t)||^2)\Delta u+(u.\nabla )u-f+\nabla p.$

The mixed problem for the operator $L$ was proposed by J. L. Lions [6]. Using an appropriate penalization, we obtain a variational inequality for the Navier-Stokes perturbed system.

-ħ$^2 \Delta v+($ ħ$^2\omega^2/|x|^2+V(x))v=f(v)$ in $\mathbf R^2.$

We give precise asymptotic profiles of solutions as ħ$\rightarrow 0$ by variational methods and ODE arguments.