American Institute of Mathematical Sciences

$\partial _t U-\Delta U=P(D)F(U), \quad (t,x) \in [0,T[ \times \mathbb{R}^n $

with initial data in Sobolev spaces of fractional order $H^s_p(\mathbb{R}^n)$. The techniques that we use allow us to consider measures but also distributions as initial data ($s<0$). We also prove some smoothing effects and $L^q([0,T[,L^p)$ estimates for the solutions of such equations.

$-\Delta u = R(x) u^p \quad \quad u > 0 x \in \Omega$

$u(x) = 0 \quad \quad x \in \partial \Omega.$

where $R(x)$ is a smooth function that may change signs. In [2], using a blowing up argument, Berestycki, Dolcetta, and Nirenberg, obtained a priori estimates and hence the existence of solutions for the problem when the exponent $1 < p < {n+2}/{n-1}$. Inspired by their result, in this article, we use the method of moving planes to fill the gap between ${n+2}/{n-1}$ and the critical Sobolev exponent ${n+2}/{n-2}$. We obtain a priori estimates for the solutions for all $1 < p < {n+2}/{n-2}$.

$ i u_t + u_{x x} + ia(|u|^2u)_x = 0, \quad (t,x) \in \mathbf{R}\times \mathbf{R},$

$ u(0,x) = u_0 (x), \quad x\in \mathbf{R},\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$(DNLS)

where $a \in \mathbf{R}$. We prove that if $ ||u_0||_{ H^{1,\gamma}} + ||u_0||_{ H^{1+\gamma,0}}$ is sufficiently small with $\gamma > 1/2$, then the solution of (DNLS) satisfies the time decay estimate

$ ||u(t)||_{L^\infty} + ||u_x(t)||_{L^\infty}\le C(1+|t|)^{-1/2}, $

where $H^{m,s}= \{f\in \mathcal{S}'; ||f||_{m,s}= ||(1+|x|^2)^{s/2}(1-\partial_x^2)^{m/2}f||_{L^2} < \infty\}$, $m,s\in \mathbf{R}$. In the previous paper [4,Theorem 1.1] we showed the same result under the condition that $\gamma \ge 2$. Furthermore we show the asymptotic behavior in time of solutions involving the previous result [4,Theorem 1.2].

$D_s\dot x(s) + \nabla V_x(x(s),s) = 0$

where $D_s\dot x(s)$ is the covariant derivative of $\dot x(s)$, $V$ is a $\mathcal{C}^2$ real function on $\mathcal{M}\times \mathbf{R}$, $T$-periodic in $s$. The manifold is allowed to be noncompact and to have boundary, so the action integral associated to the equation does not satisfy the Palais-Smale compactness condition. We overcome this problem under a assumption on the sectional curvature of $\mathcal{M}$ which allows to control the Morse index of the critical points of $f$ at "infinity". If $\mathcal{M}$ has a "rich" topology it is proved that there exist infinitely many periodic solutions.