American Institute of Mathematical Sciences

$ i \h$$ \frac{\partial\psi}{\partial t}=-$ $\frac{ \h^2}{2m}\Delta \psi+V(x)\psi-\gamma|\psi|^{p-2}\psi,$ $\gamma>0,$ $ x\in\mathbb R^{N}$

where $\h$$ >0$, $p>2$, $\psi:\mathbb R^{N}\rightarrow\mathbb C,$ and the potential $V$ satisfies some symmetric properties. In particular the cases $N=2$ with $V$ radially symmetric and $N=3$ with $V$ having a cylindrical symmetry are discussed. Our main purpose is to study the asymptotic behaviour of such solutions in the semiclassical limit (i.e. as $\hbar \rightarrow 0^+$) when a concentration phenomenon around a point of $\mathbb R^N$ appears.

$\partial_t u+A(u) \partial_x u =0,\quad\partial_t v + A(u) \partial_x v =0,$

where $u=u(x,t),\quad v=v(x,t)\in\mathbb R^N$ are the unknowns and $A(u)$ is a strictly hyperbolic matrix. Relying on the notion of weak solution proposed by Dal Maso, LeFloch, and Murat ("Definition and weak stability of nonconservative products", J. Math. Pures Appl. 74 (1995), 483--548), we establish the existence of weak solutions for the corresponding Cauchy problem, in the class of bounded functions with bounded variation. The main steps in our proof are as follows:
(i) We solve the Riemann problem based on a prescribed family of paths.
(ii) We derive a uniform bound on the total variation of corresponding wave-front tracking approximations $u^h$, $v^h$.
(iii) We justify rigorously the passage to the limit in the nonconservative product $A(u^h) \partial_x v^h$, based on the local uniform convergence properties of $u^h$, by extending an argument due to LeFloch and Liu ("Existence theory for nonlinear hyperbolic systems in nonconservative form", Forum Math. 5 (1993), 261--280). Our results provide a generalization to the existence theorem established earlier in the scalar case ($N=1$) by the second author ("An existence and uniqueness result for two nonstrictly hyperbolic systems", IMA Volumes in Math. and its Appl. 27, "Nonlinear evolution equations that change type", ed. B.L. Keyfitz and M. Shearer, Springer Verlag, 1990, pp. 126--138.)

$-L u=\lambda h(x)|x|^{\delta}(u_{+})^q-|x|^{\gamma}(u_{-})^p, \quad $ in $\Omega$,

$u_{\pm}$ ≠0, $\quad u\in E,$ $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ (P)

where $\Omega$ is either a whole space $\mathbb R^N$ or a bounded smooth domain, $Lu =:$ div$(|x|^{\alpha}|\nabla u|^{m-2}\nabla u), $ $\lambda >0, \quad0 < q < m-1 < p \leq m$*$-1,$ $\alpha, $ $\delta $ and $\gamma $ are real numbers, $ N> m-\alpha, $ $m$*$=\frac{(\gamma+N)m}{(\alpha+N-m)}$, $h:\Omega \rightarrow \mathbb R$ is a positive continuous function, $u_{\pm}=\max \{\pm u,0\}$ and $E$ is a Banach space that will be defined later on. We will show that (P) has a solution that changes sign in several situations. The proof of the main results are done by using variational methods applied to the energy functional associated to $(P)$.