We are concerned with a simplified quantum energy-transport model for bipolar semiconductors, which consists of nonlinear parabolic fourth-order equations for the electron and hole density; degenerate elliptic heat equations for the electron and hole temperature; and Poisson equation for the electric potential. For the periodic boundary value problem in the torus ${𝕋}^d$, the global existence of weak solutions is proved, based on a time-discretization, an entropy-type estimate, and a fixed-point argument....

In this paper, we study the semiclassical limit of the cubic nonlinear Schrödinger equation with the Neumann boundary condition in an exterior domain. We prove that before the formation of singularities in the limit system, the quantum density and the quantum momentum converge to the unique solution of the compressible Euler equation with the slip boundary condition as the scaling parameter approaches $0.$

In this article, the structure of semiclassical measures for solutions to the linear Schrödinger equation on the torus is analysed. We show that the disintegration of such a measure on every invariant lagrangian torus is absolutely continuous with respect to the Lebesgue measure. We obtain an expression of the Radon-Nikodym derivative in terms of the sequence of initial data and show that it satisfies an explicit propagation law. As a consequence, we also prove an observability inequality, saying...

We extend our recent results on propagation of semiclassical resolvent estimates through trapped sets when a priori polynomial resolvent bounds hold. Previously we obtained non-trapping estimates in trapping situations when the resolvent was sandwiched between cutoffs $\chi$ microlocally supported away from the trapping: $\parallel \chi R_h(E+i0)\chi \parallel =𝒪\left(h^{-1}\right)$, a microlocal version of a result of Burq and Cardoso-Vodev. We now allow one of the two cutoffs, $\tilde{\chi }$, to be supported at the trapped set, giving $\parallel \chi R_h(E+i0)\tilde{\chi }\parallel =𝒪\left(\sqrt{a\left(h\right)}{h}^{-1}\right)$ when the a priori bound is $\parallel \tilde{\chi }{R}_h(E+i0)\tilde{\chi }\parallel =𝒪\left(a\left(h\right)h^{-1}\right)$.

We consider a singularly perturbed elliptic equation ${ϵ}^2\Delta u-V\left(x\right)u+f\left(u\right)=0,u\left(x\right)>0$ on ${ℝ}^N$, ${\mathrm{𝚕𝚒𝚖}}_{\left|x\right|\to \infty }u\left(x\right)=0$, where $V\left(x\right)>0$ for any $x\in {ℝ}^N$. The singularly perturbed problem has corresponding limiting problems $\Delta U-cU+f\left(U\right)=0,U\left(x\right)>0$ on ${ℝ}^N$, ${\mathrm{𝚕𝚒𝚖}}_{\left|x\right|\to \infty }U\left(x\right)=0,c>0$. Berestycki-Lions found almost necessary and sufficient conditions on nonlinearity $f$ for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of potential $V$ under possibly general conditions on $f$. In...

We consider systems of weakly coupled Schrödinger equations with nonconstant potentials and investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials. We obtain sufficient and necessary conditions for a sequence of least energy solutions to concentrate.

Using some perturbation results in critical point theory, we prove that a class of nonlinear Schrödinger equations possesses semiclassical states that concentrate near the critical points of the potential $V$.

Let Ω be a measure space, and E, F be separable Banach spaces. Given a multifunction $f:\Omega \times E\to 2^F$, denote by $N_f\left(x\right)$ the set of all measurable selections of the multifunction $f(·,x\left(·\right)):\Omega \to 2^F$, s ↦ f(s,x(s)), for a function x: Ω → E. First, we obtain new theorems on H-upper/H-lower/lower semicontinuity (without assuming any conditions on the growth of the generating multifunction f(s,u) with respect to u) for the multivalued (Nemytskiĭ) superposition operator $N_f$ mapping some open domain G ⊂ X into $2^Y$, where X and Y are Köthe-Bochner...