In this paper we use the moving plane method to get the radial symmetry about a point x 0 ∈ R N of the positive ground state solutions of the equationWe assume f to be locally Lipschitz continuous in (0, +∞) and nonincreasing near zero but we do not require any hypothesis on the critical set of the solution. To apply the moving plane method we first prove a weak comparison theorem for solutions of differential inequalities in unbounded domains.

Abstract. For Ω ⊂ R n , n ≥ 2, a bounded domain, and for 1 < p < n, we improve the Hardy-Sobolev inequality by adding a term with a singular weight of the type () 2 . We show that this weight function is optimal in the sense that the inequality fails for any other weight function more singular than this one. Moreover, we show that a series of finite terms can be added to improve the Hardy-Sobolev inequality, which answers a question of BrezisVazquez. Finally, we use this result to analyze the behaviour of the first eigenvalue of the operator Lµu := −(div(|∇u| p−2 ∇u) +

Abstract. We investigate a model of hybrid control system in which both discrete and continuous controls are involved. In this general model, discrete controls act on the system at a given set interface. The state of the system is changed discontinuously when the trajectory hits predefined sets, namely, an autonomous jump set A or a controlled jump set C where the controller can choose to jump or not. At each jump, the trajectory can move to a different Euclidean space. We prove the continuity of the associated value function V with respect to the initial point. Using the dynamic programming principle satisfied by V , we derive a quasi-variational inequality satisfied by V in the viscosity sense. We characterize the value function V as the unique viscosity solution of the quasi-variational inequality by the comparison principle method.

In this paper we consider the one-dimensional compressible Navier-Stokes system linearized about a constant steady state (Q 0 , 0) with Q 0 > 0. We study the controllability and stabilizability of this linearized system. We establish that the linearized system is null controllable for regular initial data by an interior control acting everywhere in the velocity equation. We prove that this result is sharp by showing that the null controllability cannot be achieved by a localized interior control or by a boundary control acting only in the velocity equation. On the other hand, we show that the system is approximately controllable. We also show that the system is not stabilizable with a decay rate e −ωt for ω > ω 0 , where ω 0 is an accumulation point of the real eigenvalues of the linearized operator.

In this paper we consider the semilinear elliptic problem in a bounded domain « R n ,where · > 0, 0 6 ¬ 6 2, 2 ¤ ¬ := 2(n ¡ ¬ )=(n ¡ 2), f : « ! R + is measurable, f > 0 a.e, having a lower-order singularity than jxj ¡ 2 at the origin, and g : R ! R is either linear or superlinear. For 1 < p < n, we characterize a class of singular functions = p for which the embedding W 1;p 0 (« ) ,! L p (« ; f ) is compact. When p = 2, ¬ = 2, f 2 = 2 and 0 6 · < ( 1 2 (n ¡ 2)) 2 , we prove that the linear problem has H 1 0 -discrete spectrum. By improving the Hardy inequality we show that for f belonging to a certain subclass of = 2 , the¯rst eigenvalue goes to a positive number as · approaches ( 1 2 (n ¡ 2)) 2 . Furthermore, when g is superlinear, we show that for the same subclass of = 2 , the functional corresponding to the di® erential equation satis¯es the Palais{Smale condition if ¬ = 2 and a Brezis{Nirenberg type of phenomenon occurs for the case 0 6 ¬ < 2.