on a class of singular elliptic systems we study the semilinear elliptic system u u p v q u 0in v u r v s v 0in u v 0on where rn n 1 is a smooth and bounded domain p q r s 0 under suitable ranges of exponents we obtain various results regarding the well posedness of our system

identification of an infinite dimensional parameter for stochastic diffusion equations this paper concerns the control of diffusions under partial observations part i studies the control of the signal process dx t b t x t u t dt sigma t x t u t db t when the observation is dy t h t x t dt dw t and when the objective is to maximize a reward function e int r t k s x s u s ds g x t the existence of an optimal relaxed control is proved part ii studies the separated problem and proves the existence of an optimal markovian filter then the authors compare the two problems and prove under mild conditions that the value functions for the two problems are equal

optimal control for partially observed diffusions stochastic control problems are considered in which a state process x t and an observation process y t are governed by ito sense stochastic differential equations driven by independent brownian motions the control u t enters linearly in the dynamics of x t a separated control problem is introduced in which the state at any time t is a measure lambda t representing an unnormalized conditional distribution for x t given y s u s for s leqq t the method depends on introducing a pathwise version of lambda t which depends continuously on observation and control trajectories y u existence of an optimal control is obtained in a suitable class larger than the usual class of controls admissible in the strict sense that u t is measurable on the sigma algebra mathcal f t y generated by observations y s s leqq t the dynamics of lambda t are studied using a method of forward and backward partial differential equations under a suitable nondegeneracy condition the m

nonlinear schrodinger equations with strongly singular potentials we look for standing waves for nonlinear schrodinger equations i t g y w 0 with cylindrically symmetric potentials g vanishing at infinity and non increasing and a c 1 nonlinear term satisfying weak assumptions in particular we show the existence of standing waves with non vanishing angular momentum with prescribed l 2 norm the solutions are obtained via a minimization argument and the proof is given for an abstract functional which presents a lack of compactness as a specific case we prove the existence of standing waves with non vanishing angular momentum for the nonlinear hydrogen atom equation

existence and multiplicity of normalized solutions for the nonlinear kirchhoff type problems abstract in this paper we prove the existence and multiplicity results of solutions with prescribed l 2 norm for a class of kirchhoff type problems a b r 3 u 2 d x u u f u in r 3 where a b 0 are constants r and f c r r to obtain such solutions we look into critical points of the energy functional e b u a 2 r 3 u 2 b 4 r 3 u 2 2 r 3 f u constrained on the l 2 spheres s c u h 1 r 3 u 2 2 c here c 0 and f s 0 s f t d t under some mild assumptions on f we show that critical points of e b unbounded from below on s c exist for c 0 in addition we establish the existence of infinitely many radial critical points u n b of e b on s c provided that f is odd finally the asymptotic behavior of u n b as b 0 is analyzed these conclusions extend some known ones in previous papers

steady state solutions for a general activator inhibitor model abstract we discuss existence results for a singular gierer meinhardt elliptic system with zero dirichlet boundary conditions which originally arose in studies of pattern formation in biology and has interesting and challenging mathematical properties the mathematical difficulties are that the system becomes singular near the boundary and it is non quasimonotone we show the existence of positive solutions for the general activator inhibitor model

least energy radial sign changing solution for the schrodinger poisson system in r3 under an asymptotically cubic nonlinearity abstract in this paper we consider the following schrodinger poisson system in the whole r 3 u u u f u in r 3 u 2 in r 3 where 0 and the nonlinearity f is asymptotically cubic at infinity this implies that the nonlocal term u and the nonlinear term f u are in some sense in a strict competition we show that the system admits a least energy sign changing and radial solution obtained by minimizing the energy functional on the so called nodal nehari set

multiplicity of normalized solutions for a class of nonlinear schrodinger poisson slater equations abstract in this paper we prove a multiplicity result of solutions for the following stationary schrodinger poisson slater equations 0 1 u u x 1 u 2 u u p 2 u 0 in r 3 where r is an undetermined parameter and p 10 3 6 we obtain couple solutions u with u having a prescribed l 2 norm our proofs are mainly inspired by a recent work of bartsch and de valeriola 7

optimality in accelerated life tests we formulate a stochastic control problem arising from the optimal design of accelerated life tests the model is obtained in a natural way in the setup of point processes the cost functional depends on the total number of items under test at time t the observed number of items failed up to time t and the stress level applied to the items under test at time t the optimal policy minimizing the cost functional is characterized via the dynamic programming equation after a further specification of the model we prove that the solution is of bang bang type in some specific cases we also prove that the control problem is equivalent to an optimal stopping problem this amounts to simply running the test with the number of items and the stress level at their highest available value up to the exit time of the state process from a given region in this special case we exhibit an explicit solution and give a criterion in order to decide whether to let the experiment run or to stop it

orbital stability for the schrodinger operator involving inverse square potential abstract in this paper we prove the existence of orbitally stable standing waves for the schrodinger operator involving potential of the form c x 2 0 c c where c is the best constant in the classical hardy s inequality the approach is purely variational and it is based on the precompactness of any minimizing sequence with respect to the associated energy moreover we discuss the presence of a hardy energy term in conjunction with the behavior of the standing waves

on a singular nonlinear semilinear elliptic problem where k x c2 b v9 a p 0 1 and l is a real parameter such singular elliptic problems arise in the contexts of chemical heterogeneous catalysts nonnewtonian fluids and also the theory of heat conduction in electrically conducting materials see 3 5 8 9 for a detailed discussion obviously 1 1 cannot have a solution u c2 v9 if k x is not vanishing near v however under various appropriate assumptions on k x we will obtain classical solutions of 1 1 for l belonging to a certain range and we will also obtain some uniqueness criteria here a classical solution is a solution u of 1 1 which belongs to c2 v mc v9 with u 0 in v we also study the boundary behaviour of solutions of 1 1 and we will show that the solution u of 1 1 lies in a certain holder class the special case when k x is negative and l 0 has been studied by several authors the existence and uniqueness of the solution were established by crandall rabinowitz and tartar 6 del pino 7 gomes 10 and lazer and mckenna

scaling properties of functionals and existence of constrained minimizers in this paper we develop a new method to prove the existence of minimizers for a class of constrained minimization problems on hilbert spaces that are invariant under translations our method permits to exclude the dichotomy of the minimizing sequences for a large class of functionals we introduce family of maps called scaling paths that permits to show the strong subadditivity inequality as byproduct the strong convergence of the minimizing sequences up to translations is proved we give an application to the energy functional i associated to the schrodinger poisson equation in r3 i t x 1 2 p 2 0 when 2 0 in this way we recover the case studied in sanchez and soler 2004 20 for p 8 3 and we complete the case studied by the authors for 3 p 10 3 in bellazzini and siciliano 2011 4

a minimization method and applications to the study of solitons abstract roughly speaking a solitary wave is a solution of a field equation whose energy travels as a localized packet and which preserves this localization in time a soliton is a solitary wave which exhibits some strong form of stability so that it has a particle like behavior in this paper we prove a general abstract theorem theorem 26 which allows to prove the existence of a class of solitons such solitons are suitable minimizers of a constrained functional and they are called hylomorphic solitons then we apply the abstract theory to problems related to the nonlinear schrodinger equation nse and to the nonlinear klein gordon equation nkg

an approximation method for controlled discrete jump processes under partial observations the paper is concerned with control and filtering of a discrete jump markov process when only the total number of jumps is observed a partially observable control problem is considered and equivalence between this problem and the separated one is proved a technique to approximate the value function under some suitable assumptions is presented

singular gierer meinhardt systems of elliptic boundary value problems abstract we establish existence results for singular gierer meinhardt elliptic systems with zero dirichlet boundary conditions gierer meinhardt systems are model problems for pattern formations of spatial tissue structures of morphogenesis the mathematical difficulties are that the system becomes singular near the boundary and it is non quasimonotone we show the existence of positive solutions for the activator inhibitor model with common sources

global existence and nonexistence for some degenerate and quasilinear parabolic systems abstract the author discusses the degenerate and quasilinear parabolic system u t u v u a u p v q and v t u v v b u r v s with dirichlet boundary conditions in a bounded domain and shows that the global existence depends crucially on the sign of the difference q r 1 p 1 s and the domain

existence and orbital stability of standing waves for nonlinear schrodinger systems abstract in this paper we investigate the existence of solutions in h 1 r n h 1 r n for nonlinear schrodinger systems of the form u 1 1 u 1 1 u 1 p 1 2 u 1 r 1 u 1 r 1 2 u 1 u 2 r 2 u 2 2 u 2 2 u 2 p 2 2 u 2 r 2 u 1 r 1 u 2 r 2 2 u 2 under the constraints r n u 1 2 d x a 1 0 r n u 2 2 d x a 2 0 here n 1 0 i 0 r i 1 2 p i 2 4 n for i 1 2 and r 1 r 2 2 4 n this problem is motivated by the search of standing waves for an evolution problem appearing in several physical models our solutions are obtained as constrained global minimizers of an associated functional note that in the system 1 and 2 are unknown and will correspond to the lagrange multipliers our main result is the precompactness of the minimizing sequences up to translation assuming the local well posedness of the associated evolution problem we then obtain the orbital stability of the standing waves associated to the set of minimizers

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

regularity of an elliptic problem with a singular nonlinearity we study a semilinear boundary value problem with the feature that the vanishing boundary value makes the equation singular we prove that the positive solution is in general holder continuous up to the boundary and has even better regularity in some special cases

global existence for a singular gierer meinhardt system abstract this paper is concerned with existence results for a singular gierer meinhardt system subject to zero dirichlet boundary conditions which originally arose in studies of pattern formation in biology the mathematical difficulties are that the system becomes singular near the boundary and it lacks a variational structure we use a functional method to obtain both upper and lower bounds for the perturbed system and then use sobolev embedding theorem to prove the existence of a pair of positive solutions under suitable conditions this method is first used in a singular parabolic system and is completely different than the traditional methods of sub and super solutions

bounded solutions to a singular parabolic system abstract in this paper we are concerned with the singular parabolic system u t u f x v p v t v g x u q in a smooth bounded domain r n subject to zero dirichlet conditions with initial conditions u 0 x v 0 x 0 this problem is of interest as it is related to some problems in biology and physics under suitable assumptions on p q and f x g x some existence results of weak and classical solutions are obtained using a functional method this method is motivated by such results found in 4 and 5 when dealing with singular parabolic systems and the related references within

global and blow up solutions for compressible euler equations with time dependent damping abstract this paper deals with the cauchy problem for the compressible euler equations with time dependent damping where the time vanishing damping in the form of 1 t makes some fantastic variety of the dynamic system for 0 1 and 0 or 1 but 2 the solutions are proved to exist globally in time when the derivatives of the initial data are small but the initial data themselves can be arbitrarily large this is the so called challenging case of global solutions with large initial data while when the initial riemann invariants are monotonic and their derivatives with absolute value are large at least at one point then the solutions are still bounded but their derivatives will blow up at finite time somewhat like the singularity formed by shock waves for 1 and 0 or 1 but 0 1 the derivatives of solutions will blow up even for all initial data including the interesting case of blow up solutions with small initial data here the initial riemann invariants are monotonic in fact such a blow up phenomenon is determined by the mechanism of the dynamic system itself in order to prove the global existence of solutions with large initial data we introduce a new energy functional related to the riemann invariants which crucially enables us to build up the maximum principle for the corresponding riemann invariants and the uniform boundedness for the local solutions finally numerical simulations in different cases are carried out which further confirm our theoretical results

uniformly bounded weak and classical solutions to a singular parabolic system and applications abstract equations of both elliptic and parabolic type featuring singular nonlinearities have appeared in numerous works throughout the years in this work we consider a time dependent problem featuring nonlinearities of the form u p v q u r v s subject to homogeneous dirichlet boundary conditions and prove the existence of uniformly bounded weak and classical solutions under appropriate conditions on p q r s these results can in some ways be seen as a generalization of the results presented in 9 these results are obtained using a functional method motivated by works found in 8 22 6 etc and the boundary behaviour of a fundamental singular elliptic equation described in 16