In this paper, we consider the existence of positive solutions to the following problem ⎪⎪⎨ ⎪⎪⎩ −div(|∇u|p−2∇u) = ∂F ∂u (u,v)+ ε p−1g(x) in Ω, −div(|∇v|q−2∇v) = ∂F ∂v (u,v)+ εq−1h(x) in Ω, u,v > 0 in Ω, u = v = 0 on ∂Ω, where Ω is a bounded smooth domain in RN ; F ∈C1((R+)2,R+) is positively homogeneous of degree μ ; g,h ∈C1(Ω)\{0} ; and ε is a positive parameter. Using sub-supersolution method...

We consider an Hamilton-Jacobi equation of the form H(x,Du) = 0 x ∈ Ω ⊂ R , (1) where H(x, p) is assumed Borel measurable and quasi-convex in p. The notion of Monge solution, introduced by Newcomb and Su, is adapted to this setting making use of suitable metric devices. We establish the comparison principle for Monge sub and supersolution, existence and uniqueness for equation (1) coupled with ...

Via a sub-supersolution method and a perturbation argument, we study the Lane-Emden-Fowler equation −∆u = p(x)[g(u) + f(u) + |∇u| ] in RN (N ≥ 3), where 0 < q < 1, p is a positive weight such that R∞ 0 rφ(r)dr < ∞, where φ(r) = max|x|=r p(x), r ≥ 0. Under the hypotheses that both g and f are sublinear, which include no monotonicity on the functions g(u), f(u), g(u)/u and f(u)/u, we show the exi...

We introduce an iterative method for computing the first eigenpair (λp, ep) for the pLaplacian operator with homogeneous Dirichlet data as the limit of (μq,uq) as q → p −, where uq is the positive solution of the sublinear Lane-Emden equation −∆puq = μqu q−1 q with same boundary data. The method is shown to work for any smooth, bounded domain. Solutions to the Lane-Emden problem are obtained th...

Our goal is to resolve a problem proposed by Fernholz and Karatzas [On optimal arbitrage (2008) Columbia Univ.]: to characterize the minimum amount of initial capital with which an investor can beat the market portfolio with a certain probability, as a function of the market configuration and time to maturity. We show that this value function is the smallest nonnegative viscosity supersolution ...

We consider a portfolio optimization problem which is formulated as a stochastic control problem. Risky asset prices obey a logarithmic Brownian motion, and interest rates vary according to an ergodic Markov diffusion process. The goal is to choose optimal investment and consumption policies to maximize the infinite horizon expected discounted HARA utility of consumption. A dynamic programming ...

and Applied Analysis 3 2. Proof of Theorems Consider the more general semilinear elliptic problem −Δu f x, u , in Ω, u > 0, in Ω, u 0, on ∂Ω, 2.1 where the function f x, s is locally Hölder continuous in Ω × 0,∞ and continuously differentiable with respect to the variable s. A function u is called to be a subsolution of problem 2.1 if u ∈ C2 Ω ∩ C Ω , and −Δu ≤ fx, u, in Ω, u > 0, in Ω, u 0, on...

Abstract In this paper, the existence of a solution for an anisotropic variable exponent system is obtained and proved under general hypotheses. By considering additional conditions, it multiplicity result. The proofs are based on application appropriated $L^{\infty }$ L ∞ </mml:ma...

We consider general two-sided matching markets, so-called matching with contracts markets as introduced by Hatfield and Milgrom (2005), and analyze (Maskin) monotonic and Nash implementable solutions. We show that for matching with contracts markets the stable correspondence is monotonic and implementable (Theorems 1 and 3). Furthermore, any solution that is Pareto efficient, individually ratio...