Radial solutions of Dirichlet problems with concave-convex nonlinearities

We prove the existence of a double infinite sequence of radial solutions for a Dirichlet concave-convex problem associated with an elliptic equation in a ball of Rⁿ. We are interested in relaxing the classical positivity condition on the weights, by allowing the weights to vanish. The idea is to develop a topological method and to use the concept of rotation number. The solutions are characterized by their nodal properties.     

Quantization effects for Maxwell-Chern-Simons vortices

In this paper, we prove the quantization of potential energy of topological Maxwell-Chern-Simons vortices on R² when the gauge fields vanish. We also show the nonexistence of nontopological solutions.     

Homographic solutions of the n-body problem in E4

We prove the existence of many homographic solutions of the n-body problem in E⁴ by topological methods. Homographic solutions are associated with relative equilibria. Homothetic solutions always give rise to central configurations. In Euclidean space E⁴ central configurations are a proper subset of the relative equilibria for any n ? 3 and for any (m_i)?R₊ⁿ. We compare the existence and classification of homographic solutions of the n-body problem in E³ with the Newtonian potential and that of homographic solutions of the n-body problem in E⁴. Classifying relative equilibria leads to classifying homographic solutions.     

Multiple solutions for a nonlinear and non-homogeneous problem in Orlicz-Sobolev spaces

We study a non-homogeneous boundary value problem in a smooth bounded domain in R^N. We prove the existence of at least two non-negative and non-trivial weak solutions. Our approach relies on Orlicz-Sobolev spaces theory combined with adequate variational methods and a variant of Mountain Pass Lemma.     

On the Chern–Simons–Higgs vortex equation without gauge fields

This paper is concerned with the nonself-dual Chern–Simons–Higgs model on R²${\mathbb{R}}^2$ with vanishing gauge fields. We prove the existence of radial solutions with the topological boundary condition, and the nonexistence of radial solutions with the nontopological boundary condition. We also establish the asymptotic properties of solutions and derive the quantization of the potential energy.     

Massera type theorem for almost automorphic solutions of functional differential equations of neutral type

In this work, we study the existence of almost automorphic solutions for functional differential equations of neutral type. We prove that the existence of a bounded solution on R⁺ implies the existence of an almost automorphic solution.     

Blow-up solutions to the nonlinear Schrödinger equation with oscillating nonlinearities

In this article we investigate the possibility of finite time blow-up in H¹(R²) for solutions to critical and supercritical nonlinear Schrödinger equations with an oscillating nonlinearity. We prove that despite the oscillations some solutions blow up in finite time. Conversely, we observe that for a given initial data oscillations can extend the local existence time of the corresponding solution.     

An indefinite concave-convex equation under a Neumann boundary condition I

We investigate the problem (P _λ) ?Δu = λb(x)|u| ^q?2 u + a(x)|u| ^p?2 u in Ω, ?u/?n = 0 on ?Ω, where Ω is a bounded smooth domain in R ^N (N ≥ 2), 1 < q < 2 < p, λ ∈ R, and a, b ∈ \({C^\alpha }\left( {\overline \Omega } \right)\) with 0 < α < 1. Under certain indefinite type conditions on a and b, we prove the existence of two nontrivial nonnegative solutions for small |λ|. We then characterize the asymptotic profiles of these solutions as λ → 0, which in some cases implies the positivity and ordering of these solutions. In addition, this asymptotic analysis suggests the existence of a loop type component in the non-negative solutions set. We prove the existence of such a component in certain cases, via a bifurcation and a topological analysis of a regularized version of (P _λ).     

Existence and decay of solutions in full space to Navier-Stokes equations with delays

We consider the Navier-Stokes equations with delays in Rⁿ,2≤n≤4. We prove existence of weak solutions when the external forces contain some hereditary characteristics and uniqueness when n=2. Moreover, if the external forces satisfy a time decay condition we show that the solution decays at an algebraic rate.     

Almost automorphic solutions for some partial functional differential equations

In this work, we study the existence of almost automorphic solutions for some partial functional differential equations. We prove that the existence of a bounded solution on R⁺ implies the existence of an almost automorphic solution. Our results extend the classical known theorem by Bohr and Neugebauer on the existence of almost periodic solutions for inhomegeneous linear almost periodic differential equations. We give some applications to hyperbolic equations and Lotka-Volterra type equations used to describe the evolution of a single diffusive animal species.     

Existence and uniqueness of solutions to the GRID macroscopic growth equation

In this paper we prove the existence and uniqueness of solutions to the initial value problems associated with the GRID integro-differential equation describing macroscopic growth of an organism. We consider the general form of the macroscopic growth operator Φ and study the set of conditions on Φ that are sufficient to guarantee existence and uniqueness of solutions in Rⁿ,n=1,2,3.     

Schrödinger-Poisson equations without Ambrosetti-Rabinowitz condition

We prove the existence of ground state solutions for a stationary Schrödinger-Poisson equation in R³. The proof is based on the mountain pass theorem and it does not require the Ambrosetti-Rabinowitz condition.     

The existence and asymptotics of solutions for the Abelian Chern–Simons system with two Higgs fields and two gauge fields

In this paper, we study a system of elliptic equations in R² which arises from the self-dual equations for the Abelian Chern–Simons system with two Higgs fields and two gauge fields. We provide a new proof for the existence of topological solutions by constructing explicit supersolutions and subsolutions. We also study the asymptotic behavior of condensate solutions on the torus. It is shown that the maximal solutions converge uniformly to zero away from the vortex points, and the convergence rate is computed.     

Trivial stationary solutions to the Kuramoto-Sivashinsky and certain nonlinear elliptic equations

We show that the only locally integrable stationary solutions to the integrated Kuramoto-Sivashinsky equation in R and R² are the trivial constant solutions. We extend our technique and prove similar results to other nonlinear elliptic problems in RN.     

Global existence of solutions for one-dimensional compressible navier-stokes equations in the half space

We prove the existence of global solutions to the initial-boundary-value problem on the half space R＋ for a one-dimensional viscous ideal polytropic gas. Some suitable assumptions are made to guarantee the existence of smooth solutions. Employing the L2- energy estimate, we prove that the impermeable problem has a unique global solutionis.     

Nonlinear partial differential systems on Riemannian manifolds with their geometric applications

We make the first study of how the existence of (essential) positive supersolutions of nonlinear degenerate partial differential equations on a manifold affects the topology, geometry, and analysis of the manifold. For example, for surfaces in R³ we prove a Bernstein-type theorem that generalizes and unifies three distinct theorems. In higher dimensions, we provide topological obstructions for a minimal hypersurface in Rⁿ⁺¹ to admit an essential positive supersolution. This immediately yields information about the Gauss map of complete minimal hypersurfaces in Rⁿ⁺¹. By coping with a wider class of nonlinear partial differential equations that are involved with (p)-harmonic maps and (p)-superstrongly unstable manifolds, we derive information on the regularity of minimizers, homotopy groups, and solutions to Dirichlet problems, from the existence of essential positive supersolutions.     

Martin Kernels for Markov Processes with Jumps

We prove the existence of boundary limits of ratios of positive harmonic functions for a wide class of Markov processes with jumps and irregular (possibly disconnected) domains of harmonicity, in the context of general metric measure spaces. As a corollary, we prove the uniqueness of the Martin kernel at each boundary point, that is, we identify the Martin boundary with the topological boundary. We also prove a Martin representation theorem for harmonic functions. Examples covered by our results include: strictly stable Lévy processes in R ^d with positive continuous density of the Lévy measure; stable-like processes in R ^d and in domains; and stable-like subordinate diffusions in metric measure spaces.     

Topological structure of solution sets for impulsive differential inclusions in Fréchet spaces

In this paper, we consider the existence of solutions as well as the topological and geometric structure of solution sets for first-order impulsive differential inclusions in some Fréchet spaces. Both the initial and terminal problems are considered. Using ingredients from topology and homology, the topological structures of solution sets (closedness and compactness) as well as some geometric properties (contractibility, acyclicity, AR and R_δ) are investigated. Some of our existence results are obtained via the method of taking the inverse system limit on noncompact intervals.     

Regularity of ground state solutions of dispersion managed nonlinear schrödinger equations

We consider the dispersion managed nonlinear Schrödinger equation (DMNLS) in the case of zero residual dispersion. Using dispersive properties of the equation and estimates in Bourgain spaces we show that the ground state solutions of DMNLS are smooth. The existence of smooth solutions in this case matches the well-known smoothness of the solutions in the case of nonzero residual dispersion. In the case x∈R² we prove that the corresponding minimization problem with zero residual dispersion has no solution.