Global behavior for Whitham type equations on a finite interval

We study the initial-boundary value problem for nonlinear nonlocal equations on a finite interval where λ > 0 and pseudodifferential operator is defined by the inverse Laplace transform. The aim of this paper is to prove the global existence of solutions to the inital-boundary value problem (0.1) and to find the main term of the asymptotic representation in the case of the large initial data.     

Bubbling solutions for an anisotropic Emden–Fowler equation

We consider the following anisotropic Emden–Fowler equation where is a bounded smooth domain and a(x) is a positive smooth function. We investigate the effect of anisotropic coefficient a(x) on the existence of bubbling solutions. We show that at given local maximum points of a(x), there exists arbitrarily many bubbles. As a consequence, the quantity can approach to as . These results show a striking difference with the isotropic case [ Constant].     

Solutions of elliptic problems with nonlinearities of linear growth

In this paper, we study existence of nontrivial solutions to the elliptic equation

From 1-homogeneous supremal functionals to difference quotients: relaxation and Γ-convergence

In this paper we consider positively 1-homogeneous supremal functionals of the type . We prove that the relaxation $\bar{F}$ is a difference quotient, that is where is a geodesic distance associated to F. Moreover we prove that the closure of the class of 1-homogeneous supremal functionals with respect to Γ-convergence is given exactly by the class of difference quotients associated to geodesic distances. This class strictly contains supremal functionals, as the class of geodesic distances strictly contains intrinsic distances. Mathematics Subject Classification (2000) 47J20, 58B20, 49J45     

On the existence of solutions to a special variational problem

In this paper we establish an existence and regularity result for solutions to the problem

Stabilization of the damped wave equation with Cauchy–Ventcel boundary conditions

This paper is devoted to the study of uniform energy decay rates of solutions to the wave equation with Cauchy–Ventcel boundary conditions:

Bubble accumulations in an elliptic Neumann problem with critical Sobolev exponent

We consider the following critical elliptic Neumann problem on , Ω; being a smooth bounded domain in is a large number. We show that at a positive nondegenerate local minimum point Q ₀ of the mean curvature (we may assume that Q ₀ = 0 and the unit normal at Q ₀ is − e _N ) for any fixed integer K ≥ 2, there exists a μ _K > 0 such that for μ > μ _K , the above problem has K−bubble solution u _μ concentrating at the same point Q ₀. More precisely, we show that u _μ has K local maximum points Q ₁^μ, ... , Q _K ^μ ∈∂Ω with the property that and approach an optimal configuration of the following functional (*) Find out the optimal configuration that minimizes the following functional: where are two generic constants and φ (Q) = Q ^T G Q with G = (∇ _ij H(Q ₀)). Research supported in part by an Earmarked Grant from RGC of HK.     

Asymptotic behaviour of renormalized solutions to elliptic equations with measure data and G-converging operators

We study the limit as n goes to +∞ of the renormalized solutions u ⁿ to the nonlinear elliptic problems

Generalised twists,stationary loops,and the Dirichlet energy over a space of measure preserving maps

Let be a bounded Lipschitz domain and consider the Dirichlet energy functional

Solutions to nonlinear Neumann problems with an inverse square potential

Let Ω be an open bounded domain in with smooth boundary . We are concerned with the critical Neumann problem

Bound states for a coupled Schrödinger system

We consider the existence of bound states for the coupled elliptic system

The Dirichlet problem for constant mean curvature surfaces in Heisenberg space

We study constant mean curvature graphs in the Riemannian three- dimensional Heisenberg spaces . Each such is the total space of a Riemannian submersion onto the Euclidean plane with geodesic fibers the orbits of a Killing field. We prove the existence and uniqueness of CMC graphs in with respect to the Riemannian submersion over certain domains taking on prescribed boundary values. L. J. Alías was partially supported by MEC/FEDER project MTM2004-04934-C04-02 and Fundación Séneca project 00625/PI/04, Spain.     

A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems

We consider the 2m-th order elliptic boundary value problem Lu = f (x, u) on a bounded smooth domain with Dirichlet boundary conditions on ∂Ω. The operator L is a uniformly elliptic operator of order 2m given by . For the nonlinearity we assume that , where are positive functions and q > 1 if N ≤ 2m, if N > 2m. We prove a priori bounds, i.e, we show that for every solution u, where C > 0 is a constant. The solutions are allowed to be sign-changing. The proof is done by a blow-up argument which relies on the following new Liouville-type theorem on a half-space: if u is a classical, bounded, non-negative solution of ( − Δ) ^m u  =  u ^q in with Dirichlet boundary conditions on and q > 1 if N ≤ 2m, if N > 2m then .      

Regularity of minimizers of W1,p -quasiconvex variational integrals with (p,q)-growth

We consider autonomous integrals

A mixed problem for the infinity Laplacian via Tug-of-War games

In this paper we prove that a function is the continuous value of the Tug-of-War game described in Y. Peres et al. (J. Am. Math. Soc., 2008, to appear) if and only if it is the unique viscosity solution to the infinity Laplacian with mixed boundary conditions

On <Emphasis Type="Italic">t</Emphasis><Superscript>2</Superscript>-like solutions of certain second order differential equations

Via an integral transformation, we establish two embedding results between the Emden-Fowler type equation , t ≥ t ₀ > 0, with solutions x such that as , , and the equation , u > 0, with solutions y such that for given k > 0. The conclusions of our investigation are used to derive conditions for the existence of radial solutions to the elliptic equation , , that blow up as in the two dimensional case.      

Reiterated homogenization of monotone parabolic problems

Reiterated homogenization is studied for divergence structure parabolic problems of the form . It is shown that under standard assumptions on the function a(y ₁,y ₂,t,ξ) the sequence of solutions converges weakly in to the solution u of the homogenized problem .      

Brake orbits type solutions to some class of semilinear elliptic equations

We consider a class of semilinear elliptic equations of the form

Fast and slow decay solutions for supercritical elliptic problems in exterior domains

We consider the elliptic problem Δu  +  u ^p  =  0, u  >  0 in an exterior domain, under zero Dirichlet and vanishing conditions, where is smooth and bounded in , N ≥ 3, and p is supercritical, namely . We prove that this problem has infinitely many solutions with slow decay at infinity. In addition, a solution with fast decay O(|x|^2-N ) exists if p is close enough from above to the critical exponent.     

A note on equi-integrability in dimension reduction problems

In the framework of the asymptotic analysis of thin structures, we prove that, up to an extraction, it is possible to decompose a sequence of ‘scaled gradients’ (where is the gradient in the k-dimensional ‘thin variable’ x _β) bounded in (1 < p < + ∞) as a sum of a sequence whose p-th power is equi-integrable on Ω and a ‘rest’ that converges to zero in measure. In particular, for k = 1 we recover a well-known result for thin films by Bocea and Fonseca (ESAIM: COCV 7:443–470; 2002).