Asymptotic symmetries for conformal scalar curvature equations with singularity

We give conditions on a positive Hölder continuous function C²such that every C ² positive solution u((x)) of the conformal scalar curvature equation in a punctured neighborhood of the origin in R ⁿ either has a removable singularity at the origin or satisfies for some positive singular solution u ₀(|x|) of where is the Hölder exponent of K.Mathematics Subject Classification (2000) Primary 35J60, 53C21     

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

Existence and non-existence of solutions to elliptic equations related to the Caffarelli–Kohn–Nirenberg inequalities

We investigate elliptic equations related to the Caffarelli–Kohn–Nirenberg inequalities: and such that . For various parameters α, β and various domains Ω, we establish some existence and non-existence results of solutions in rather general, possibly degenerate or singular settings.     

Solutions of elliptic problems with nonlinearities of linear growth

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

Critical Nonlinear Nonlocal Equations on a Half-Line

We study nonlinear nonlocal equations on a half-line in the critical case

Partial regularity for polyconvex functionals depending on the Hessian determinant

We prove a C ^2,α partial regularity result for local minimizers of polyconvex variational integrals of the type , where Ω is a bounded open subset of , and is a convex function, with subquadratic growth.     

Nonradial solutions for a conformally invariant fourth order equation in \mathbb {R}^4

We consider the following Liouville equation in

Ground state alternative for <Emphasis Type="Italic">p</Emphasis>-Laplacian with potential term

Let Ω be a domain in , d ≥ 2, and 1 < p < ∞. Fix . Consider the functional Q and its Gateaux derivative Q′ given by If Q ≥ 0 on, then either there is a positive continuous function W such that for all, or there is a sequence and a function v > 0 satisfying Q′ (v) = 0, such that Q(u _k ) → 0, and in . In the latter case, v is (up to a multiplicative constant) the unique positive supersolution of the equation Q′ (u) = 0 in Ω, and one has for Q an inequality of Poincaré type: there exists a positive continuous function W such that for every satisfying there exists a constant C > 0 such that . As a consequence, we prove positivity properties for the quasilinear operator Q′ that are known to hold for general subcritical resp. critical second-order linear elliptic operators.     

Some remarks on systems of elliptic equations doubly critical in the whole $${\mathbb{R}^N}$$

We study the existence of different types of positive solutions to problem

Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem

We study the following system of Maxwell-Schrödinger equations $ \Delta u - u - \delta u \psi+ f(u)=0, \quad \Delta \psi + u^2 = 0 \mbox{in} {\mathbb R}^N , u, \;\psi > 0, \quad u, \;\psi \to 0 \ \mbox{as} \ |x| \to + \infty, $ where δ > 0, u, ψ : $\psi: {\mathbb R}^N \to {\mathbb R}We study the following system of Maxwell-Schr?dinger equations where δ > 0, u, ψ : , f : , N ≥ 3. We prove that the set of solutions has a rich structure: more precisely for any integer K there exists δ_K > 0 such that, for 0 < δ < δ_K, the system has a solution (u_δ, ψ_δ) with the property that u_δ has K spikes centered at the points . Furthermore, setting , then, as δ → 0, approaches an optimal configuration for the following maximization problem: Subject class: Primary 35B40, 35B45; Secondary 35J55, 92C15, 92C40     

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

We consider autonomous integrals

Bound states for a coupled Schrödinger system

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

Brake orbits type solutions to some class of semilinear elliptic equations

We consider a class of semilinear elliptic equations of the form

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].     

Sign-changing stationary solutions and blowup for the nonlinear heat equation in a ball

In this paper, we prove that if is a radially symmetric, sign-changing stationary solution of the nonlinear heat equation

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.     

Calderón–Zygmund estimates for higher order systems with p(x) growth

For weak solutions of higher order systems of the type , for all , with variable growth exponent p : Ω → (1,∞) we prove that if with , then . We should note that we prove this implication both in the non-degenerate (μ > 0) and in the degenerate case (μ = 0).     

Multiple positive solutions for a quasilinear system of Schrödinger equations

We consider the quasilinear system

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: