Solutions of elliptic problems with nonlinearities of linear growth

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

Global regularity for almost minimizers of nonconvex variational problems

We prove some global, up to the boundary of a domain $\Omega \subset {\mathbb{R}}^{n}We prove some global, up to the boundary of a domain , continuity and Lipschitz regularity results for almost minimizers of functionals of the form The main assumption for g is that it be asymptotically convex with respect its third argument. For the continuity results, the integrand is allowed to have some discontinuous behavior with respect to its first and second arguments. For the global Lipschitz regularity result, we require g to be H?lder continuous with respect to its first two arguments.      

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

Infinitely many solutions of some nonlinear variational equations

The aim of this paper is investigating the existence of one or more critical points of a family of functionals which generalizes the model problem

Poincaré inequality and Palais–Smale condition for the <Emphasis Type="Italic">p</Emphasis>-Laplacian

An improved Poincaré inequality and validity of the Palais-Smale condition are investigated for the energy functional on , 1 < p < ∞, where Ω is a bounded domain in , is a spectral (control) parameter, and is a given function, in Ω. Analysis is focused on the case λ = λ₁, where −λ₁ is the first eigenvalue of the Dirichlet p-Laplacian Δ _p on , λ₁ > 0, and on the “quadratization” of within an arbitrarily small cone in around the axis spanned by , where stands for the first eigenfunction of Δ _p associated with −λ₁.     

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

We consider autonomous integrals

Noncoercive convection–diffusion elliptic problems with Neumann boundary conditions

We study the existence and uniqueness of solutions of the convective–diffusive elliptic equation

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

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

We consider the following Liouville equation in

The boundary growth of superharmonic functions and positive solutions of nonlinear elliptic equations

We investigate the boundary growth of positive superharmonic functions u on a bounded domain Ω in , n ≥ 3, satisfying the nonlinear elliptic inequality

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].     

$C^\infty$ regularity of the free boundary for a two-dimensional optimal compliance problem

We study the regularizing effect of perimeter penalties for a problem of optimal compliance in two dimensions. In particular, we consider minimizers of

Stationary measures and rectifiability

For integers , we consider -valued Radon measures on an open set which satisfy

Geometric second derivative estimates in Carnot groups and convexity

We prove some new a priori estimates for H ₂-convex functions which are zero on the boundary of a bounded smooth domain Ω in a Carnot group . Such estimates are global and are geometric in nature as they involve the horizontal mean curvature of ∂Ω. As a consequence of our bounds we show that if has step two, then for any smooth H ₂-convex function in vanishing on ∂Ω one has

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

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     

Morse indices and the number of maximum points of some solutions to a two-dimensional elliptic problem

In this note, we consider the problem

Every Biregular Function Is a Biholomorphic Map

We study Fueter-biregular functions of one quaternionic variable. We consider left-regular functions in the kernel of the Cauchy–Riemann operator

Invariants and spaces of zero solutions of linear partial differential operators

It is shown that for open convex , d > 1 and a nontrivial polynomial P the space does not have property . If P is elliptic or homogeneous, then this holds for every open Ω. For even cannot occur and if it occurs for some Ω, then P must be hypoelliptic. Received: 18 July 2005     

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.