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

Variational integrals of splitting-type: higher integrability under general growth conditions

Besides other things we prove that if , , locally minimizes the energy

Regularity results for the gradient of solutions linear elliptic equations with L 1, λ data

We prove that if f belongs to the Morrey space , with λ ∊ [0, n−2], and u is the solution of the problem

A remark on global existence of solutions of shadow systems

Let Ω be an open, bounded domain in \mathbbRⁿ  (n ? \mathbbN){\mathbb{R}^n\;(n \in \mathbb{N})} with smooth boundary ∂Ω. Let p, q, r, d ₁, τ be positive real numbers and s be a non-negative number which satisfies 0 < \fracp-1r < \fracqs+1{0 < \frac{p-1}{r} < \frac{q}{s+1}}. We consider the shadow system of the well-known Gierer–Meinhardt system:

Renormalized solutions of nonlinear parabolic equations with general measure data

Let a bounded open set, N ≥  2, and let p > 1; we prove existence of a renormalized solution for parabolic problems whose model is

Relaxation of variational problems in two-dimensional nonlinear elasticity

Consider a plate occupying in a reference configuration a bounded open set Ω ⊂ ℝ ² , and let be its stored-energy function. In this paper we are concerned with relaxation of variational problems of type:

Bounds for Zeros of Entire Functions

Let f be an entire function. Denote by z ₁(f),z ₂(f),… the zeros of f with their multiplicities. In the paper, estimates for the sums

On Existence and Multiplicity of Solutions for Elliptic Equations Involving the p-Laplacian

In the present paper, the following Dirichlet problem and Neumann problem involving the p-Laplacian

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     

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

We consider autonomous integrals

Solutions of elliptic problems with nonlinearities of linear growth

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

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.      

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 .      

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

On a class of singular elliptic problems with the perturbed Hardy–Sobolev operator

In this paper, firstly, we investigate a class of singular eigenvalue problems with the perturbed Hardy–Sobolev operator, and obtain some properties of the eigenvalues and the eigenfunctions. (i.e. existence, simplicity, isolation and comparison results). Secondly, applying these properties of eigenvalue problem, and the linking theorem for two symmetric cones in Banach space, we discuss the following singular elliptic problem $$\left\{\begin{array}{ll}-\Delta_{p}u-a(x)\frac{|u|^{p-2}u}{|x|^{p}}= \lambda \eta(x)|u|^{p-2}u+ f(x,u) \quad x \in \Omega, \\ u =0 \quad\quad\quad\quad\quad\quad\quad x\in\partial \Omega, \end{array} \right.$$ where ${a(x)=(\frac{n-p}{p})^{p}q(x),}$ if 1 < p < n, ${a(x)=(\frac{n-1}{n})^{n} \frac{q(x)}{({\rm log}\frac{R}{|x|})^{n}},}$ if p = n, and prove the existence of a nontrivial weak solution for any ${\lambda \in \mathbb{R}.}$      

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:

On the asymptotics of solutions of the Lane-Emden problem for the p-Laplacian

In this paper we consider the Lane–Emden problem adapted for the p-Laplacian

Bifurcations of Positive and Negative Continua in Quasilinear Elliptic Eigenvalue Problems

The main result of this work is a Dancer-type bifurcation result for the quasilinear elliptic problem

On the number of moduli of plane sextics with six cusps

Let be the variety of irreducible sextics with six cusps as singularities. Let be one of irreducible components of . Denoting by the space of moduli of smooth curves of genus 4, we consider the rational map sending the general point [Γ] of Σ, corresponding to a plane curve , to the point of parametrizing the normalization curve of Γ. The number of moduli of Σ is, by definition the dimension of Π(Σ). We know that , where ρ(2, 4, 6) is the Brill–Noether number of linear series of dimension 2 and degree 6 on a curve of genus 4. We prove that both irreducible components of have number of moduli equal to seven.