On the stability of limit cycles for planar differential systems

We consider a planar differential system , , where P and Q are C¹ functions in some open set U⊆R², and . Let γ be a periodic orbit of the system in U. Let f(x,y):U⊆R²→R be a C¹ function such that

     

A symmetry result for a general class of divergence form PDEs in fibered media

In R^m×R^n−m, endowed with coordinates X=(x,y), we consider the PDE

     

Entire solutions of completely coercive quasilinear elliptic equations

A famous theorem of Sergei Bernstein says that every entire solution u=u(x), x∈R², of the minimal surface equation

     

On a Sobolev-type inequality related to the weighted p-Laplace operator

In this paper we deal with some Sobolev-type inequalities with weights that were proved by Maz'ya in [V.G. Maz'ja, Sobolev Spaces, Springer-Verlag, Berlin, 1980] and by Caffarelli, Kohn and Nirenberg in [L. Caffarelli, R. Kohn, L. Nirenberg, First order interpolation inequalities with weight, Compos. Math. 53 (1984) 259-275]. For integers 1?k?N denote points ξ∈RN=Rk×RN−k as pairs (x,y). Let p∈(1,N), q∈(p,p^∗] and assume . Then there exists c>0 such that

     

Multiple positive solutions for a quasilinear elliptic equation with critical exponential nonlinearity

Let Ω be a bounded domain in R^N,N≥2, with C² boundary. In this work, we study the existence of multiple positive solutions of the following problem:

     

Quasilinear parabolic equations with nonlinear Wentzell-Robin type boundary conditions

Let Ω⊂RN be a bounded domain with Lipschitz boundary, with a>0 on . Let σ be the restriction to ∂Ω of the (N−1)-dimensional Hausdorff measure and let be σ-measurable in the first variable and assume that for σ-a.e. x∈∂Ω, B(x,⋅) is a proper, convex, lower semicontinuous functional. We prove in the first part that for every p∈(1,∞), the operator Ap:=div(a|∇u|^p−2∇u) with nonlinear Wentzell-Robin type boundary conditions

     

Solutions to critical elliptic equations with multi-singular inverse square potentials

Let Ω be an open-bounded domain in R^N(N?3) with smooth boundary ∂Ω. We are concerned with the multi-singular critical elliptic problem

     

Infinitely many solutions for some nonlinear scalar system of two elliptic equations

In this paper, we consider the elliptic system of two equations in H¹(RN)×H¹(RN):

     

On the equality of quasi-arithmetic and Lagrangian means

The paper deals with the equality problem of quasi-arithmetic and Lagrangian means which is to determine all pairs of continuous strictly monotone functions φ,ψ:I→R such that, for all x,y∈I,

     

Uniformly continuous superposition operators in the Banach space of Hölder functions

Let I,J⊂R be intervals. One of the main results says that if a superposition operator H generated by a two place ,

H(φ)(x):=h(x,φ(x)),     

A global multiplicity result for N-Laplacian with critical nonlinearity of concave-convex type

Let Ω⊂RN, N?2, be a bounded domain. We consider the following quasilinear problem depending on a real parameter λ>0:

     

Interior estimates for solutions of a fourth order nonlinear partial differential equation

Let be a locally strongly convex hypersurface, given by the graph of a convex function xn+1=f(x₁,…,xn) defined in a convex domain Ω⊂Rn. M is called a α-extremal hypersurface, if f is a solution of

     

Standing waves for supercritical nonlinear Schrödinger equations

Let V(x) be a non-negative, bounded potential in RN, N?3 and p supercritical, . We look for positive solutions of the standing-wave nonlinear Schrödinger equation Δu−V(x)u+up=0 in RN, with u(x)→0 as |x|→+∞. We prove that if V(x)=o(⁻²|x|) as |x|→+∞, then for N?4 and this problem admits a continuum of solutions. If in addition we have, for instance, V(x)=O(|x|^−μ) with μ>N, then this result still holds provided that N?3 and . Other conditions for solvability, involving behavior of V at ∞, are also provided.     

Asymptotic properties of generalized Laguerre orthogonal polynomials

In the present paper we deal with the polynomials L_n^(α,M,N) (x) orthogonal with respect to the Sobolev inner product

     

Existence of positive solutions for the Hénon equation involving critical Sobolev terms

Let N?3, ^2*=2N/(N−2) and Ω⊂RN be a bounded domain with a smooth boundary ∂Ω and 0∈Ω. Our purpose in this paper is to consider the existence of solutions of Hénon equation:

     

On positive solutions concentrating on spheres for the Gierer-Meinhardt system

We consider the stationary Gierer-Meinhardt system in a ball of R^N: