On the structure of positive radial solutions to an equation containing a p-Laplacian with weight

Let A,B:(0,∞)?(0,∞) be two given weight functions and consider the equation

     

Weighted weak type inequalities for modified Hardy operators and geometric means operators in dimensions one and greater

We characterize the pairs of weights (u,v) such that the geometric mean operator G₁, defined for positive functions f on (0,∞) by

     

On the singular set for Lipschitzian critical points of polyconvex functionals

Partial regularity is proved for Lipschitzian critical points of polyconvex functionals provided ‖DuL^∞_‖ is small enough. In particular, the singular set for a Lipschitzian critical point has Hausdorff dimension strictly less than n when ‖DuL^∞_‖ is small enough. Model problems treated include

     

Bernstein widths of Hardy-type operators in a non-homogeneous case

Let I=[a,b]⊂R, let 1<p?q<∞, let u and v be positive functions with u∈Lp^′(I), v∈Lq(I) and let be the Hardy-type operator given by

     

On the spectral radius of positive operators on Banach sequence spaces

Let K₁,…,K_n be (infinite) non-negative matrices that define operators on a Banach sequence space. Given a function f:[0,∞)×…×[0,∞)→[0,∞) of n variables, we define a non-negative matrix and consider the inequality

     

Deduction of L. Hörmander's extension of Ásgeirsson's mean value theorem

L. Hörmander's extension of Ásgeirsson's mean value theorem states that if u is a solution of the inhomogeneous ultrahyperbolic equation (Δ_x−Δ_y)u=f, , , then

     

Positive solutions of reaction diffusion equations with super-linear absorption: Universal bounds, uniqueness for the Cauchy problem, boundedness of stationary solutions

Consider classical solutions u∈C²(Rⁿ×(0,∞))∩C(Rⁿ×[0,∞)) to the parabolic reaction diffusion equation

     

Existence, uniqueness, and quenching properties of solutions for degenerate semilinear parabolic problems with second boundary conditions

Let q?0, p?0, T?∞, D=(0,a), , Ω=D×(0,T), and Lu=xqut−uxx. This article considers the following degenerate semilinear parabolic initial-boundary value problem,

     

Bounded positive entire solutions of singular quasilinear elliptic equations

Suppose that β?0 is a constant and that is a continuous function with R₊:=(0,∞). This paper investigates N-dimensional singular, quasilinear elliptic equations of the form

     

On blow-up at space infinity for semilinear heat equations

A nonnegative blowing up solution of the semilinear heat equation ut=Δu+up with p>1 is considered when initial data u₀ satisfies

     

Explicit formulas for Laplace transforms of certain functionals of some time inhomogeneous diffusions

We consider a process given by the SDE , t∈[0,T), with initial condition , where T∈(0,∞], α∈R, (Bt)_t∈[0,T) is a standard Wiener process, b:[0,T)→R?{0} and σ:[0,T)→(0,∞) are continuously differentiable functions. Assuming , t∈[0,T), with some K∈R, we derive an explicit formula for the joint Laplace transform of and for all t∈[0,T) and for all α∈R. Our motivation is that the maximum likelihood estimator (MLE) of α can be expressed in terms of these random variables. As an application, we show that in case of α=K, K≠0,

     

Localization for the evolution p-Laplacian equation with strongly nonlinear source term

In this paper we study the strict localization for the p-Laplacian equation with strongly nonlinear source term. Let u:=u(x,t) be a solution of the Cauchy problem

     

The asymptotic behaviour of solutions with blow-up at the boundary for semilinear elliptic problems

By constructing the comparison functions and the perturbed method, it is showed that any solution u∈C²(Ω) to the semilinear elliptic problems Δu=k(x)g(u), x∈Ω, u_|∂Ω=+∞ satisfies , where Ω is a bounded domain with smooth boundary in RN; , −2<σ, c₀>0, ; g∈C¹[0,∞), g?0 and is increasing on (0,∞), there exists ρ>0 such that , ∀ξ>0, , .     

Multiplicity of solutions for the plasma problem in two dimensions

Let Ω be a bounded domain in R², u₊=u if u?0, u₊=0 if u<0, u₋=u₊−u. In this paper we study the existence of solutions to the following problem arising in the study of a simple model of a confined plasma

     

Invariant Banach limits and applications

Let ?_∞ be the space of all bounded sequences x=(x₁,x₂,…) with the norm