Positive Continuous Function Research Articles

Indentation of a hard transversely isotropic functionally graded coating by a conical indenter

An axisymmetric contact problem on indentation of a rigid conical punch into an elastic transversely isotropic half-space with a functionally graded transversely isotropic coating is considered. Elastic moduli of the coating vary in depth according to arbitrary continuous positive functions, independent of each other. Mathematical statement of the problem is made in terms of the linear theory of elasticity. Using integral transformation technique the problem is reduced to the solution of a dual integral equation. Kernel transform of the integral equation, which is calculated numerically from a two-point boundary value problem for a system of ordinary differential equations with variable coefficients, is approximated by a product of fractional quadratic functions. Using these approximations, an approximated solution of the problem is constructed in analytical form. The solution is asymptotically exact both for small and big values of the characteristic geometrical parameter of the problem (ratio of thickness of the coating to radius of the contact area). Approximated analytical expressions relating the displacement of the punch, indentation force acting on the punch and the size of the contact area are obtained. Correlation between the contact normal stresses arising on surface of the coated half-space and on surface of the homogeneous half-space without a coating is studied. Some relations are obtained analytically using asymptotic analysis and illustrated numerically. Results on numerical simulation of an indentation of a conical punch into a hard homogeneous or functionally graded (with linearly varying elastic moduli in depth) transversely isotropic coating are provided. The materials widely used in electronics are chosen for numerical examples. Qualitative differences in process of elastic deformation of bodies with homogeneous and functionally graded coatings are illustrated.

Definability of semifields of continuous positive functions by the lattices of their subalgebras

We consider the lattice of subalgebras of a semifield of continuous positive functions on an arbitrary topological space and its sublattice of subalgebras with unity. The main result of the paper is the proof of the definability of any semifield both by the lattice and by its sublattice . Bibliography: 12 titles.

Multiplicity of positive solutions to boundary blow‐up problem with variable exponent and sign‐changing weights

In this paper, we consider the elliptic boundary blow‐up problem urn:x-wiley:mma:media:mma4119:mma4119-math-0001 where Ω is a bounded smooth domain of are positive continuous functions supported in disjoint subdomains Ω+,Ω− of Ω, respectively, a+ vanishes on the boundary of satisfies p(x)≥1 in Ω,p(x) > 1 on ∂Ω and , and ε is a parameter. We show that there exists ε∗>0 such that no positive solutions exist when ε > ε∗, while a minimal positive solution uε exists for every ε∈(0,ε∗). Under the additional hypotheses that is a smooth N − 1‐dimensional manifold and that a+,a− have a convenient decay near Γ, we show that a second positive solution vε exists for every ε∈(0,ε∗) if , where N∗=(N + 2)/(N − 2) if N > 2 and if N = 2. Our results extend that of Jorge Garcá‐Melián in 2011, where the case that p > 1 is a constant and a+>0 on ∂Ω is considered. Copyright © 2016 John Wiley & Sons, Ltd.

Asymptotic formulae for solutions of half-linear differential equations

We establish asymptotic formulae for regularly varying solutions of the half-linear differential equation(r(t)|y′|α−1sgny′)′=p(t)|y|α−1sgny,where r, p are positive continuous functions on [a, ∞) and α ∈ (1, ∞). The results can be understood in several ways: Some open problems posed in the literature are solved. Results for linear differential equations are generalized; some of the observations are new even in the linear case. A refinement on information about behavior of solutions in standard asymptotic classes is provided. A precise description of regularly varying solutions which are known to exist is given. Regular variation of all positive solutions is proved.

Solutions of half-linear differential equations in the classes Gamma and Pi

We study asymptotic behavior of (all) positive solutions of the non\-oscillatory half-linear differential equation of the form $$ (r(t)|y'|^ {\alpha-1}\text{sgn}\, y')'=p(t)|y|^{\alpha-1}\text{sgn}\, y , $$ where $\alpha\in(1,\infty)$ and $r,p$ are positive continuous functions on $[a,\infty)$, with the help of the Karamata theory of regularly varying functions and the de Haan theory. We show that increasing resp. decreasing solutions belong to the de Haan class $\Gamma$ resp. $\Gamma_-$ under suitable assumptions. Further we study behavior of slowly varying solutions for which asymptotic formulas are established. Some of our results are new even in the linear case $\alpha=2$.

A Singular Semilinear Elliptic Equation with a Variable Exponent

Abstract In this paper we consider singular semilinear elliptic equations with a variable exponent whose model problem is - Δ ⁢ u = f ⁢ ( x ) u γ ⁢ ( x ) in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω . $-\Delta u=\frac{f(x)}{u^{\gamma(x)}}\quad\text{in }\Omega,\qquad u=0\quad\text% {on }\partial\Omega.$ Here Ω is an open bounded set of ℝ N ${\mathbb{R}^{N}}$ , γ ⁢ ( x ) ${\gamma(x)}$ is a positive continuous function and f ⁢ ( x ) ${f(x)}$ is a positive function that belongs to a certain Lebesgue space. We prove that there exists a solution to this problem in the natural energy space H 0 1 ⁢ ( Ω ) ${H^{1}_{0}(\Omega)}$ when γ ⁢ ( x ) ≤ 1 ${\gamma(x)\leq 1}$ in a strip around the boundary. For another case, we prove that the solution belongs to H loc 1 ⁢ ( Ω ) ${H^{1}_{\mathrm{loc}}(\Omega)}$ and that it is zero on the boundary in a suitable sense.

A variable exponent diffusion problem of concave-convex nature

We deal with the problem $$ \begin{cases} -\Delta u = \lambda u^{q(x)} & \text{if } x\in \Omega,\\ u = 0 & \text{if } x\in \p\Omega, \end{cases} $$ where $\Omega\subset \R^N$ is a bounded smooth domain, $\lambda\gt 0$ is a parameter and the exponent $q(x)$ is a continuous positive function that takes values both greater than and less than one in $\overline{\Omega}$. It is therefore a kind of concave-convex problem where the presence of the interphase $q=1$ in $\overline{\Omega}$ poses some new difficulties to be tackled. The results proved in this work are the existence of $\lambda^* \gt 0$ such that no positive solutions are possible for $\lambda \gt \lambda^*$, the existence and structural properties of a branch of minimal solutions, $u_\lambda$, $0 \lt \lambda \lt \lambda^*$, and, finally, the existence for all $ \lambda \in (0,\lambda^*)$ of a second positive solution.

Approximation of a Function and its Derivatives by Entire Functions

Abstract A simple proof is given for the fact that for m a non-negative integer, a function f ∊ C (m)(ℝ), and an arbitrary positive continuous function ∊, there is an entire function g such that |g(i)(x) − f (i)(x)| < ∊(x), for all x ∊ ℝ and for each i = 0, 1 …, m. We also consider the situation where ℝ is replaced by an open interval.

Radial continuous rotation invariant valuations on star bodies

We characterize the positive radial continuous and rotation invariant valuations V defined on the star bodies of Rn as the applications on star bodies which admit an integral representation with respect to the Lebesgue measure. That is,V(K)=∫Sn−1θ(ρK)dm, where θ is a positive continuous function, ρK is the radial function associated to K and m is the Lebesgue measure on Sn−1. As a corollary, we obtain that every such valuation can be uniformly approximated on bounded sets by a linear combination of dual quermassintegrals.

Some counterexamples related to the stationary Kirchhoff equation

In this note we consider the stationary Kirchhoff equation \[ { − M ( ‖ u ‖ 2 ) Δ u = f ( x , u ) a m p ; in Ω , u = 0 a m p ; on ∂ Ω , \left \{ \begin {array}{ll} -M(\| u\|^2) \Delta u = f(x,u) & \hbox {in }\Omega ,\\ \ \ u=0 & \hbox {on }\partial \Omega , \end {array} \right . \] where M M is a continuous positive function and ‖ ⋅ ‖ \| \cdot \| is the standard norm in H 0 1 ( Ω ) H_0^1(\Omega ) . We show that the equation does not enjoy the usual comparison principles (both weak or strong) nor the sub and supersolutions method.

Multiple solutions to a pure supercritical problem for the p-Laplacian

Under some symmetry assumptions, we establish existence and multiplicity of positive and sign changing solutions to the quasilinear problem $$\begin{aligned} -\Delta _{p}v=\left| v\right| ^{q-2}v\quad \text {in }\Omega ,\quad v=0\quad \text {on }\partial \Omega , \end{aligned}$$ in some bounded smooth domains \(\Omega \) in \(\mathbb {R}^{N}\), for \(1 p^{*}:=\frac{Np}{N-p}\). Related to this question is the existence of solutions to the anisotropic critical problem $$\begin{aligned} -\text {div}(a(x)\left| \nabla u\right| ^{p-2}\nabla u)=c(x)\left| u\right| ^{p^{*}-2}u\quad \text {in } \Omega ,\quad u=0\quad \text {on } \partial \Omega , \end{aligned}$$ where a and c are positive continuous functions on \(\overline{\Omega } .\) We study the lack of compactness for this problem in a symmetric setting, and establish existence and multiplicity of positive and sign changing solutions by means of a suitable variational principle.

Ground states of critical and supercritical problems of Brezis–Nirenberg type

We study the existence of symmetric ground states to the supercritical problem $$\begin{aligned} -\Delta v=\lambda v+\left| v\right| ^{p-2}v\;\text {in}\;\Omega ,\quad v=0\;\text {on}\;\partial \Omega , \end{aligned}$$ in a domain of the form $$\begin{aligned} \Omega =\{(y,z)\in \mathbb {R}^{k+1}\times \mathbb {R}^{N-k-1}:\left( \left| y\right| ,z \right) \in \Theta \}, \end{aligned}$$ where \(\Theta \) is a bounded smooth domain such that \(\overline{\Theta } \subset \left( 0,\infty \right) \times \mathbb {R}^{N-k-1}\), \(1\le k\le N-3\), \(\lambda \in \mathbb {R}\), and \(p=\frac{2(N-k)}{N-k-2}\) is the \((k+1)\)-st critical exponent. We show that symmetric ground states exist for \(\lambda \) in some interval to the left of each symmetric eigenvalue and that no symmetric ground states exist in some interval \((-\infty ,\lambda _{*})\) with \(\lambda _{*}>0\) if \(k\ge 2\). Related to this question is the existence of ground states to the anisotropic critical problem $$\begin{aligned} -\text {div}(a(x)\nabla u)=\lambda b(x)u+c(x)\left| u\right| ^{2^{*}-2}u\;\text {in}\; \Theta ,\quad u=0\;\text {on}\; \partial \Theta , \end{aligned}$$ where a, b, c are positive continuous functions on \(\overline{\Theta }\). We give a minimax characterization for the ground states of this problem, study the ground state energy level as a function of \(\lambda \), and obtain a bifurcation result for ground states.

Hilbert and Thompson geometries isometric to infinite-dimensional Banach spaces

We study the horofunction boundaries of Hilbert and Thompson geome-tries, and of Banach spaces, in arbitrary dimension. By comparing the boundaries of these spaces, we show that the only Hilbert and Thompson geometries that are isometric to Banach spaces are the ones defined on the cone of positive continuous functions on a compact space.

Properties of composite positive continuous functions in $\mathbb{C}^n$

The properties of positive continuous functions with $Q^n_{\mathbf{b}}$ and $Q$ are investigated. We prove that some composite functions with $Q$ belong to class $Q^n_{\mathbf{b}}.$ A relation between functions with these classes are established.

The entire large solutions for a quasilinear Schrödinger elliptic equation by the dual approach

In this paper, we consider a Modified Nonlinear Schrödinger equation Δu−Δ(|u|2γ)|u|2γ−2u=q(x)F(u) on Rn(n≥3), where γ>12, q is a nonnegative continuous function on Rn, F is a continuous positive and non-decreasing function on [0,∞). We establish sufficient conditions for the existence and nonexistence of positive entire large solutions for the given equation by using the dual approach.

An inhomogeneous nonlocal diffusion problem with unbounded steps

We consider the following nonlocal equation $$\int J\left(\frac{x-y}{g(y)} \right) \frac{u(y)}{g(y)} dy -u(x)=0\qquad x\in \mathbb{R},$$ where J is an even, compactly supported, Holder continuous kernel with unit integral and g is a continuous positive function. Our main concern will be with unbounded functions g, contrary to previous works. More precisely, we study the influence of the growth of g at infinity on the integrability of positive solutions of this equation, therefore determining the asymptotic behavior as \({t\to +\infty}\) of the solutions to the associated evolution problem in terms of the growth of g.

Elliptic equations with general singular lower order term and measure data

In this paper we study a nonlinear elliptic boundary value problem with a general singular lower order term, whose model is {−Δu=H(u)μinΩ,u=0on∂Ω,u>0onΩ, where Ω is an open bounded subset of RN, μ is a nonnegative bounded Radon measure on Ω and H is a continuous positive function outside the origin such that lims→0+H(s)=+∞. We do not require any monotonicity property on the singular function H.

Existence of solutions for a class of p-Laplacian type equation with critical growth and potential vanishing at infinity

In this paper, we study the existence of positive solution for the following p-Laplacain type equations with critical nonlinearity \begin{equation*} \left\{ \renewcommand{\arraystretch}{1.25} \begin{array}{ll} -\Delta_p u + V （x)|u|^{p-2}u = K(x)f(u)+P(x)|u|^{p^*-2}u, \quad x\in\mathbb{R}^N,\\ u \in \mathcal{D}^{1,p}(\mathbb{R}^N), \end{array} \right. \end{equation*} where $\Delta_p u = div(|\nabla u|^{p-2} \nabla u), 1 < p < N, p^* =\frac {Np}{N-p}$, $V(x)$, $K(x)$ are positive continuous functions which vanish at infinity, $f$ is a function with a subcritical growth, and $P(x)$ is a bounded, nonnegative continuous function. By working in the weighted Sobolev spaces, and using variational method, we prove that the given problem has at least one positive solution.

Extension of the example by Moore-Nehari

Abstract R. Moore and Z. Nehari developed the variational theory for superlinear boundary value problems of the form x'' = p(t) |x|2εx, x(a) = 0 = x(b), where ε &gt; 0 and p(t) is a positive continuous function. They constructed simple example of the equation considered in the interval [0, b] so that the problem had three positive solutions. We show that this example can be extended so that the respective BVP has infinitely many groups of solutions with a presribed number of zeros.

Positive Solutions with Nonpower Asymptotic Behavior and Quasiperiodic Solutions to an Emden–Fowler Type Higher Order Equations

We consider the differential equation y (n) = p 0|y| k sgn y, where p 0 > 0 and 12 ≤ n ≤ 14, and prove that there exists k > 1 such that the equation has positive solutions with nonpower asymptotics y(x) = (x ∗ -x) -α h(ln (x ∗ -x)), x < x ∗ , where h is a nonconstant continuous positive periodic function. For n ≥ 2 we prove that such a solution exists, but with an oscillating periodic function h. Bibliography: 8 titles. Illustrations: 1 figure.