Abstract:
We establish a 3G-theorem for the iterated Green function of (−∆)pm, (p≥1,m≥1), on the unit ball B of ℝn(n≥1) with boundary conditions (∂/∂ν)j(−∆)kmu=0 on ∂B, for 0≤k≤p−1 and 0≤j≤m−1. We exploit thisresult to study a class of potentials 𝒥m,n(p). Next, we aim at proving the existence of positive continuoussolutions for the following polyharmonic nonlinear problems (−∆)pmu=h(‧,u), in D (in the sense of distributions), lim|x|→1((−∆)kmu(x)/(1−|x|)m−1)=0, for 0≤k≤p−1, where D=B or B\{0} and h is a Borel measurablefunction on D×(0,∞) satisfying some appropriateconditions related to 𝒥m,n(p).

Abstract:
We establish a 3 G -theorem for the iterated Green function of ( ) p m , ( p ≥ 1 , m ≥ 1 ), on the unit ball B of n ( n ≥ 1 ) with boundary conditions ( / ν ) j ( ) k m u = 0 on B , for 0 ≤ k ≤ p 1 and 0 ≤ j ≤ m 1 . We exploit this result to study a class of potentials m , n ( p ) . Next, we aim at proving the existence of positive continuous solutions for the following polyharmonic nonlinear problems ( ) p m u = h ( , u ) , in D (in the sense of distributions), lim | x | → 1 ( ( ) k m u ( x ) / ( 1 | x | ) m 1 ) = 0 , for 0 ≤ k ≤ p 1 , where D = B or B { 0 } and h is a Borel measurable function on D × ( 0 , ∞ ) satisfying some appropriate conditions related to m , n ( p ) .

Abstract:
We present existence result for the polyharmonic nonlinear problem $$displaylines{ (-Delta )^{pm} u=varphi (.,u)+psi (.,u),quad hbox{in }B cr u>0,quad hbox{in }B cr lim_{|x|o 1} frac{(-Delta )^{jm}u(x)}{(1-|x|)^{m-1}}=0, quad 0leq jleq p-1, }$$ in the sense of distributions. Here $m,p$ are positive integers, $B$ is the unit ball in $mathbb{R}^{n}(ngeq 2)$ and the nonlinearity is a sum of a singular and sublinear terms satisfying some appropriate conditions related to a polyharmonic Kato class of functions $mathcal{J}_{m,n}^{(p)}$.

Abstract:
We prove the existence of positive continuous solutions to the nonlinear fractional system $$displaylines{ (-Delta|_D) ^{alpha/2}u+lambda g(.,v) =0, cr (-Delta|_D) ^{alpha/2}v+mu f(.,u) =0, }$$ in a bounded $C^{1,1}$-domain $D$ in $mathbb{R}^n$ $(ngeq 3)$, subject to Dirichlet conditions, where $0

Abstract:
We aim at giving a rich class of quasi-metrics from which we obtain as an application an interesting inequality for the Greens function of the fractional Laplacian in a smooth domain in .

Abstract:
We study the existence and the asymptotic behaviour of positive solutions of the nonlinear equation $Delta u+f(.,u)=0$, in the domain $D={(x_1,x_2)in mathbb{R}^2:x_2>0}$, with $u=0$ on the boundary. The aim is to prove some existence results for the above equation in a general setting by using a fixed-point argument.

Abstract:
This paper concerns nonlinear elliptic equations in the half space $mathbb{R}_{+}^{n}:={ x=(x',x_{n})in mathbb{R}^{n}:x_{n}$ greater than 0 $}$, $ngeq 2$, with a nonlinear term satisfying some conditions related to a certain Kato class of functions. We prove some existence results and asymptotic behaviour for positive solutions using a potential theory approach.

Abstract:
We aim at giving a rich class of quasi-metrics from which we obtain as an application an interesting inequality for the Greens function of the fractional Laplacian in a smooth domain in n.

Abstract:
We establish a new form of the 3G theorem for polyharmonic Green function on the unit ball of ℝn(n≥2) corresponding to zero Dirichlet boundary conditions. This enables us to introduce a new class of functions Km,n containing properly the classical Kato class Kn. We exploit properties of functions belonging to Km,n to prove an infinite existence result of singular positive solutions for nonlinear elliptic equation of order 2m.

Abstract:
This paper discusses the dilemma facing the art teacher as a liberal educator. The author first reviews the evolution of liberal education from ancient times, through the Renaissance to modern times and discusses, through an extensive bibliography, ancient and modern theories which have impacted on the concept of “liberalism,” examining notions such as tolerance, individualism and autonomy which constitute the pillars of liberalism. The author discusses the contributions of philosophers and sociologists such as Thomas Hobbes, John Stuart Mill, Isaiah Berlin, and Will Kymlicka and then examines the two main approaches to liberalism: philosophical liberalism and political liberalism. The different emphases placed by these two approaches, on the individual and the group respectively, form the basis of the dilemma which faces the art teacher as a liberal educator. In order to understand the dilemma, the author draws a parallel between the two approaches and the role of the traditional art studio master and art educator. The goals of the studio master, who is devoted to the development of the individual, accord with philosophical liberalism, while those of the art educator, who is obliged to adhere to the demands of a school system, accord with political liberalism, which stresses equality for all. The ideal, says the author, resides in an amalgam of the two approaches and is symbolized in the term “artist-teacher” but she asks whether it is possible to truly merge the two approaches, at the same level, in the teaching process.