We compare several definitions of weak solutions to second order partial differential equations with nonnegative characteristic form.

On étudie la classification des solutions du problème elliptique$${\left({\left|u^{\prime }\right|}^{p-2}{u}^{\prime }\right)}^{\prime }\left(t\right)+{\left({\left|u\right|}^{q-1}u\right)}^{\prime }\left(t\right)-f\left(t\right){\left|u\right|}^{m-1}u\left(t\right)=0,t\>0,$$où $q\>1,p\ge m+1\>2$et $f$ une fonction changeant de signe. En utilisant une méthode de tire, On montre qu’en partant avec une dérivée initiale nulle toutes les solutions sont globales. De plus si $p\>m+1$ et $q\>(p-1)(m+1)/p$ l’ensemble des solutions est constitué d’une seule solution à support compact et de deux familles de solutions ; celles qui sont strictement positives et celles qui changent de signes. On montre aussi que ces deux familles tendent vers l’infini quand...

Consider a (1,1) tensor field J, defined on a real or complex m-dimensional manifold M, whose Nijenhuis torsion vanishes. Suppose that for each point p ∈ M there exist functions $f_1,...,f_m$, defined around p, such that $(d{f}_1\wedge ...\wedge d{f}_m)\left(p\right)\ne 0$ and $d\left(d{f}_j\left(J\left(\right)\right)\right)\left(p\right)=0$, j = 1,...,m. Then there exists a dense open set such that we can find coordinates, around each of its points, on which J is written with affine coefficients. This result is obtained by associating to J a bihamiltonian structure on T*M.

This paper deals with the classification of hyperbolic Monge-Ampère equations on a two-dimensional manifold. We solve the local equivalence problem with respect to the contact transformation group assuming that the equation is of general position nondegenerate type. As an application we formulate a new method of finding symmetries. This together with previous author's results allows to state the solution of the classical S. Lie equivalence problem for the Monge-Ampère equations.

We give a structure result for the positive radial solutions of the following equation: $${\Delta }_{p}u+K\left(r\right)u{\left|u\right|}^{q-1}=0$$ with some monotonicity assumptions on the positive function $K\left(r\right)$. Here $r=\left|x\right|$, $x\in {ℝ}^n$; we consider the case when $n>p>1$, and $q>p_{*}=\frac{n(p-1)}{n-p}$. We continue the discussion started by Kawano et al. in [KYY], refining the estimates on the asymptotic behavior of Ground States with slow decay and we state the existence of S.G.S., giving also for them estimates on the asymptotic behavior, both as $r\to 0$ and as $r\to \infty$. We make use of a Emden-Fowler transform...

This paper is devoted to the study of cloaking via anomalous localized resonance (CALR) in the two- and three-dimensional quasistatic regimes. CALR associated with negative index materials was discovered by Milton and Nicorovici [21] for constant plasmonic structures in the two-dimensional quasistatic regime. Two key features of this phenomenon are the localized resonance, i.e., the fields blow up in some regions and remain bounded in some others, and the connection between the localized resonance...

We present a new continuous selection theorem, which unifies in some sense two well known selection theorems; namely we prove that if F is an H-upper semicontinuous multivalued map on a separable metric space X, G is a lower semicontinuous multivalued map on X, both F and G take nonconvex $L_p(T,E)$-decomposable closed values, the measure space T with a σ-finite measure μ is nonatomic, 1 ≤ p < ∞, $L_p(T,E)$ is the Bochner-Lebesgue space of functions defined on T with values in a Banach space E, F(x) ∩ G(x) ≠ ∅...