We show the existence of infinitely many positive solutions \(u \in H^1(\mathbb {R}^2)\) to the equation \(-\Delta u+a(x)u=u^p\), with \(p>1\) , without asking, on the positive potential \(a(x)\), any symmetry assumption as in Wei and Yan (Calc Var Partial Differ Equ 37, 423–439, 2010) or Devillanova and Solimini (Adv Nonlinear Studies 12, 173–186, 2012) or small oscillation assumption as in Cerami et al. (Commun Pure Appl Math, doi: 10.1002/cpa.21410, 2012) and in Weiwei and Wei (Infinitely many positive solutions for Nonlinear equations with non-symmetric Potential, 2012).

In this paper the problem of determining if a given-measure is irrigable, in the sense of [4], or not is addressed. A notion of irrigability dimension of a measure is given and lower and upper bounds are proved in terms of the minimal Hausdorff and respectively Minkowski dimension of a set on which the measure is concentrated. A notion of resolution dimension of a measure based on its discrete approximations is also introduced and its relation with the irrigation dimension is studied.

In this paper we follow the approach in Maddalena et al. (Interfaces and Free Boundaries 5, 391–415, 2003) to the study of the ramified structures and we identify some geometrical properties enjoyed by optimal irrigation patterns. These properties are “elementary” in the sense that they are not concerned with the regularity at the ending points of such structures, where the presumable selfsimilarity properties should take place. This preliminary study already finds an application in G. Devillanova and S. Solimini (Math. J. Univ. Padua, to appear), where it is used in order to discuss the irrigability of a given measure.

In this paper we consider the problem \(-\Delta u + a(x)u = \vert u\vert ^{p-2}u\) in \(\mathbb{R}^N\), where p > 2 and \(p < 2^* = \frac{2N}{N-2} \) if N > 2. Assuming that the potential a(x) is a regular function such that \(\liminf_{\vert x\vert\rightarrow + \infty} a(x) = a_\infty > 0\) and that verifies suitable decay assumptions, but not requiring any symmetry property on it, we prove that the problem has infinitely many solutions.

Irrigation and draining systems, plants and trees together with their root systems, lungs and cardiovascular systems have a common morphology which seems to derive from topological constraints together with energy saving requirements. All of these systems look like spatial trees and succeed in spreading out a fluid from a source onto a volume. The associated morphology is a tree of bifurcating vessels. Their intuitive explanation is that transport energy is saved by using broad vessels as long as possible rather than thin spread out vessels. In this paper, we define a general formalism dealing with irrigation patterns. Related to martingale theory, this formalism permits one to define irrigation trees and their vessels, to give a generic form to their energy, and to show compactness for the irrigation patterns with bounded energy as well as a lower semicontinuity result for the cost functional. As a consequence, we show that a variety of source to volume irrigation problems are well posed.

In this paper, we consider the problem $-\Delta u =|u| ^{2^*-2}u+\lambda u$ in $\Omega$, $u= 0$ on $\partial \Omega$, where $\Omega$ is an open regular bounded subset of $\mathbb R^N$ $(N\geq 3)$, $2^*=\frac{2N}{N-2}$ is the critical Sobolev exponent and $\lambda>0$. Our main result asserts that, if $N\geq 7$, the problem has infinitely many solutions and, from the point of view of the compactness arguments employed here, the restriction on the dimension $N$ cannot be weakened.

The paper deals with the problem of minimizing a free discontinuity functional under Dirichlet boundary conditions. An existence result was known so far for C-1 (partial derivative Omega) boundary data u. We show here that the same result holds for (u) over cap epsilon C-0.u(partial derivative Omega) if it > 1/2 and it cannot be extended to cover the case mu = 1/2 The proof is based on some geometric measure theoretic properties, in part introduced here, which are proved a priori to hold for all the possible minimizers. (C) 2001 editions scientifiques et medicales Elsevier SAS.

Some properties of ''concentration-compactness type'' are proved to the aim of characterizing the behaviour of bounded sequences of functions in a Sobolev Space with respect to Lorentz norms. Such properties are shown to exist as far as the embedding is not optimal with respect to the secundary index.

Let g be a real or vectorial, bounded and measurable function defined on a smooth open set-OMEGA of R(N). We consider the problem of finding a "segmentation" of g obtained by minimizing an energy functional of the kind [GRAPHICS] where (O(i))i-epsilon-I is a Borel partition of OMEGA, P (O(i), OMEGA) denotes the perimeter in OMEGA of each set of the partition, and u is constant (and therefore equal to the mean value of g) on each set O(i) of the partition. We give an elementary proof of the fact that the boundaries of the regions of an optimal segmentation have a local density which is bounded from below by an explicit constant.

We are concerned with the problem of searching for T-periodic solutions, T prescribed, for the equation -x=⊇V(x), where x(t) ∈ C 2 (R,R n ) and V ∈ C 1 (R/{0},R), in the case when V is «singular», i.e. V behaves like -k/|x|α near the origin

On considere le probleme −Δu=λ k u+g(u) dans Ω, u=0 sur δΩ, ou Ω est un domaine borne lisse dans R m et λ 0 <Λ 1 ≤λ 2 ≤… est la suite des valeurs propres de −Δ repetees autant de fois que leur multiplicite

Wc give a constructive proob that given a bounded Iunction on a rectangle R, the minimum of the following bunctional is achieved:where B is a finite set ob C' curves lii R and u is locally constant in R -B.In image processing, g can be interpreted as an image (g(x,y) is the grey level a! (x,y)) and the curves ob B as the contours of the image, u representing the mean value inside each contour.Following ideas ob Mumbord and Shah, our proob suggests a method br transborming an image into a cartoon.

On considere le probleme −Δu=g(u)+h dans Ω, u=0 sur ∂Ω ou Ω⊂R N est un ouvert borne a frontiere lisse ∂Ω en se restreignant au cas −u″=u 2 −t sin x dans (0,π), u(0)=u(π)=0. Etant donne N 0 ∈N il existe t Nd0 tel que pour tout t≥t Nd0 l'equation consideree a au moins N 0 solutions