In this paper the equation $ -Delta u+a(x)u=|u|^{p-1}u mbox{ in }R^N$ is considered, when $N ge2$, $p>1, p<{frac{N+2}{N-2}},$ if $Nge 3.$ Assuming that the potential $a(x)$ is a positive function belonging to $L^{N/2}_ {loc}(R^N),$ such that $a(x) o a_infty > 0, mbox{as} |x| ightarrow infty$, and that satisfies slow decay assumptions, but not requiring any symmetry property, the existence of infinitely many positive solutions, by purely variational methods, is proved. The shape of the solutions is described and, furthermore, their asymptotic behavior when $|a(x) - a_infty|_ {L^ {N/2}_ {loc}(R^N)} o 0$.

In this paper we consider the equation {equation presented}. During last thirty years the question of the existence and multiplicity of solutions to (E) has been widely investigated mostly under symmetry assumptions on a. The aim of this paper is to show that, differently from those found under symmetry assumption, the solutions found in [6] admit a limit configuration and so (E) also admits a positive solution of infinite energy having infinitely many "bumps".

Abstract We show the variational structure of a multiplicity result of positive solutions u is an element of H(1) (R(N)) to the equation -Delta u + a(x)u = u(p), where N >= 2, p > 1 with p < 2* - 1 = N+2/N-2 if N >= 3 and the potential a(x) is a positive function enjoying a planar symmetry. We require suitable decay assumptions which are widely implied by those in [6], in which Wei and Yan have obtained an analogous multiplicity result by using different techniques.

We show the existence of infinitely many positive solutions u ∈ H1(R2) 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 inWei 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) 6 and in Weiwei and Wei (Infinitely many positive solutions for Nonlinear equations with non-symmetric Potential, 2012).

In this paper we complete our work started in [31], where the present paper was announced; in order to set a unified theory of the irrigation problem. The main result of the paper is the equivalence of the various formulations introduced so far as well as a new one introduced here. To this aim we introduce several geometric and analytical concepts which are essential for reaching our final goal even if they may deserve an intrinsic interest in themselves.

Dans cet article, le problème de la régularité, c'est-à-dire du comportement fractal , des minima du problème de transport branché est considéré. On montre que, dans des conditions appropriées sur la mesure irriguée, les minima présentent une régularité fractale, à savoir sur une branche de longueur l le nombre de branches de bifurcation de celle-ci dont la longueur est comparable à ε peut être estimé à la fois supérieurement et inférieurement en fonction de l/ε.

We study the H¨older regularity of the landscape function introduced by Santambrogio in [S]. We develop a new technique which both extends Santambrogio’s result to lower Ahlfors regular measures in general dimension h and simplifies its proof.