In this paper we consider the problem −∆uAssuming that the potential a(x) is a regular function such that lim inf |x|→+∞ a(x) = a∞ > 0 and that verifies suitable decay assumptions, but not requiring any symmetry property on it, we prove that the problem has infinitely many solutions.

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We consider the problem -Δu = |u|2*-2u + λu in Ω, u = 0 on ∂Ω, where Ω is an open regular subset of ℝN (N ≥ 3), [Formula: see text] is the critical Sobolev exponent and λ is a constant in ]0, λ1[ where λ1 is the first eigenvalue of -Δ. In this paper we show that, when N ≥ 4, the problem has at least [Formula: see text] (pairs of) solutions, improving a result obtained in [4] for N ≥ 6.

In this paper we follow the approach in Maddalena et al. (Interfaces and 1 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.

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