The main purpose of this paper is to construct families of positive solutions for the equationon ∂Ω which blow-up and concentrate in k ≥ 1 different points of Ω as ε goes to 0. We exhibit some examples of contractible domains where a large number of solutions exists.

We consider the boundary value problem Δu+u^p=0 in a bounded, smooth domain Ω in R^2 with homogeneous Dirichlet boundary condition and p a large exponent. We find topological conditions on Ω which ensure the existence of a positive solution up concentrating at exactly m points as p → ∞. In particular, for a non simply connected domain such a solution exists for any given m\geq 1

Abstract. We consider the sinh-Poisson equationwhere Ω is a smooth bounded domain in R 2 and λ is a small positive parameter.If 0 ∈ Ω and Ω is symmetric with respect to the origin, for any integer k if λ is small enough, we construct a family of solutions to (P ) λ which blows-up at the origin whose positive mass is 4πk(k − 1) and negative mass is 4πk(k + 1).It gives a complete answer to an open problem formulated by Jost-Wang-Ye-Zhou in [Calc. Var. PDE (2008) 31: 263-276].

In conformal geometry, the Compactness Conjecture asserts that the set of Yamabe metrics on a smooth, compact Riemannian manifold (M,g) is compact. Established in the locally conformally flat case [41,42] and for n\leq 24 [23], it has revealed to be generally false for n\geq 25 [8,9]. A stronger version of it, the compactness under perturbations of the Yamabe equation, is addressed here with respect to the linear geometric potential n-2/4(n-1)Scal_g, Scal_g being the Scalar curvature of (M,g). Even tough the Yamabe equation is compact in some cases, surprisingly we show that a-priori L^\infty-bounds fail on all manifolds with n\geq 4 as well as H_1^2-bounds do in the locally conformally flat case when n\geq 7. In several situations, the results are optimal

We study a perturbed semilinear problem with Neumann boundary conditionwhere Ω is a bounded smooth domain of R N , N ≥ 2, ε > 0, 1 < p < N +2

We prove the existence of equilibria of the N-vortex Hamiltonian in a bounded domain Omega subset of R(2), which is not necessarily simply connected. On an arbitrary bounded domain we obtain new equilibria for N = 3 or N = 4. If Omega has an axial symmetry we obtain a symmetric equilibrium for each N is an element of N. We also obtain new stream functions solving the sinh-Poisson equation -Delta psi = rho sinh psi in Omega with Dirichlet boundary conditions for rho > 0 small. The stream function psi(rho) induces a stationary velocity field v(rho) solving the Euler equation in Omega. On an arbitrary bounded domain we obtain velocitiy fields having three or four counter-rotating vortices. If Omega has an axial symmetry we obtain for each N a velocity field v(rho) that has a chain of N counter-rotating vortices, analogous to the Mallier-Maslowe row of counter-rotating vortices in the plane. Our methods also yield new nodal solutions for other semilinear Dirichlet problems, in particular for the Lane-Emden-Fowler equation -Delta u = vertical bar u vertical bar(p-1)u in Omega with p large

Given (M, g) a smooth, compact Riemannian n-manifold, we consider equations like ∆ g u + hu = u 2 * −1−ε , where h is a C 1-function on M , the exponent 2 * = 2n/ (n − 2) is critical from the Sobolev viewpoint, and ε is a small real parameter such that ε → 0. We prove the existence of blowing-up families of positive solutions in the subcritical and supercritical case when the graph of h is distinct at some point from the graph of n−2 4(n−1) Scal g .

We establish the existence of a positive fully nontrivial solution (u, v) to the weakly coupled elliptic systemwhere N ≥ 4, 2 * := 2N N−2 is the critical Sobolev exponent, α, β ∈ (1, 2], α + β = 2 * , µ 1 , µ 2 > 0, and λ < 0. We show that these solutions exhibit phase separation as λ → −∞, and we give a precise description of their limit domains.If µ 1 = µ 2 and α = β, we prove that the system has infinitely many fully nontrivial solutions, which are not conformally equivalent.Pitaevskii equations. This type of systems arises, e.g., in the Hartree-Fock theory for double condensates, that is, Bose-Einstein condensates of two different hyperfine states which overlap in space; see [9]. The sign of µ i reflects the interaction of the particles within each single state. If µ i is positive, this interaction is attractive. The sign of λ, on the other hand, reflects the interaction of particles in different states. This interaction is attractive if λ > 0 and it is repulsive if λ < 0. If the condensates Date: September 29, 2018. M. Clapp was partially supported by CONACYT grant 237661 (Mexico) and UNAM-DGAPA-PAPIIT grant IN104315 (Mexico). A. Pistoia was partially supported by Fondi di Ateneo "Sapienza" Universitá di Roma (Italy).1 2 * 2 for all λ > 0 if N ≥ 5 and for a wide range of λ > 0 if N = 4; see [5,6]. Peng, Peng and Wang [20] studied the system for µ 1 = µ 2 = 1, λ = 1 2 * and different values of α and β, and they obtained uniqueness and nondegeneracy results for positive synchronized solutions. Guo, Li and Wei studied the critical system (1.1) in dimension N = 3 for λ < 0 and they established the existence of positive solutions with k peaks for k sufficiently large in [12]. In [10,11] Gladiali, Grossi and Troestler obtained radial and nonradial solutions to some critical systems using bifurcation methods.Here we focus our attention to the competitive case, i.e., to λ < 0. In this case, the system (1.1) does not have a least energy fully nontrivial solution; see Proposition 2.2 below. This behavior showcases the lack of compactness of the variational functional, which comes from the fact that system is invariant under