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Persistence of Coron’s solution in nearly critical problems

Monica MussoAngela Pistoia — 2007

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider the problem - Δ u = u N + 2 N - 2 + λ in Ω ε ω , u > 0 in Ω ε ω , u = 0 on Ω ε ω , where Ω and ω are smooth bounded domains in N , N 3 , ε > 0 and λ . We prove that if the size of the hole ε goes to zero and if, simultaneously, the parameter λ goes to zero at the appropriate rate, then the problem has a solution which blows up at the origin.

Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation

Monica MussoFrank PacardJuncheng Wei — 2012

Journal of the European Mathematical Society

We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations Δ u - u + f ( u ) = 0 in N , u H 1 ( N ) , where N 2 . Under natural conditions on the nonlinearity f , we prove the existence of 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒𝑙𝑦𝑚𝑎𝑛𝑦𝑛𝑜𝑛𝑟𝑎𝑑𝑖𝑎𝑙𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠 in any dimension N 2 . Our result complements earlier works of Bartsch and Willem ( N = 4 𝚘𝚛 N 6 ) and Lorca-Ubilla ( N = 5 ) where solutions invariant under the action of O ( 2 ) × O ( N - 2 ) are constructed. In contrast, the solutions we construct are invariant under the action of D k × O ( N - 2 ) where D k O ( 2 ) denotes the dihedral group...

Morse index and bifurcation of -geodesics on semi Riemannian manifolds

Monica MussoJacobo PejsachowiczAlessandro Portaluri — 2007

ESAIM: Control, Optimisation and Calculus of Variations

Given a one-parameter family { g λ : λ [ a , b ] } of semi Riemannian metrics on an -dimensional manifold , a family of time-dependent potentials { V λ : λ [ a , b ] } and a family { σ λ : λ [ a , b ] } of trajectories connecting two points of the mechanical system defined by ( g λ , V λ ) , we show that there are trajectories bifurcating from the trivial branch σ λ if the generalized Morse indices μ ( σ a ) and μ ( σ b ) are different. If the data are analytic we obtain estimates for the number of bifurcation points on the branch and, in particular, for the number of strictly conjugate...

Bubbling on boundary submanifolds for the Lin–Ni–Takagi problem at higher critical exponents

Manuel del PinoFethi MahmudiMonica Musso — 2014

Journal of the European Mathematical Society

Let Ω be a bounded domain in n with smooth boundary Ω . We consider the equation d 2 Δ u - u + u n - k + 2 n - k - 2 = 0 in Ω , under zero Neumann boundary conditions, where Ω is open, smooth and bounded and d is a small positive parameter. We assume that there is a k -dimensional closed, embedded minimal submanifold K of Ω , which is non-degenerate, and certain weighted average of sectional curvatures of Ω is positive along K . Then we prove the existence of a sequence d = d j 0 and a positive solution u d such that d 2 | u d | 2 S δ K as d 0 in the sense of measures, where δ K ...

Bubbling along boundary geodesics near the second critical exponent

Manuel del PinoMonica MussoFrank Pacard — 2010

Journal of the European Mathematical Society

The role of the second critical exponent p = ( n + 1 ) / ( n - 3 ) , the Sobolev critical exponent in one dimension less, is investigated for the classical Lane–Emden–Fowler problem Δ u + u p = 0 , u > 0 under zero Dirichlet boundary conditions, in a domain Ω in n with bounded, smooth boundary. Given Γ , a geodesic of the boundary with negative inner normal curvature we find that for p = ( n + 1 ) / ( n - 3 - ε ) , there exists a solution u ε such that | u ε | 2 converges weakly to a Dirac measure on Γ as ε 0 + , provided that Γ is nondegenerate in the sense of second variations of...

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