Browsing by Author "del Pino, Manuel"

Now showing 1 - 13 of 13
  • No Thumbnail Available
    Item
    Beyond the Trudinger-Moser supremum
    (2012) del Pino, Manuel; Musso, Monica; Ruf, Bernhard
    Let Omega be a bounded, smooth domain in R-2. We consider the functional
  • No Thumbnail Available
    Item
    Bistable Boundary Reactions in Two Dimensions
    (2011) Davila, Juan; del Pino, Manuel; Musso, Monica
    In a bounded domain Omega subset of R(2) with smooth boundary we consider the problem Delta U =0 in Omega, du/dv = i/epsilon f(u) on d Omega
  • Thumbnail Image
    Item
    Boundary singularities for weak solutions of semilinear elliptic problems
    (ACADEMIC PRESS INC ELSEVIER SCIENCE, 2007) del Pino, Manuel; Musso, Monica; Pacard, Frank
    Let Omega be a bounded domain in R-N, N >= 2, with smooth boundary partial derivative Omega. We construct positive weak solutions of the problem Delta u + u(p) = 0 in Omega, which vanish in a suitable trace sense on partial derivative Omega, but which are singular at prescribed isolated points if p is equal or slightly above N+1/N-1. Similar constructions are carried out for solutions which are singular at any given embedded submanifold of partial derivative Omega of dimension k epsilon [0, N -2], if p equals or it is slightly above N-k-1/N-k-1, and even on countable families of these objects, dense on a given closed set. The role of the exponent N+1/N-1 (first discovered by Brezis and Turner [H. Brezis, R. Turner, N-1 On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977) 601-614]) for boundary regularity, parallels that of N/N-2 for interior singularities. (c) 2007 Elsevier Inc. All rights reserved.
  • No Thumbnail Available
    Item
    Fast and slow decay solutions for supercritical elliptic problems in exterior domains
    (2008) Davila, Juan; del Pino, Manuel; Musso, Monica; Wei, Juncheng
    We consider the elliptic problem Delta u + u(p) = 0, u > 0 in an exterior domain, Omega = R(N)\D under zero Dirichlet and vanishing conditions, where D is smooth and bounded in R(N), N >= 3, and p is supercritical, namely p > N+2/N-2. We prove that this problem has infinitely many solutions with slow decay O(vertical bar x vertical bar(-2/p-1)) at infinity. In addition, a solution with fast decay O(vertical bar x vertical bar(2- N)) exists if p is close enough from above to the critical exponent.
  • No Thumbnail Available
    Item
    Interior bubbling solutions for the critical Lin-Ni-Takagi problem in dimension 3
    (2019) del Pino, Manuel; Musso, Monica; Roman, Carlos; Wei, Juncheng
    We consider the problem of finding positive solutions of the problem u -.u + u5 = 0 in a bounded, smooth domain in R3, under zero Neumann boundary conditions. Here. is a positive number. We analyze the role of Green's function of - +. in the presence of solutions exhibiting single bubbling behavior at one point of the domain when. is regarded as a parameter. As a special case of our results, we find and characterize a positive value.* such that if. -. * > 0 is sufficiently small, then this problem is solvable by a solution u. which blows-up by bubbling at a certain interior point of lambda down arrow lambda(*).
  • Thumbnail Image
    Item
    Large energy entire solutions for the Yamabe equation
    (ACADEMIC PRESS INC ELSEVIER SCIENCE, 2011) del Pino, Manuel; Musso, Monica; Pacard, Frank; Pistoia, Angela
    We consider the Yamabe equation Delta u + n(n-2_/4 vertical bar u vertical bar 4/n-2 u = 0 in R(n), n >= 3. Let k >= 1 and xi(k)(j) = (e(2j pi u/k), 0) is an element of R(n) = C x R(n-2). For all large k we find a solution of the form u(k)(x)= u(x) - Sigma(k)(j=1) mu(k) (-n-2/2) U X (mu(-1)(k) (x - xi(j)) +o(1), where U(x) = (2/1+vertical bar x vertical bar(2)) (n-2/2), mu(k) = c(n)/k(2) for n >= 4, mu k = c/k(2)(logk)(2) for n =3 and o(1) -> 0 uniformly as k -> +infinity. (C) 2011 Elsevier Inc. All rights reserved.
  • Thumbnail Image
    Item
    New solutions for Trudinger-Moser critical equations in R-2
    (ACADEMIC PRESS INC ELSEVIER SCIENCE, 2010) del Pino, Manuel; Musso, Monica; Ruf, Bernhard
    Let Omega be a bounded, smooth domain in R-2. We consider critical points of the Trudinger-Moser type functional J(lambda) (u) = 1/2 integral(Omega)vertical bar del u vertical bar(2) - lambda/2 integral(Omega)e(u2) in H-0(1)(Omega), namely solutions of the boundary value problem Delta u + lambda ue(u2) = 0 with homogeneous Dirichlet boundary conditions, where lambda > 0 is a small parameter. Given k >= 1 we find conditions under which there exists a solution u(lambda) which blows up at exactly k points in Omega as lambda -> 0 and J(lambda)(u(lambda)) -> 2k pi. We find that at least one such solution always exists if k = 2 and Omega is not simply connected. If Omega has d >= 1 holes, in addition d + 1 bubbling solutions with k = 1 exist. These results are existence counterparts of one by Druet in [O. Druet, Multibump analysis in dimension 2: Quantification of blow-up levels, Duke Math. J. 132 (2) (2006) 217-269] which classifies asymptotic bounded energy levels of blow-up solutions for a class of nonlinearities of critical exponential growth, including this one as a prototype case. (C) 2009 Elsevier Inc. All rights reserved.
  • Thumbnail Image
    Item
    NONDEGENERACY OF ENTIRE SOLUTIONS OF A SINGULAR LIOUVILLLE EQUATION
    (AMER MATHEMATICAL SOC, 2012) del Pino, Manuel; Esposito, Pierpaolo; Musso, Monica
    We establish nondegeneracy of the explicit family of finite mass solutions of the Liouvillle equation with a singular source of integer multiplicity, in the sense that all bounded elements in the kernel of the linearization correspond to variations along the parameters of the family.
  • No Thumbnail Available
    Item
    Solutions of the Allen-Cahn equation which are invariant under screw-motion
    (2012) del Pino, Manuel; Musso, Monica; Pacard, Frank
    We study entire solutions of the Allen-Cahn equation which are defined in the 3-dimensional Euclidean space and which are invariant under screw-motion. In particular, we discuss the existence and non existence of nontrivial solutions whose nodal set is a helicoid of .
  • Thumbnail Image
    Item
    Standing waves for supercritical nonlinear Schrodinger equations
    (ACADEMIC PRESS INC ELSEVIER SCIENCE, 2007) Davila, Juan; del Pino, Manuel; Musso, Monica; Wei, Juncheng
    Let V (x) be a non-negative, bounded potential in R-N, N >= 3 and p supercritical, p > N+2/N-2. We look for positive solutions of the standing-wave nonlinear Schrodinger equation Delta u - V(x)u + u(P) = 0 in R-N, with u(x) -> 0 as vertical bar x vertical bar -> +infinity. We prove that if V(x) = 0(vertical bar x vertical bar(-2)) as vertical bar x vertical bar -> +infinity, then for N >= 4 and p > N+1/N-3 this problem admits a continuum of solutions. If in addition we have, for instance, V (x) = 0 (vertical bar x vertical bar-mu) with mu > N, then this result still holds provided that N >= 3 and p > N+2/N-2. Other conditions for solvability, involving behavior of V at infinity, are also provided. (C) 2007 Elsevier Inc. All rights reserved.
  • Thumbnail Image
    Item
    The supercritical Lane-Emden-Fowler equation in exterior domains
    (TAYLOR & FRANCIS INC, 2007) Davila, Juan; del Pino, Manuel; Muss, Monica
    We consider the exterior problem
  • No Thumbnail Available
    Item
    Torus action on Sn and sign-changing solutions for conformally invariant equations
    (2013) del Pino, Manuel; Musso, Monica; Pacard, Frank; Pistoia, Angela
    We construct sequences of sign-changing solutions for some conformany invariant semilinear elliptic equation which is defined S-n, when n >= 4. The solutions we obtain have large energy and concentrate along some special submanifolds of S-n. For example, for n >= 4 we obtain sequences of solutions whose energy concentrates along one great circle or finitely many great circles which are linked to each other (and they correspond to Hopf links embedded in S-3 x {0) subset of S-n). In dimension n >= 5 we obtain sequences of solutions whose energy concentrates along a two-dimensional torus (which corresponds to a Clifford torus embedded in S-3 x {0} subset of S-n).
  • Thumbnail Image
    Item
    TWO-DIMENSIONAL EULER FLOWS WITH CONCENTRATED VORTICITIES
    (AMER MATHEMATICAL SOC, 2010) del Pino, Manuel; Esposito, Pierpaolo; Musso, Monica
    For a planar model of Euler flows proposed by Tur and Yanovsky (2004), we construct a family of velocity fields w(e) for it fluid in a bounded region Omega, with concentrated vorticities w(e) for epsilon > 0 small. More precisely, given a positive integer a and a sufficiently small complex number a, we find a family of stream functions psi(epsilon) which solve the Liouville equation with Dirac mass source,