"Bubble-tower" radial solutions in the slightly supercritical Brezis–Nirenberg problem

By Manuel del Pino, Jean Dolbeault, Monica Musso. 2003

89 citations recorded by the source corpus.

Abstract In this paper, we consider the Brezis–Nirenberg problem in dimension N⩾4, in the supercritical case. We prove that if the exponent gets close to N+2 N−2 and if, simultaneously, the bifurcation parameter tends to zero at the appropriate rate, then there are radial solutions which behave like a superposition of bubbles, namely solutions of the form γ ∑ j=1 k 1 1+M j 4 N−2 |y| 2 (N−2)/2 M j (1+o(1)), γ=(N(N−2)) (N−2)/4 , where Mj→+∞ and Mj=o(Mj+1) for all j. These solutions lie close to turning points “to the right” of the associated bifurcation diagram.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9ce36c504ad977a8272c40e30aa58f1d7aba3bdc

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In Applied Mathematics

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Citations in the corpus

Multiple Blow-Up Phenomena for the Sinh-Poisson Equation (2012, background): “In particular, towers of positive bubbles were found by Del Pino-DolbeaultMusso [6, 7], Ge-Jing-Pacard [15] and Del Pino-Musso-Pistoia [12], while towers of signchanging bubbles were built by Pistoia-Weth [26], Musso-Pistoia [23, 24] and Ge-MussoPistoia [16].”
Singular limits in Liouville-type equations (2005, methodology): “L∞-weighted spaces have been used in [17,18] to detect bubbling from above the critical exponent in higher dimensional problems improving the method in [1,32], both in lifting criticality (or subcriticality) required there, and non-degeneracy of critical points of the analogue of φm in that context.”
THE BREZIS-NIRENBERG PROBLEM NEAR CRITICALITY IN DIMENSION 3 ? (2004, methodology): “On the other hand, an oscillatin behavior has been observed from a variational point of vein in [11] and with ODE in [12].”
The Toda System and Clustering Interfaces in the Allen–Cahn equation (2008, background): “In the one-dimensional or radial cases, concentration in the form of clusters of transition layers or spikes has already been observed in various problems in the literature, see [4,10,11,13,27,28,32,33], phenomena in accordance with higherdimensional multi-spike clustering as predicted in [20].”
Sign Changing Tower of Bubbles for an Elliptic Problem at the Critical Exponent in Pierced Non-Symmetric Domains (2010, background): “The first contribution in this direction is due to [17], where the authors generalize the construction of tower of bubbles for the slightly super critical Brezis–Nirenberg problem obtained in [14] for a general non-symmetric domain.”
Geometry of phase space and solutions of semilinear elliptic equations in a ball (2007, background): “for p − N+2 N−2 > 0, small [10, 12], which also has an ODE interpretation [11].”
Sharp quantitative estimates of Struwe’s decomposition (2021, background): “[10], Musso and Pistoia [20], Pistoia and Vétois [21] and the references therein.”
Tower of bubbles for almost critical problems in general domains (2010, methodology): “The method used to prove the result in [5] strongly relies on the symmetry of the problem.”
ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH THE FRACTIONAL LAPLACIAN (2013, background): “[16, 33, 32, 22] which studied the existence of bubble-towers for the related local problems.”
Super-critical boundary bubbling in a semilinear Neumann problem (2005, methodology): “The proofs of Theorems 1 and 2 rely on a form of Lyapunov–Schmidt procedure which reduces the construction of the searched solutions to a finite-dimensional variational problem, in a general scheme already followed in the study of interior bubbling in [10, 8 ].”