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Asymptotic analysis and sign-changing bubble towers for Lane–Emden problems

Francesca De Marchis, Isabella Ianni, Filomena Pacella (2015)

Journal of the European Mathematical Society

We consider the semilinear Lane–Emden problem where p > 1 and Ω is a smooth bounded domain of 2 . The aim of the paper is to analyze the asymptotic behavior of sign changing solutions of ( p ) , as p + . Among other results we show, under some symmetry assumptions on Ω , that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as p + , and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of the Liouville...

Asymptotic properties of ground states of scalar field equations with a vanishing parameter

Vitaly Moroz, Cyrill B. Muratov (2014)

Journal of the European Mathematical Society

We study the leading order behaviour of positive solutions of the equation - Δ u + ϵ u - | u | p - 2 u + | u | q - 2 u = 0 , x N , where N 3 , q > p > 2 and when ϵ > 0 is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of p , q and N . The behavior of solutions depends sensitively on whether p is less, equal or bigger than the critical Sobolev exponent 2 * = 2 N N - 2 . For p < 2 * the solution asymptotically coincides with the solution of the equation in which the last term is absent. For p > 2 * the solution asymptotically coincides...

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

Manuel del Pino, Fethi Mahmudi, Monica 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 ...

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

Monica Musso, Frank Pacard, Juncheng 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...

Generalized n -Laplacian: semilinear Neumann problem with the critical growth

Robert Černý (2013)

Applications of Mathematics

Let Ω n , n 2 , be a bounded connected domain of the class C 1 , θ for some θ ( 0 , 1 ] . Applying the generalized Moser-Trudinger inequality without boundary condition, the Mountain Pass Theorem and the Ekeland Variational Principle, we prove the existence and multiplicity of nontrivial weak solutions to the problem u W 1 L Φ ( Ω ) , - div Φ ' ( | u | ) u | u | + V ( x ) Φ ' ( | u | ) u | u | = f ( x , u ) + μ h ( x ) in Ω , u 𝐧 = 0 on Ω , where Φ is a Young function such that the space W 1 L Φ ( Ω ) is embedded into exponential or multiple exponential Orlicz space, the nonlinearity f ( x , t ) has the corresponding critical growth, V ( x ) is a continuous potential,...

Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential

Jaeyoung Byeon, Kazunaga Tanaka (2013)

Journal of the European Mathematical Society

We consider a singularly perturbed elliptic equation ϵ 2 Δ u - V ( x ) u + f ( u ) = 0 , u ( x ) > 0 on N , 𝚕𝚒𝚖 x u ( x ) = 0 , where V ( x ) > 0 for any x N . The singularly perturbed problem has corresponding limiting problems Δ U - c U + f ( U ) = 0 , U ( x ) > 0 on N , 𝚕𝚒𝚖 x U ( x ) = 0 , c > 0 . Berestycki-Lions found almost necessary and sufficient conditions on nonlinearity f for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of potential V under possibly general conditions on f . In...

Singularly perturbed elliptic equations with solutions concentrating on a 1-dimensional orbit

Bernhard Ruf, P.N. Srikanth (2010)

Journal of the European Mathematical Society

We consider a singularly perturbed elliptic equation with superlinear nonlinearity on an annulus in 4 , and look for solutions which are invariant under a fixed point free 1-parameter group action. We show that this problem can be reduced to a nonhomogeneous equation on a related annulus in dimension 3. The ground state solutions of this equation are single peak solutions which concentrate near the inner boundary. Transforming back, these solutions produce a family of solutions which concentrate...

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