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Large energy entire solutions for the yamabe equation
Abstract We consider the Yamabe equation Δ u + n ( n − 2 ) 4 | u | 4 n − 2 u = 0 in R n , n ⩾ 3 . Let k ⩾ 1 and ξ j k = ( e 2 j π i k , 0 ) ∈ R n = C × R n − 2 . For all large k we find a solution ofExpand
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Concentration on curves for nonlinear Schrödinger Equations
We consider the problem where p > 1, e > 0 is a small parameter, and V is a uniformly positive, smooth potential. Let Γ be a closed curve, nondegenerate geodesic relative to theExpand
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Variational reduction for Ginzburg–Landau vortices
Let Ω be a bounded domain with smooth boundary in R2. We construct non-constant solutions to the complex-valued Ginzburg–Landau equation e2Δu+(1−|u|2)u=0 in Ω, as e→0, both under zero Neumann andExpand
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Trans-Pacific range extension by rafting is inferred for the flat oyster Ostrea chilensis.
Stretches of deep ocean are potent barriers to the dispersion of nearshore, benthic marine taxa. Such obstacles can be overcome, however, by species that have either a protracted pelagic larvalExpand
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Solutions with multiple catenoidal ends to the Allen–Cahn equation in R3
Abstract We consider the Allen–Cahn equation Δ u + u ( 1 − u 2 ) = 0 in R 3 . We construct two classes of axially symmetric solutions u = u ( | x ′ | , x 3 ) such that the (multiple) components ofExpand
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Concentrating solutions in a two-dimensional elliptic problem with exponential Neumann data
Abstract We consider the elliptic equation - Δ u + u = 0 in a bounded, smooth domain Ω in R 2 subject to the nonlinear Neumann boundary condition ∂ u ∂ ν = ɛ e u . Here ɛ > 0 is a small parameter. WeExpand
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Bubbling blow-up in critical parabolic problems
These lecture notes are devoted to the analysis of blow-up of solutions for some parabolic equations that involve bubbling phenomena. The term bubbling refers to the presence of families of solutionsExpand
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Standing waves for supercritical nonlinear Schrödinger equations
Let V(x) be a non-negative, bounded potential in RN, N⩾3 and p supercritical, p>N+2N−2. We look for positive solutions of the standing-wave nonlinear Schrodinger equation Δu−V(x)u+up=0 in RN, withExpand
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Green's function and infinite-time bubbling in the critical nonlinear heat equation
Let $\Omega$ be a smooth bounded domain in $\R^n$, $n\ge 5$. We consider the semilinear heat equation at the critical Sobolev exponent $$ u_t = \Delta u + u^{\frac{n+2}{n-2}} \inn \Omega\timesExpand
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A counterexample to a conjecture by De Giorgi in large dimensions
We consider the Allen–Cahn equation Δu+u(1−u2)=0in RN. A celebrated conjecture by E. De Giorgi (1978) states that if u is a bounded solution to this problem such that ∂xNu>0, then the level setsExpand
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