EUDML

Let $Ω$ be a bounded domain in $ℝ^{n}$ with smooth boundary $\partial Ω$ . We consider the equation $d^{2} Δ u - u + u^{\frac{n - k + 2}{n - k - 2}} = 0 in Ω$ , 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 $\partial Ω$ , which is non-degenerate, and certain weighted average of sectional curvatures of $\partial Ω$ is positive along $K$ . Then we prove the existence of a sequence $d = d_{j} \to 0$ and a positive solution $u_{d}$ such that $d^{2} {| \nabla u_{d} |}^{2} ⇀ S δ_{K} as d \to 0$ in the sense of measures, where $δ_{K}$ ...

Bubbling along boundary geodesics near the second critical exponent

Manuel del Pino; Monica Musso; Frank 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 $ℝ^{n}$ 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 $| \nabla u_{ε} |^{2}$ converges weakly to a Dirac measure on $Γ$ as $ε \to 0^{+}$ , provided that $Γ$ is nondegenerate in the sense of second variations of...

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Description of regional blow-up in a porous-medium equation.

Supercritical elliptic problems in domains with small holes

Multi-peak bound states for nonlinear Schrödinger equations

Type II collapsing of maximal solutions to the Ricci flow in $R^{2}$

Solutions of elliptic equations with indefinite nonlinearities via Morse theory and linking

Super-critical boundary bubbling in a semilinear Neumann problem

Ground states of semilinear elliptic equations : a geometric approach

Uniqueness and stability of regional blow-up in a porous-medium equation

Multi-bump ground states of the Gierer–Meinhardt system in R2

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

Bubbling along boundary geodesics near the second critical exponent