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...

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

Cortázar, Carmen; del Pino, Manuel; Elgueta, Manuel — 2002

Electronic Journal of Differential Equations (EJDE) [electronic only]

Page 1

Download Results (CSV)

Advanced Search

Formula preview

Currently displaying 1 – 11 of 11

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

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