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Displaying 1 – 4 of 4

Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion

Moshe Marcus, Laurent Véron (2004)

Journal of the European Mathematical Society

Let $Ω$ be a bounded domain of class $C^{2}$ in $ℝ$ N and let $K$ be a compact subset of $\partial Ω$ . Assume that $q \geq (N + 1) / (N - 1)$ and denote by $U_{K}$ the maximal solution of $- Δ u + u^{q} = 0$ in $Ω$ which vanishes on $\partial Ω ∖ K$ . We obtain sharp upper and lower estimates for $U_{K}$ in terms of the Bessel capacity $C_{2 / q, q^{'}}$ and prove that $U_{K}$ is $σ$ -moderate. In addition we describe the precise asymptotic behavior of $U_{K}$ at points $σ \in K$ , which depends on the “density” of $K$ at $σ$ , measured in terms of the capacity $C_{2 / q, q^{'}}$ .

Computation of radial solutions of semilinear equations.

Korman, P. (2007)

Electronic Journal of Qualitative Theory of Differential Equations [electronic only]

Concentration points of least energy solutions to the Brezis-Nirenberg equation with variable coefficients

Futoshi Takahashi (2006)

Banach Center Publications

Continuum spectrum for the linearized extremal eigenvalue problem with boundary reactions

Futoshi Takahashi (2014)

Mathematica Bohemica

We study the semilinear problem with the boundary reaction $- Δ u + u = 0 in Ω, \frac{\partial u}{\partial ν} = λ f (u) on \partial Ω,$ where $Ω \subset ℝ^{N}$ , $N \geq 2$ , is a smooth bounded domain, $f : [0, \infty) \to (0, \infty)$ is a smooth, strictly positive, convex, increasing function which is superlinear at $\infty$ , and $λ > 0$ is a parameter. It is known that there exists an extremal parameter $λ^{*} > 0$ such that a classical minimal solution exists for $λ < λ^{*}$ , and there is no solution for $λ > λ^{*}$ . Moreover, there is a unique weak solution $u^{*}$ corresponding to the parameter $λ = λ^{*}$ . In this paper, we continue to study the spectral properties of $u^{*}$ and show...

Currently displaying 1 – 4 of 4

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