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On Positive Solutions of Semilinear Elliptic Equations [PDF]
Proceedings of the American Mathematical Society, 1987 This paper is concerned with necessary conditions for the existence of positive solutions of the semilinear problem Δ u + f ( u ) = 0 , x ∈ Ω , u = 0 , x ∈ ∂ Ω \Delta u + f(u) = 0,x \in \Omega ,u = 0,x ...
E. N. Dancer, Klaus Schmitt
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A dynamical approach to semilinear elliptic equations [PDF]
Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2021 A characterization of a semilinear elliptic partial differential equation (PDE) on a bounded domain in \mathbb{R}^{n} is given in terms of an infinite-dimensional dynamical system. The dynamical system is on the space of boundary data for the PDE.
Yuri Latushkin +4 more
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Semilinear fractional elliptic equations involving measures [PDF]
Journal of Differential Equations, 2013 We study the existence of weak solutions of (E) $ (-\Delta)^\alpha u+g(u)=\nu $ in a bounded regular domain $\Omega$ in $\R^N (N\ge2)$ which vanish on $\R^N\setminus\Omega$, where $(-\Delta)^\alpha$ denotes the fractional Laplacian with $\alpha\in(0,1)$,
Chen, Huyuan, Veron, Laurent
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Generalized Harnack inequality for semilinear elliptic equations
Journal de Mathématiques Pures et Appliquées, 2016 This paper is concerned with semilinear equations in divergence form \[ \diver(A(x)Du) = f(u) \] where $f :\R \to [0,\infty)$ is nondecreasing. We prove a sharp Harnack type inequality for nonnegative solutions which is closely connected to the classical
Julin, Vesa
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Solutions of Semilinear Elliptic Equations in Tubes [PDF]
Journal of Geometric Analysis, 2012 Given a smooth compact k-dimensional manifold embedded in $\mathbb {R}^m$, with m\geq 2 and 1\leq k\leq m-1, and given >0, we define B_ ( ) to be the geodesic tubular neighborhood of radius about . In this paper, we construct positive solutions of the semilinear elliptic equation u + u^p = 0 in B_ ( ) with u = 0 on \partial B_ ...
Frank Pacard +2 more
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Topological Derivatives for Semilinear Elliptic Equations [PDF]
International Journal of Applied Mathematics and Computer Science, 2009 Topological Derivatives for Semilinear Elliptic EquationsThe form of topological derivatives for an integral shape functional is derived for a class of semilinear elliptic equations. The convergence of finite element approximation for the topological derivatives is shown and the error estimates in theL∞norm are obtained.
Iguernane, Mohamed +4 more
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A Concentration Phenomenon for Semilinear Elliptic Equations [PDF]
Archive for Rational Mechanics and Analysis, 2012 For a domain $ \subset\dR^N$ we consider the equation $ - u + V(x)u = Q_n(x)\abs{u}^{p-2}u$ with zero Dirichlet boundary conditions and $p\in(2,2^*)$. Here $V\ge 0$ and $Q_n$ are bounded functions that are positive in a region contained in $ $ and negative outside, and such that the sets $\{Q_n>0\}$ shrink to a point $x_0\in $ as $n\to\infty ...
Andrzej Szulkin, Nils Ackermann
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On a Model Semilinear Elliptic Equation in the Plane [PDF]
Journal of Mathematical Sciences, 2016 Assume that Ω is a regular domain in the complex plane ℂ, and A(z) is a symmetric 2×2 matrix with measurable entries, det A = 1, and such that 1/K|ξ|2 ≤ 〈A(z)ξ, ξ〉 ≤ K|ξ|2, ξ ∈ ℝ2, 1 ≤ K < ∞. We study the blow-up problem for a model semilinear equation div (A(z)∇u) = eu in Ω and show that the well-known Liouville–Bieberbach function solves the problem ...
Gutlyanskii, V.Y. +2 more
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The structure of solutions of a semilinear elliptic equation [PDF]
Transactions of the American Mathematical Society, 1992 We give a complete classification of solutions of the elliptic equation Δ u + K ( x ) e 2 u = 0 \Delta u + K(x){e^{2u}} = 0 in R n
Kuo-Shung Cheng, Tai-Chia Lin
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On Singular Semilinear Elliptic Equations [PDF]
Journal of Mathematical Analysis and Applications, 1993 Abstract For the semilinear elliptic equation Δ u + p ( x ) u −γ = 0, x ∈ R n , n ≥ 3, γ > 0, we show via the barrier method the existence of a positive entire solution behaving like | x | 2 − n near ∞.
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