Criteria for the existence of positive solutions to the equation \(\rho(x)\Delta u=u^2\) in \(\mathbb{R}^d\) for all \(d\geq 1\) -- a new probabilistic approach

DOI10.1023/A:1009847108602zbMath0962.35048OpenAlexW2913858072MaRDI QIDQ1590223

János Engländer

Publication date: 13 June 2001

Published in: Positivity (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1023/a:1009847108602

zbMATH Keywords

diffusion processsuperprocessbranching\(h\)-transformmeasure-valued processlog-Laplace equationpositive solutions to nonlinear PDEs

Mathematics Subject Classification ID

Nonlinear elliptic equations (35J60) Diffusion processes (60J60) Second-order elliptic equations (35J15) Branching processes (Galton-Watson, birth-and-death, etc.) (60J80)

Related Items (5)

Uniqueness/nonuniqueness for nonnegative solutions of second-order parabolic equations of the form \(u_{t}=Lu+Vu-\gamma u^{p}\) in \(\mathbb R^{n}\). ⋮ Finite time extinction of super-Brownian motions with deterministic catalyst ⋮ An alternative approach to super-Brownian motion with a locally infinite branching mass. ⋮ Support properties of super-Brownian motions with spatially dependent branching rate. ⋮ Positive solutions of reaction diffusion equations with superlinear absorption: universal bounds, uniqueness for the Cauchy problem, boundedness of stationary solutions

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