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The existence problem is solved, and global pointwise estimates of solu-tions are obtained for quasilinear and Hessian equations of Lane-Emden type, including the following two model problems:
The Lane–Emden equation was the first to describe the internal structure of a self-gravitating polytrophic body. When temperature effects are also combined, such as in the study of isothermal gas spheres, the astrophysical phenomenon is described by a Lane–Emden equation
SMS Emden, a light cruiser of the German Kaiserliche Marine, was built in 1908 in the Danzig Kaiserliche Werft (Imperial Shipyard). In 1913, the Emden was based in Tsing Tao (China) as part of the German East Asiatic Squadron commanded by Vice Admiral Maximilian Reichsgraf von Spee (1861-1914).
The Lane-Emden equation was first studied by the astrophysicists Jonathan Homer Lane and Robert Emden, where they considered the thermal behavior of a spherical cloud of gas acting under the mutual attraction of its molecules and subject to the classical laws of thermodynamics.
This problem can be thought of as a natural extension of the Lane–Emden equation, Δ𝑤𝑤+𝑤𝑤𝑝𝑝= 0, 𝑥𝑥∈ℝ𝑛𝑛. (2) Equation (2) is strongly related to Sobolev inequality.
the Lane-Emden type equations due to their important applicability in a wide range of scientific ields. The Lane-Emden diferential equation was used to represent a wide range of phenomena in mathematical physics and astronomy, such as the theory of star structure, the the
Table 1 presents all solvable Emden–Fowler equations whose solutions are outlined in Handbook of Exact Solutions for Ordinary Differential Equations by Polyanin & Zaitsev.
Applying the Lane-Emden equation to stars I construct polytropes for n < 5 and given M, R ! possible as long as K not fixed because of definition of from (r) = c n(r) (Eq. (10)) and r = rn …
the extension of the Lane-Emden equation to the System of Lane-Emden equations. We presented the complete Lie symmetry group classification of a generalized Lane-Emden-Fowl
Since that time, the Lane–Emden equation and its generalisations were applied in many di erent branches of physics, ranging from astrophysics to kinetic theory and quantum me-chanics (see, e.g., [4] and references therein).
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