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Positive solutions of nonlinear elliptic equations--existence and nonexistence of solutions with radial symmetry in $ L\sb{p}({\bf R}\sp{N})$


Author: J. F. Toland
Journal: Trans. Amer. Math. Soc. 282 (1984), 335-354
MSC: Primary 35J60; Secondary 35B32, 35B45, 58E07
DOI: https://doi.org/10.1090/S0002-9947-1984-0728716-3
MathSciNet review: 728716
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Abstract: It is shown that when $ r$ is nonincreasing, radially symmetric, continuous and bounded below by a positive constant, the solution set of the nonlinear elliptic eigenvalue problem

$\displaystyle - \Delta u = \lambda u + r{u^{1 + \sigma }},\qquad u > 0\qquad {\... ...athbf{R}^N},\qquad u \to 0\qquad {\text{as}}\,{\text{\vert x\vert}} \to \infty $

, contains a continuum $ \mathcal{C}$ of nontrivial solutions which is unbounded in $ \mathbf{R}\, \times \,{L_p}({\mathbf{R}^N})$ for all $ p \geq 1$. Various estimates of the $ {L_p}$ norm of $ u$ are obtained which depend on the relative values of $ \sigma$ and $ p$, and the Pohozaev and Sobolev embedding constants.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0728716-3
Keywords: Global bifurcation, singular elliptic problem, a priori estimates
Article copyright: © Copyright 1984 American Mathematical Society

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