Classification of the Solutions of Semilinear Elliptic Problems in a Ball

By R. Benguria, Jean Dolbeault, Maria J. Esteban. 2000

47 citations recorded by the source corpus.

Abstract In this paper we fully describe the set of the positive and nodal (regular and singular) radial solutions of the superlinear elliptic PDE. −Δu=λu+|u| p −1 uinB 1 ,u=0on∂B 1 ,p>1, (1) without restriction on the range of λ∈ R . Here, B1 is the unit ball in R N. More precisely, in all subcritical, critical and supercritical cases, we analyze the possible singularities of radial solutions at the origin and the number of bounded and unbounded solutions. The solutions will be of three different types: bounded with a finite number of zeroes in (0, 1), singular at the origin, still with a finite number of zeroes and singular with sign changing oscillations at the origin.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/eeff2f3ae136c7725710dd5d437fdabdacf8f9d8

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Geometry of phase space and solutions of semilinear elliptic equations in a ball (2007, background): “Although in a slightly different context, phase space analysis has been the issue in several related works on radial solutions to semilinear elliptic equations; see for instance [5, 2, 8, 21].”
Classification of radial solutions to the Emden–Fowler equation on the hyperbolic space (2011, background): “We refer to [3,8,16,26] for information on the roles played by these exponents in Rn .”
"Bubble-tower" radial solutions in the slightly supercritical Brezis–Nirenberg problem (2003, methodology): “According to the theorem of Gidas, Ni and Nirenberg [11], all solutions are radial and decreasing along any radius, so there exists a unique branch of solutions as shown by the following parametrization method (see for instance [3] for more details).”