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Weak Eigenfunctions for the Linearization of Extremal Elliptic Problems

✍ Scribed by Xavier Cabré; Yvan Martel

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
466 KB
Volume
156
Category
Article
ISSN
0022-1236

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✦ Synopsis

We consider the semilinear elliptic problem

where * is a nonnegative parameter and g is a positive, nondecreasing, convex nonlinearity. There exists a value ** of the parameter which is extremal in terms of existence of solution. We study the linearization of the semilinear problem at the extremal weak solution corresponding to the parameter *=**. In some cases, this linearized problem has discrete and positive H 1 0 -spectrum. However, we prove that there always exists a positive weak eigenfunction in L 1 (0) with eigenvalue zero for this linearized problem. The zero L 1 -eigenvalue is coherent with the nonexistence of solutions of the semilinear problem for *>**. Finally, we find all weak eigenfunctions and eigenvalues for the linearization of the extremal problem when 0 is the unit ball and g(u)=e u or g(u)=(1+u) p .

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