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Uniqueness of positive radial solutions for quasilinear elliptic equations in an annulus. (English) Zbl 1184.35144

Summary: Extending a previous result of M. Tang [J. Differ. Equations 189, No. 1, 148–160 (2003; Zbl 1158.35366)] we prove the uniqueness of positive radial solutions of \(\varDelta _pu+f(u)=0\), subject to Dirichlet boundary conditions on an annulus in \(\mathbb R^n\) with \(2<p\leq n\), under suitable hypotheses on the nonlinearity \(f\). This argument also provides an alternative proof for the uniqueness of positive solutions of the same problem in a finite ball (see [A. Aftalion and F. Pacella, J. Differ. Equations 195, No. 2, 380–397 (2003; Zbl 1109.35039)]), in the complement of a ball or in the whole space \(\mathbb R^n\) [see B. Franchi, E. Lanconelli and J. Serrin, Adv. Math. 118, No. 2, 177–243 (1996; Zbl 0853.35035); P Pucci and J. Serrin, Indiana Univ. Math. J. 47, No. 2, 501–528 (1998; Zbl 0920.35054); J. Serrin and M. Tang, Indiana Univ. Math. J. 49, No. 3, 897–923 (2000; Zbl 0979.35049)].

MSC:

35J70 Degenerate elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
Full Text: DOI

References:

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