Extending a previous result of Tang
1 we prove the uniqueness of positive radial solutions of
Δ
p
u
+
f
(
u
)
=
0
, subject to Dirichlet boundary conditions on an annulus in
R
n
with
2
<
p
≤
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
9), in the complement of a ball or in the whole space
R
n
(see
10,3,11).
We continue and completely set up the spectral theory initiated in Castorina et al. (2009)
5 for the linearized operator arising from
Δ
p
u
+
f
(
u
)
=
0
. We establish existence and variational ...characterization of all the eigenvalues, and by a weak Harnack inequality we deduce Hölder continuity for the corresponding eigenfunctions, this regularity being sharp. The Morse index of a positive solution can be now defined in the classical way, and we will illustrate some qualitative consequences one should expect to deduce from such information. In particular, we show that zero Morse index (or more generally, non-degenerate) solutions on the annulus are radial.
We prove nondegeneracy of extremals for some Hardy–Sobolev–Maz'ya inequalities and present applications to scalar curvature-type problems, including the Webster scalar curvature equation in a ...cylindrically symmetric setting. The main theme is hyperbolic symmetry.
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