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Zbl 1058.34114
Brown, B.M.; Reichel, W.
Eigenvalues of the radially symmetric $p$-Laplacian in $\bbfR^n$.
(English)
[J] J. Lond. Math. Soc., II. Ser. 69, No. 3, 657-675 (2004). ISSN 0024-6107; ISSN 1469-7750/e

For the $p$-Laplacian $\Delta _pv=\text {div}(\vert \nabla v\vert ^{p-2}\nabla v)$, $p>1$, the eigenvalue problem $$-\Delta _pv+q(\vert x\vert )\vert v\vert ^{p-2}v=\lambda \vert v\vert ^{p-2}v \text { on } \Bbb R^n,\quad v\ne 0,$$ is investigated, where the potential $q$ is radially symmetric. This leads to the eigenvalue problem $$-(r^{n-1}u'{^{(p-1)})}'+r^{n-1}q(r)u^{(p-1)}=\lambda r^{n-1}u^{(p-1)}$$ on $(0,\infty )$ with the initial condition $u'(0)=0$. With the aid of the generalized sine function, a generalized Prüfer transformation is introduced, which -- as for the linear case $p=2$ -- is used to find eigenvalues. The first main result deals with potentials $q(r)\ge \alpha r^\beta$ for large $r$, $\alpha >0$, $\beta >\max\{(p-n)/(p-1),0\}$ with $q'(r)/q(r)^{1+1/p}\to0$ as $r\to\infty$. Then the spectrum consists of an infinite number of simple eigenvalues $\lambda _1<\lambda _2<\dots$, with eigenfunctions having $k-1$ zeros. This limit-point type theorem is complemented by the second main result, a limit-circle type result. Here, the potential is of the form $q(r)=-r^\alpha$ for large $r$ with $\alpha >p/p-1$. Imposing a suitable boundary condition at $\infty$, the eigenvalues $(\lambda _i)_{i\in \Bbb Z}$ are simple with $\lim_{k\to\pm\infty }\lambda _k=\pm \infty$. The corresponding eigenfunctions have infinitely many zeros.
[Manfred Möller (Johannesburg)]
MSC 2000:
*34L15 Estimation of eigenvalues for OD operators
34L40 Particular ordinary differential operators
35P30 Nonlinear eigenvalue problems for PD operators

Keywords: limit-point; limit-circle; $p$-Laplacian; tamed solution; Prüfer transformation

Cited in: Zbl 1133.34043 Zbl 1133.35076

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