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Separable solutions of quasilinear Lane-Emden equations. (English) Zbl 1272.35105
Let \(S\) be a smooth subset of the unit sphere \(S^{N-1}\) of \(\mathbb{R}^N\) and let \(p,q\in \mathbb{R}\) with \(q>p-1>0\). Moreover, let \(C_S=\{x\in \mathbb{R}^N\setminus \{0\}: \frac{x}{|x|}\in S\}\) be the cone with vertex \(0\) and opening \(S\).
The authors investigate the existence and nonexistence of positive solutions vanishing on \(\partial C_S\setminus\{0\}\) and of the separable form \(u(x)=|x|^{-\beta}\omega\left(\frac{x}{|x|}\right)\) for the quasilinear Lane-Emden equations
\[ -{\text{div}} \,(|\nabla u|^{p-2}\nabla u)=\epsilon u^q \quad { \text{in}} \;\;C_S, \] where \(\epsilon=\pm 1\).
A function \(u\) of the above form satisfies this problem provided that \(\beta_q=\frac{p}{q+p-1}\) and \(\omega\) is a positive solution, vanishing on \(\partial S\), of the following quasilinear elliptic equation on \(S\) \[ -{\text{div}}((\beta_q^2\omega^2+|\nabla'\omega|^2)^{\frac{p-2}{2}}\nabla'\omega)- \beta_q\lambda(\beta_q)(\beta_q^2\omega^2+|\nabla'\omega|^2)^{\frac{p-2}{2}}\omega=\epsilon \omega^q, \tag{1} \] where \(\lambda(\beta_q)=\beta_q(p-1)+p-N\) and \(\nabla'\) is the covariant derivative on \(S^{N-1}\).
From L. Vèron [Colloq. Math. Soc. János Bolyai. 62, 317–352 (1991; Zbl 0822.58052)], it is known that there exists a unique positive constant \(\beta_S\) such that, for \(\beta=\beta_S\), equation \((1)\) with \(\epsilon=0\) has a positive solution \(\omega\) vanishing on \(\partial S\). The constant \(\beta_S\) is involved in the existence results established by the authors. Indeed, they prove that:
for the reaction problem (i.e. \(\epsilon=1\)), a solution of \((1)\) vanishing on \(\partial S\) exists provided that \(q<\frac{(N-1)p}{N-1-p}-1\) if \(p<N-1\), and \(\beta_q<\beta_S\).
for the absorption problem (i.e. \(\epsilon=-1\)), a solution of \((1)\) vanishing on \(\partial S\) exists provided that \(\beta_q>\beta_S\).
Actually, the authors prove more general existence results where the unit sphere is replaced by a \(d\)-dimensional Riemannian manifold \((M,g)\) and \(S\) is a relatively compact smooth open domain of M.
Nonexistence results are known for \(\epsilon=1\) and \(\beta_q\geq \beta_S\) and for \(\epsilon=-1\) and \(\beta_q\leq \beta_S\). Therefore, the constrains \(\beta_q<\beta_S\) and \(\beta_q>\beta_S\) in the above existence results are sharp.
A nonexistence result is proved by the authors in the case in which \(\epsilon=1\), \(p<N-1\), \(S\nsubseteq S_+^{N-1}\) is a starshaped domain, and \(q=\frac{(N-1)p}{N-1-p}-1\). Therefore, in the case \(\epsilon=1\), the existence result is optimal for \(q=\frac{(N-1)p}{N-1-p}-1\).

MSC:
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35J60 Nonlinear elliptic equations
47H11 Degree theory for nonlinear operators
58C30 Fixed-point theorems on manifolds
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