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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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