On a prescribed mean curvature equation in Lorentz-Minkowski space (Q335874)

Let \(N\geq 3\) and let \(p>1\). The author studies the existence, multiplicity and behavior at infinity of radial solutions to the following problem \((P)\): NEWLINE\[NEWLINE\nabla \biggl(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\biggr)+u^p=0\text{ in }\mathbb{R}^N,\, u>0\text{ in }\mathbb R^N,\quad u(x)\rightarrow 0\text{ as }|x|\rightarrow +\infty.NEWLINE\]NEWLINE The following results are established: if \(p\) is subcritical (i.e., \(p<\frac{N+2}{N-2}\)), then problem \((P)\) admits no radial solutions, while if \(p\) is supercritical (i.e. \(p>\frac{N+2}{N-2}\)), then there exist infinitely many radial solutions which does not belong to the space \(\mathcal{D}^{1,2}(\mathbb{R}^N):=\{u\in L^\frac{2N}{N-2}(\mathbb{R}^N): |\nabla u|\in L^2(\mathbb{R}^N)\}\). Moreover, in the case of \(p\) supercritical, the author proves that if \(u\) is a radial solution, then:NEWLINENEWLINE1) there are no \(\alpha>\frac{2}{p-1}\) and \(c,M>0\) such that \(u(|x|)\leq c|x|^{-\alpha}\), for \(|x|\geq M\);NEWLINENEWLINE2) one of the following two conditions holds:NEWLINENEWLINE(i) \(u\in \mathcal{D}^{1,2}(\mathbb{R}^N)\) and \(u(x)=O(|x|^{2-N})\) as \(|x|\rightarrow +\infty\),NEWLINENEWLINE(ii) \(u\notin \mathcal{D}^{1,2}(\mathbb{R}^N)\) and there exist \(c_1,c_2,M>0\) such that \(c_1|x|^{-\frac{2N}{(N-1)(p+1)-2N}}\leq u(x)\leq c_1|x|^{-\frac{1}{p-1}}\), for \(|x|\geq M\).NEWLINENEWLINEThe multiplicity of sing-changing solutions is also investigated. In particular, the author proves that, for \(p>1\), the equation \(\nabla \biggl(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\biggr)+u^p=0\), in \(\mathbb{R}^N\), admits infinitely many sign-changing solutions vanishing at infinity.NEWLINENEWLINEThe method of proof consists in studying the related Cauchy problem NEWLINE\[NEWLINE\biggl(\frac{u'}{\sqrt{1-|u'|^2}}\biggr)'+\frac{N-1}{r}\frac{u'}{\sqrt{1-|u'|^2}}+|u|^{p-1}u=0,\quad u'(0)=0,\quad u(0)=\xi,NEWLINE\]NEWLINE on varying of the initial datum \(\xi\). A generalized Erbe-Tang identity and an ``intersection point theorem'' are among the main tools used in the proof.