The main purpose of the paper is to prove the existence of a positive and smooth radial solution of the quasilinear equation \(-\mathrm{div} (a_{0}(\left| \nabla u\right| ^{2})\nabla u)=\left| u\right| ^{\alpha -2}u\) in \(\mathbb{R}^{N}\), \(N\geq 3\), for every \(\alpha >2^{\ast }= \frac{2N}{N-2}\). Here \(a_{0}(s)=(1-s)^{-1/2}\), \(s<1\). The quasilinear operator \(Q(u)=-\mathrm{div}(\frac{\nabla u}{\sqrt{1-\left| \nabla u\right| ^{2}}})\) represents the mean curvature of a \(N\)-dimensional hypersurface in the Lorentz-Minkowski space \(\mathbb{L}^{N+1}=\{(x,t)\in \mathbb{R}^{N}\times \mathbb{R}\}\) equipped with the flat metric \( \sum_{j}(dx_{j})^{2}-dt^{2}\). To this quasilinear problem, the authors associate the energy functional \(I_{0}(u)=\frac{1}{2}\int_{\mathbb{R} ^{N}}A_{0}(\left| \nabla u\right| ^{2})\) with \(A_{0}(t)= \int_{0}^{t}a_{0}(s)ds\), \(t\leq 1\). They introduce the perturbed function \( a_{\theta }(s)=a_{0}(s)\) if \(0\leq s\leq 1-\theta \) and \(a_{\theta }(s)=\gamma s^{p}+\delta \) if \(s>1-\theta \) where \(\gamma \) and \(\delta \) are chosen so that \(a_{\theta }\) is \(C^{1}\) and \(p\) depends on \(\alpha \). They also introduce the space \(\mathcal{D}_{rad}^{1;(2,q)}(\mathbb{R}^{N})\) as the completion of radial functions in \(C_{c}^{\infty }(\mathbb{R}^{N})\) with respect to the norm \((\int_{\mathbb{R}^{N}}\left| \nabla u\right| ^{2})^{1/2}+(\int_{\mathbb{R}^{N}}\left| \nabla u\right| ^{q})^{1/q}\) and the perturbed functional \(I_{\theta }:\mathcal{ D}_{rad}^{1;(2,q)}(\mathbb{R}^{N})\rightarrow \mathbb{R}^{+}\) defined through \(I_{\theta }(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}A_{\theta }(\left| \nabla u\right| ^{2})\) with \(A_{\theta }(t)=\int_{0}^{t}a_{\theta }(s)ds\).
The authors first prove some properties of the space \(\mathcal{D}_{rad}^{1;(2,q)}(\mathbb{R}^{N})\) among which is a compactness result for bounded sequences of functions of this space. The first part of the proof of the main result consists to study the critical points of \(I_{\theta }\) constrained on the manifold \(\mathcal{M}=\{u\in \mathcal{D}_{rad}^{1;(2,q)}(\mathbb{R}^{N})\mid \int_{\mathbb{R} ^{N}}\left| u\right| ^{\alpha }=1\}\). The authors prove the existence of \(u_{\theta }\in \mathcal{D}_{rad}^{1;(2,q)}(\mathbb{R}^{N})\) which realizes the minimum of \(I_{\theta }\) on \(\mathcal{M}\) for \(\theta \) small enough. They then prove some properties of \(u_{\theta }\). The second tool for the proof of the main result is a comparison argument involving the symmetric rearrangement \(u^{\bigstar }\) of a function \(u\in \mathcal{X} =\{u\in \mathcal{D}_{rad}^{1;(2,q)}(\mathbb{R}^{N})\mid \nabla u\in L^{\infty }(\mathbb{R}^{N})\), \(\left\| \nabla u\right\| _{L^{\infty }}\leq 1\}\). As they consider critical points for the energy functional \( I_{0}\) associated to the original problem, the authors prove that the infimum \(C(\alpha )=\inf_{u\in \mathcal{X}\setminus \{0\}}\frac{\int_{ \mathbb{R}^{N}}(1-\sqrt{1-\left| \nabla u\right| ^{2}})}{(\int_{ \mathbb{R}^{N}}\left| u\right| ^{\alpha })^{N/(N+2)}}\) is achieved by a radial function. The authors also prove that \(C(2^{\ast })\) is not achieved.
In the last part of their paper, the authors prove that for \( \alpha >2^{\ast }\) there exists a sequence \(u^{k}\) such that \(\int_{\mathbb{R }^{N}}(1-\sqrt{1-\left| \nabla u^{k}\right| ^{2}})\) goes to \(+\infty \). They here prove that \(I_{\theta }\) satisfies the Palais-Smale condition and they apply the Lusternik-Schnirelmann theory.