Classification of radial blow-up at the first critical exponent for the Lin-Ni-Takagi problem in the ball (Q6634471)

The authors analyze the behavior of radial solutions \(u_{p}\) to the Lin-Ni-Takagi problem posed in the ball \(B_{R}\subset \mathbb{R}^{N}\), \( N\geq 3\) and written as: \(-\Delta u_{p}+u_{p}=\left\vert u_{p}\right\vert ^{p-2}u_{p}\), in \(B_{R}\), \(\partial _{\nu }u_{p}=0\), on \(\partial B_{R}\), when \(p\) is close to the first critical Sobolev exponent \(2^{\ast }=\frac{2N }{N-2}\).\ The first main result considers the case \(N\geq 7\). The authors prove that if \((p_{k})_{k\geq 0}\) is a sequence of positive real numbers such that \(p_{k}\geq 2^{\ast }\) for all \(k\geq 0\) and \(\lim_{k\rightarrow +\infty }p_{k}=2^{\ast }\), then any sequence \((u_{k})_{k\geq 0}\) of \( C^{2}(B_{R})\) radial solutions to the above problem satisfying \(\left\Vert u_{k}\right\Vert _{L^{\alpha _{k}}(B_{R})}\leq C\), with \(\alpha _{k}=\max(2^{\ast },\frac{N}{2}p_{k}-N)\), is precompact in \(C^{2}(B_{R})\). The second main result focuses on the case \(p_{k}=2^{\ast }\) for all \(k\geq 0 \). Assuming that \((u_{k})_{k\geq 0}\) is a sequence of radial solutions to the above problem that is uniformly bounded in \(L^{2^{\ast }}(B_{R})\). If \( N\notin \{3,6\}\), the authors prove that \((u_{k})_{k\geq 0}\) is precompact in \(C^{2}(B_{R})\). If \(N=3\), there exists an exceptional radius \(R^{\ast }>0\) such that if \(R=R^{\ast }\), then \((u_{k})_{k\geq 0}\) is precompact in \( C^{2}(B_{R})\). If \(N=6\) there exists as increasing sequence of exceptional radii \((R_{\ell })_{\ell \geq 1}\), with \(R_{\ell }\rightarrow +\infty \) and \( \liminf_{\ell \rightarrow +\infty }(R_{\ell }-R_{\ell -1})>0\), such that if \( R\notin \{R_{\ell }\}_{\ell \geq 1}\) then \((u_{k})_{k\geq 0}\) is precompact in \(C^{2}(\overline{B_{R}})\). The precompacity in \(C^{2}(\overline{B_{R}})\) of an uniformly (energy) bounded sequence \((u_{k})_{k\geq 0}\) of radial solutions to the problem with \(p_{k}=2^{\ast }\) for all \(k\geq 0\), is not yet proved. From the preceding main theorems, the authors deduce necessary conditions for blow-up results, that is \(\lim_{k\rightarrow +\infty }\left\Vert u_{k}\right\Vert _{L^{\infty }(B_{R})}=+\infty \), for positive solutions to the above problem in the subcritical case, in the supercritical case in dimension \(3\leq N\leq 6\), and finally in the subcritical case in dimensions \(N\geq 4\). They also define a type \(B\) blowup that involves a function \(u_{0}\in C^{2,\theta }(B_{R})\), \(\theta \in (0,1)\), \(u_{0}\geq 0\), and a sequence \((B_{\lambda _{k}})_{k}\) of bubbles such that \(u_{k}=u_{0}+B_{ \lambda _{k}}+o(1)\) in \(H^{1}(B_{R})\). They prove type \(B\) results in the subcritical case in dimensions \(N\geq 4\) and in the supercritical case in dimensions \(3\leq N\leq 6\). The key tool of the proof of these results is a sharp pointwise asymptotic description of finite-energy solutions to the above problem. The authors obtain this description, first proving a global scale-invariant estimate on \(u_{k}\). They build the blowup scales for \( (u_{k})_{k\geq 0}\) through properties that they prove by induction and contradiction. They prove that the number of concentration points is finite. They finally prove interpolation and pointwise estimates. The proofs of the different main results are give with details.