Almgren-type monotonicity methods for the classification of behaviour at corners of solutions to semilinear elliptic equations (Q2871078)

In this paper, the authors study the behavior near corners of solutions to semilinear elliptic equations in domains with conical boundary points. More precisely, they consider a domain \(\Omega\subset\mathbb{R}^N\) (\(N\geq 2\)) with the following geometric property: there exist a positive number \(R\), a cone \(\mathcal{C}=\{(x',x_N)\in \mathbb{R}^{N-1}\times \mathbb{R}: x_N>|x'|g(x'/|x'|)\), if \(x'\neq 0\), and \(x_N>0\), if \(x'=0\}\), where \(g\in C^1(\mathbb{S}^{N-2})\) if \(N\geq 3\), and a function \(\varphi\in C^2(\mathbb{R}^{N-1}\setminus\{0\})\), with \(\varphi(0)=0\), and satisfying, for some \(\delta>0\),NEWLINENEWLINE(1) \(\sup_{\nu \in \mathbb{S}^{N-2}}|\varphi(t\nu)/t-g(\nu)|=O(t^\delta)\) \ as \ \(t\rightarrow 0^+\),NEWLINENEWLINE(2) \(\sup_{\nu \in \mathbb{S}^{N-2}}|\nabla \varphi(t\nu)-g(\nu)\nu-\nabla_{\mathbb{S}^{N-2}}g(\nu)|=O(t^\delta)\) \ as \ \(t\rightarrow 0^+\), where \(\mathbb{S}^{N-2}=\{-1,1\}\) and \(\nabla_{\mathbb{S}^{N-2}}g(\nu)=0\) if \(N=2\),NEWLINENEWLINE(3) \(|D^2\varphi(x')|=O(|x'|^{-1})\), \ as \ \(|x'|\rightarrow 0\),NEWLINENEWLINE\noindent such that \(\Omega \cap B(0,R)=\{(x',x_N)\in B_R: x_N>\varphi(x')\}\), where \(B(0,R)\) is the ball in \(\mathbb{R}^N\) centered at \(0\) with radius \(R\). The above conditions express the fact that, near the origin, the domain \(\Omega\) is a perturbation of the cone \(\mathcal{C}\), and the origin is a conical boundary point of \(\Omega\). Then, the authors study the behavior near the origin of solutions to the equationNEWLINENEWLINE\(-\)div\(\displaystyle{(A(x)\nabla u(x))+{\mathbf b}(x)\cdot\nabla u(x)-\frac{V(x/|x|)}{|x|^2}u(x)=h(x)u(x)+f(x,u(x))}\), \ \ \(x\in \Omega\),NEWLINENEWLINE\noindent with the boundary condition \(u=0\) on \(\partial \Omega \cap B(0,R)\), whereNEWLINENEWLINE- \(A:\Omega \rightarrow \mathcal{M}_{N\times N}\) is such that, for some \(C>0\), \(A(x)\xi\cdot\xi\geq C|\xi|^2\) , \((A(x))_{ij}=(A(x))_{ji}\), and \((A)_{ij}\in W^{1,\infty}(\Omega)\), for all \(\xi \in \mathbb{R}^N\), \(x\in \Omega\), and \(i,j=1,\dots,N\);NEWLINENEWLINE- the coefficients \({\mathbf b}\) and \(h\) are locally bounded and may be singular at \(0\) provided that suitable decay conditions are satisfied;NEWLINENEWLINE- the nonlinearity \(f\) is continuous in \(\Omega \times \mathbb{R}\) and has subcritical growth;NEWLINENEWLINE- \(V\equiv 0\) if \(N=2\), and \(V\) is Lipschitz continuous in \(\mathbb{S}^{N-1}\) and satisfiesNEWLINENEWLINE\(\displaystyle{\sup_{v\in \mathcal{D}(\mathcal{C})\setminus\{0\}}\frac{\int_\mathcal{C}|x|^{-2}V(x/|x|)v^2(x)dx}{\int_\mathcal{C}|\nabla v(x)|^2dx}<1}\),NEWLINENEWLINE\noindent where \(\mathcal{D}(\mathcal{C})\) is the completion of \(C_c^\infty(\mathcal{C})\) with respect to the norm \((\int_\mathcal{C}|\nabla(\cdot)|^2dx)^{\frac{1}{2}}\), if \(N\geq 3\);NEWLINENEWLINE\noindent The main result of this paper states that if \(u\in H^1(\Omega)\) is a non-trivial weak solution to the above problem, which is trivially extended outside \(\Omega\), then there exists an eigenvalue \(\mu(V)\) of the operator \(-\Delta_{\mathbb{S}^{N-1}}-V\) on the spherical cap \(\mathcal{C}\cap \mathbb{S}^{N-1}\), under null boundary conditions, such that, for any \(\alpha\in (0,1)\),NEWLINENEWLINE\(\displaystyle{\lim_{\lambda\rightarrow 0^+}\lambda^\gamma u(\lambda x)= |x|^{-\gamma}\psi\biggl(\frac{x}{|x|}\biggr)}\),NEWLINENEWLINE\noindent in \(H^1(B_1)\), in \(C_{\text{ loc}}^{1,\alpha}(\mathcal{C}\cap B_1)\) and in \(C_{\text{loc}}^{0,\alpha}(B_1\setminus \{0\})\), where \(\gamma={\frac{N-2}{2}}-\sqrt{(\frac{N-2}{2})^2+\mu(V)}\), and \(\psi\) is the eigenfunction associated to \(\mu(V)\), normalized with respect to the \(L^2\) norm.NEWLINENEWLINETwo direct consequences of the main result are:NEWLINENEWLINE- \ the following point-wise upper bound for the solution \(u\): \(u(x)=O(|x|^{-\gamma})\), as \(|x|\rightarrow 0^+\),NEWLINENEWLINE- \ the following unique continuation principle: if, for any \(k\in \mathbb{N}\), the solution \(u\) is such that \(u(x)=O(|x|^{k})\), as \(|x|\rightarrow 0^+\), then \(u\equiv0\) in \(\Omega\).NEWLINENEWLINEThe proof is based on the monotonicity method introduced in [\textit{F. J. Almgren, jun.}, Bull. Am. Math. Soc., New Ser. 8, 327--328 (1983; Zbl 0557.49021)] and extended to elliptic operators with variable coefficients in [\textit{N. Garofalo} and \textit{F.-H. Lin}, Indiana Univ. Math. J. 35, 245--268 (1986; Zbl 0678.35015)].