A complete classification of the isolated singularities for nonlinear elliptic equations with inverse square potentials (Q2925682)

Let \(\Omega\) be an open subset of \(\mathbb{R}^N\), with \(N\geq 3\) and \(0\in \Omega\). In this paper, the author studies the behavior near \(0\) of the positive solutions to the following semilinear elliptic equation NEWLINE\[NEWLINE-\Delta u-\frac{\lambda}{|x|^2}u+b(x)h(u)=0 \mathrm{ in}\, \Omega^*:=\Omega \setminus \{0\}.\tag{1}NEWLINE\]NEWLINE Here, \(\lambda \in (-\infty, (N-2)^2/4]\), \(b:\overline{\Omega}\setminus \{0\}\rightarrow (0,+\infty)\) is a continuous function, and \(h:\mathbb{R}\rightarrow \mathbb{R}\) is a continuous function which is positive on \((0,+\infty)\) and such that \(h(t)/t\) is bounded in a right-neighborhood of \(0\).NEWLINENEWLINEThe solutions to \((1)\) are understood in the distributional sense, that is a function \(u\in C^1(\Omega^*)\) is a solution of \((1)\) if and only if NEWLINE\[NEWLINE\displaystyle{\int_\Omega \biggl(\nabla u \nabla \phi-\frac{\lambda}{|x|^2}u\phi+b(x)h(u)\phi\biggr)dx=0},\tag{2}NEWLINE\]NEWLINE \noindent for all \(\phi\in C_c^1(\Omega^*)\).NEWLINENEWLINESeveral results which give the exact asymptotic behavior of the positive solutions near 0 are established in this paper. The key assumptions in these results are certain structure conditions imposed on the functions \(b,h\) and involving the regular variation of these latter at \(0\) and at \(\infty\), respectively. More precisely, the following assumption is common to all the main results: there exist a positive function \(L_h\) slowly varying at \(\infty\), a positive function \(L_b\) slowly varying at \(0\), and two numbers \(q\in (1,\infty)\), \(\theta \in (-2,\infty)\) such that: \(h(t)\sim t^qL_h(t)\) as \(t\rightarrow +\infty\) and \(b(x)\sim |x|^{\theta}L_b(|x|)\) as \(x\rightarrow 0\). When the previous conditions hold, the functions \(h,b\) are said regularly varying at \(\infty\) with index \(q\), and regularly varying at \(0\) with index \(\theta\), respectively.NEWLINENEWLINELet \(p=(N-2)/2-\sqrt{(N-2)^2/4-\lambda}\). For the subcritical case \(\lambda<(N-2)^2/4\), the fundamental solutions \(\Phi_\lambda^+(|x|)=|x|^{2-N+p}\) and \(\Phi_\lambda^-(|x|)=|x|^{-p}\) of the equation \(-\Delta u-\frac{\lambda}{|x|^2}u=0\) in \(\mathbb{R}^N\setminus \{0\}\) play a key role in the main results. In particular, after having introduced the following functions NEWLINE\[NEWLINE\mathcal{I}^*(\tau,\varpi)=\int_\tau^\varpi r^{N+\theta-p-1-(N-2-p)q}L_b(r)L_h(\Phi_\lambda^+(r))dr,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\mathcal{I}^{**}(\tau,\varpi)=\int_\tau^\varpi r^{\theta+1-p(q-1)}L_b(r)L_h(\Phi_\lambda^-(r))drNEWLINE\]NEWLINE \noindent for all \(\tau \in ]0,\varpi[\), where \(\varpi>0\) is sufficiently small, and after having observed that \(\lim_{\tau\rightarrow 0^+}\mathcal{I}^*(\tau,\varpi)<+\infty\) implies \(\lim_{\tau\rightarrow 0^+}\mathcal{I}^{**}(\tau,\varpi)<+\infty\), and that the finiteness of latter limit always holds when \(\lambda\leq 0\), the author considers the following three exhaustive cases: NEWLINE\[NEWLINE\lim_{\tau\rightarrow 0^+}\mathcal{I}^*(\tau,\varpi)=+\infty, \mathrm{and}\, \lim_{\tau\rightarrow 0^+}\mathcal{I}^{**}(\tau,\varpi)<+\infty;NEWLINE\]NEWLINE NEWLINE\[NEWLINE\lim_{\tau\rightarrow 0^+}\mathcal{I}^{**}(\tau,\varpi)=+\infty,\mathrm{ and}\, \lambda>0;NEWLINE\]NEWLINE NEWLINE\[NEWLINE\lim_{\tau\rightarrow 0^+}\mathcal{I}^*(\tau,\varpi)<\infty.NEWLINE\]NEWLINE \noindent For each of these cases, the author establishes the behavior of the positive solutions near \(0\) by giving precise information about the limit of the ratio \(u(x)/\Phi_\lambda^\pm(|x|)\) as \(x\rightarrow 0\).NEWLINENEWLINE\noindent The critical case \(\lambda=(N-2)^2/4\) is analyzed in a similar way. In this case, the fundamental solutions of the equation \(-\Delta u-\frac{(N-2)^2}{4|x|^2}u=0\) in \(\mathbb{R}^N\setminus\{0\}\) are \(\Psi^+(|x|)=-|x|^{-(N-2)/2}\log(|x|)\) and \(\Psi^-(|x|)=-|x|^{-(N-2)/2}\) and the roles of the functions \(\mathcal{I}^*(\tau,\varpi), \mathcal{I}^{**}(\tau,\varpi)\) are now played, respectively, by the functions NEWLINE\[NEWLINE\mathcal{F}^*(\tau,\varpi)=\int_\tau^\varpi r^{N+2\theta-(N-2)q}L_b(r)L_h(\Psi^+(r))(-\log r)^qdr,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\mathcal{F}_{*}(\tau,\varpi)=\int_\tau^\varpi r^{N+2\theta-(N-2)q}L_b(r)L_h(\Psi^-(r))(-\log r)dr.NEWLINE\]NEWLINE \noindent After having observed that \(\lim_{\tau\rightarrow 0^+}\mathcal{F}_*(\tau,\varpi)<+\infty\) implies \(\lim_{\tau\rightarrow 0^+}\mathcal{F}^{*}(\tau,\varpi)<+\infty\), the author considers the following three exhaustive situations: NEWLINE\[NEWLINE\lim_{\tau\rightarrow 0^+}\mathcal{F}_*(\tau,\varpi)=+\infty \mathrm{and}\, \lim_{\tau\rightarrow 0^+}\mathcal{F}^{*}(\tau,\varpi)<+\infty;NEWLINE\]NEWLINE NEWLINE\[NEWLINE\lim_{\tau\rightarrow 0^+}\mathcal{F}^{*}(\tau,\varpi)=+\infty;NEWLINE\]NEWLINE NEWLINE\[NEWLINE\lim_{\tau\rightarrow 0^+}\mathcal{F}_*(\tau,\varpi)<+\infty.NEWLINE\]NEWLINE \noindent As for the subcritical case, for each of these cases, the behavior of the positive solutions near \(0\) is established by giving precise information about the limit of the ratio \(u(x)/\Psi_\lambda^\pm(|x|)\) as \(x\rightarrow 0\).NEWLINENEWLINE\noindent When \(\Omega\) is a ball and \(h(t)/t\) is increasing in \((0,\infty)\), the author also finds necessary and sufficient conditions for the existence of positive solutions which are comparable with the fundamental solutions \(\Phi_\lambda^\pm,\Psi^\pm\).NEWLINENEWLINE\noindent Several known results are improved and generalized in this paper. Moreover, as a consequence of the main results, the author finds that the following condition NEWLINE\[NEWLINE\displaystyle{\int_1^\infty h(t)t^{-2\frac{N-1}{N-2}}dt=+\infty}NEWLINE\]NEWLINE \noindent is optimal for the existence of solutions to the equation \(-\Delta u+h(u)=0\) in \(\Omega^*\) having at 0 a removable singularity, that is such that they satisfy equation \((2)\) (with \(\lambda=0\) and \(b(x)=1\)) for all \(\phi\in C_c^1(\Omega)\). Therefore, for regularly varying functions of index greater than 1, the author gives an answer to the question proposed in [\textit{J. L. Vázquez} and \textit{L. Véron}, J. Differ. Equations 60, 301--321 (1985; Zbl 0549.35043)].