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In this paper, we study the following problem (P) { ( − Δ ) s u = | u | 2 s ⁎ − 2 u + λ u , in Ω , u = 0 , in R N ∖ Ω , where 2 s ⁎ = 2 N N − 2 s , s > 1 2 , Ω is a bounded domain in R N , N > 2 s . We first show that if λ > 0 is small, single bubbling solutions of (P) concentrating at a non-degenerate critical point of the Robin function is non-degenerate provided N ≥ 4 s + 1 . Then, using this result, we prove that if N ∈ [ 4 s + 1 , 6 s ] and Ω is a ball, (P) has infinitely many sign-changing bubbling solutions, whose energy can be arbitrarily large.

We prove the existence of solutions $(\lambda, v)\in \mathbb{R}\times H^{1}(\Omega)$ of the elliptic problem \[ \begin{cases} -\Delta v+(V(x)+\lambda) v =v^{p}\ &\text{ in $ \Omega, $} \ v>0,\qquad \int_\Omega v^2\,dx =\rho. \end{cases} \] Any $v$ solving such problem (for some $\lambda$) is called a normalized solution, where the normalization is settled in $L^2(\Omega)$. Here $\Omega$ is either the whole space $\mathbb R^N$ or a bounded smooth domain of $\mathbb R^N$, in which case we assume $V\equiv0$ and homogeneous Dirichlet or Neumann boundary conditions. Moreover, $11$ if $N=1,2$. Normalized solutions appear in different contexts, such as the study of the Nonlinear Schr\"odinger equation, or that of quadratic ergodic Mean Field Games systems. We prove the existence of solutions concentrating at suitable points of $\Omega$ as the prescribed mass $\rho$ is either small (when $p1+\frac 4N$) or it approaches some critical threshold (when $p=1+\frac 4N$)., Comment: 34 pages

We prove the non-degeneracy for the critical Lane–Emden system − Δ U = V p , − Δ V = U q , U , V > 0 in R N \begin{equation*} -\Delta U = V^p,\quad -\Delta V = U^q,\quad U, V > 0 \quad \text {in } \mathbb {R}^N \end{equation*} for all N ≥ 3 N \ge 3 and p , q > 0 p,q > 0 such that 1 p + 1 + 1 q + 1 = N − 2 N \frac {1}{p+1} + \frac {1}{q+1} = \frac {N-2}{N} . We show that all solutions to the linearized system around a ground state must arise from the symmetries of the critical Lane–Emden system provided that they belong to the corresponding energy space or they tend to zero at infinity.

In this paper we consider a two component system of coupled non linear Schr\"odinger equations modeling the phase separation in the binary mixture of Bose-Einstein condensates and other related problems. Assuming the existence of solutions in the limit of large interspecies scattering length $\beta$ the system reduces to a couple of scalar problems on subdomains of pure phases. Here we show that given a solution to the limiting problem under some additional non degeneracy assumptions there exists a family of solutions parametrized by the parameter $\beta\gg 1$.

We consider the elliptic system $$-\Delta u_i = u_i^3+\sum\limits_{j=1\atop j\not=i}^{q+1}{ \beta_{ij}}u_i u_j^2\ \hbox{in}\ \mathbb R^4, \ i=1,\dots,q+1.$$ when $\alpha:=\beta_{ij}$ and $\beta:=\beta_{i(q+1)}=\beta_{(q+1)j}$ for any $i,j=1,\dots,q.$ If $\beta

In this paper we consider the Liouville equation Δ u + λ 2 e u = 0 with Dirichlet boundary conditions in a two dimensional, doubly connected domain Ω. We show that there exists a simple, closed curve γ ⊂ Ω such that for a sequence λ n → 0 and a sequence of solutions u n it holds u n log ⁡ 1 λ n → H , where H is a harmonic function in Ω ∖ γ and λ n 2 log ⁡ 1 λ n ∫ Ω e u n d x → 8 π c Ω , where c Ω is a constant depending on the conformal class of Ω only.

We study the existence of fully nontrivial solutions to the system − Δ u i + λ i u i = ∑ j = 1 l β i j | u j | p | u i | p − 2 u i i n Ω , i = 1 , … , l , in a bounded or unbounded domain Ω in R N , N ≥ 3 . The λ i ’s are real numbers, and the nonlinear term may have subcritical ( 1 p N N − 2 ), critical ( p = N N − 2 ), or supercritical growth ( p > N N − 2 ). The matrix ( β i j ) is symmetric and admits a block decomposition such that the diagonal entries β i i are positive, the interaction forces within each block are attractive (i.e., all entries β i j in each block are non-negative) and the interaction forces between different blocks are repulsive (i.e., all other entries are non-positive). We obtain new existence and multiplicity results of fully nontrivial solutions, i.e., solutions where every component u i is nontrivial. We also find fully synchronized solutions (i.e., u i = c i u 1 for all i = 2 , … , l ) in the purely cooperative case whenever p ∈ ( 1 , 2 ) .

We study the existence of solutions to the cubic Schr\"odinger system $$ -\Delta u_i = \sum_{j =1}^m \beta_{ij} u_j^2u_i + \lambda_i u_i\ \hbox{in}\ \Omega,\ u_i=0\ \hbox{on}\ \partial\Omega,\ i =1,\dots,m, $$ when $\Omega$ is a bounded domain in $\mathbb R^4, $ $\lambda_i$ are positive small numbers, $\beta_{ij}$ are real numbers so that $\beta_{ii}>0$ and $\beta_{ij}=\beta_{ji}$, $i\neq j$. We assemble the components $u_i$ in groups so that all the interaction forces $\beta_{ij}$ among components of the same group are attractive, i.e. $\beta_{ij}>0$, while forces among components of different groups are repulsive or weakly attractive, i.e. $\beta_{ij}, Comment: 18 pages

We prove the existence of regular optimal \begin{document}$ G $\end{document}-invariant partitions, with an arbitrary number \begin{document}$ \ell\geq 2 $\end{document} of components, for the Yamabe equation on a closed Riemannian manifold \begin{document}$ (M,g) $\end{document} when \begin{document}$ G $\end{document} is a compact group of isometries of \begin{document}$ M $\end{document} with infinite orbits. To this aim, we study a weakly coupled competitive elliptic system of \begin{document}$ \ell $\end{document} equations, related to the Yamabe equation. We show that this system has a least energy \begin{document}$ G $\end{document}-invariant solution with nontrivial components and we show that the limit profiles of its components separate spatially as the competition parameter goes to \begin{document}$ -\infty $\end{document}, giving rise to an optimal partition. For \begin{document}$ \ell = 2 $\end{document} the optimal partition obtained yields a least energy sign-changing \begin{document}$ G $\end{document}-invariant solution to the Yamabe equation with precisely two nodal domains.

We consider a slightly subcritical Dirichlet problem with a non-power nonlinearity in a bounded smooth domain. For this problem, standard compact embeddings cannot be used to guarantee the existence of solutions as in the case of power-type nonlinearities. Instead, we use a Ljapunov-Schmidt reduction method to show that there is a positive solution which concentrates at a non-degenerate critical point of the Robin function. This is the first existence result for this type of generalized slightly subcritical problems., Comment: 25 pages

We construct singular solutions to the Yamabe equation using a reduction of the problem in an equivariant setting. This provides a non-trivial geometric example for which the analysis is simpler than in Mazzeo-Pacard program. Our construction provides also a non-trivial example of a weak solution to the Yamabe problem involving an equation with (smooth) coefficients., arXiv admin note: text overlap with arXiv:1911.11891

We consider generic 2 x 2 singular Liouville systems on a smooth bounded domain in the plane having some symmetry with respect to the origin. We construct a family of solutions to which blow-up at the origin and whose local mass at the origin is a given quantity depending on the parameters of the system. We can get either finitely many possible blow-up values of the local mass or infinitely many. The blow-up values are produced using an explicit formula which involves Chebyshev polynomials., 35 pages, accepted on Rev. Mat. Iberoam

In this paper, we build infinitely many non-radial sign-changing solutions to the critical problem: ( P ) { − Δ u = | u | 4 N − 2 u , in Ω , u = 0 , on ∂ Ω , on the annulus Ω : = { x ∈ R N : a | x | b } , N ≥ 3 . In particular, for any integer k large enough, we build a non-radial solution which look like the unique positive solution u 0 to (P) crowned by k negative bubbles arranged on a regular polygon with radius r 0 such that r 0 N − 2 2 u 0 ( r 0 ) = : max a ≤ r ≤ b ⁡ r N − 2 2 u 0 ( r ) .

We consider equations of the form $\Delta u +\lambda^2 V(x)e^{\,u}=\rho$ in various two dimensional settings. We assume that $V>0$ is a given function, $\lambda>0$ is a small parameter and $\rho=\mathcal O(1)$ or $\rho\to +\infty$ as $\lambda\to 0$. In a recent paper we prove the existence of the maximal solutions for a particular choice $V\equiv 1$, $\rho=0$ when the problem is posed in doubly connected domains under Dirichlet boundary conditions. We related the maximal solutions with a novel free boundary problem. The purpose of this note is to derive the corresponding free boundary problems in other settings. Solvability of such problems is, viewed formally, the necessary condition for the existence of the maximal solution.

We consider the following critical weakly coupled elliptic system −Δui=μi|ui|2∗−2ui+∑j≠iβij|uj|2∗2|ui|2∗−42uiinΩeui=0on∂Ωe,i=1,⋯,m, in a domain Ωe⊂RN, N=3,4, with small shrinking holes as the parameter e→0. We prove the existence of positive solutions of two different types: either each density concentrates around a different hole, or we have groups of components such that all the components within a single group concentrate around the same point, and different groups concentrate around different points.

We show that the classical Brezis-Nirenberg problem $$ -\Delta u=u|u| + \lambda u\ \hbox{in}\ \Omega, u=0\ \hbox{on}\ \partial\Omega, $$ when $\Omega$ is a bounded domain in $\mathbb R^6$ has a sign-changing solution which blows-up at a point in $\Omega$ as $\lambda$ approaches a suitable value $\lambda_0>0.$

We study the existence of sign-changing solutions for a non-local version of the sinh-Poisson equation on a bounded one-dimensional interval $I$, under Dirichlet conditions in the exterior of $I$. This model is strictly related to the mathematical description of galvanic corrosion phenomena for simple electrochemical systems. By means of the finite-dimensional Lyapunov-Schmidt reduction method, we construct bubbling families of solutions developing an arbitrarily prescribed number sign-alternating peaks. With a careful analysis of the limit profile of the solutions, we also show that the number of nodal regions coincides with the number of blow-up points., Comment: 37 pages

We consider the two-dimensional mean field equation of the equilibrium turbulence with variable intensities and Dirichlet boundary condition on a pierced domain $$\left\{ \begin{array}{ll} -\Delta u=\lambda_1\dfrac{V_1 e^{u}}{ \int_{\Omega_{\boldsymbol\epsilon}} V_1 e^{u} dx } - \lambda_2\tau \dfrac{ V_2 e^{-\tau u}}{ \int_{\Omega_{\boldsymbol\epsilon}}V_2 e^{ - \tau u} dx}&\text{in $\Omega_{\boldsymbol\epsilon}=\Omega\setminus \displaystyle \bigcup_{i=1}^m \overline{B(\xi_i,\epsilon_i)}$}\\ \ \ u=0 &\text{on $\partial \Omega_{\boldsymbol\epsilon}$}, \end{array} \right. $$ where $B(\xi_i,\epsilon_i)$ is a ball centered at $\xi_i\in\Omega$ with radius $\epsilon_i$, $\tau$ is a positive parameter and $V_1,V_2>0$ are smooth potentials. When $\lambda_1>8\pi m_1$ and $\lambda_2 \tau^2>8\pi (m-m_1)$ with $m_1 \in \{0,1,\dots,m\}$, there exist radii $\epsilon_1,\dots,\epsilon_m$ small enough such that the problem has a solution which blows-up positively and negatively at the points $\xi_1,\dots,\xi_{m_1}$ and $\xi_{m_1+1},\dots,\xi_{m}$, respectively, as the radii approach zero., Comment: 23 pages

In this work we study the existence of nodal solutions for the problem $$\begin{aligned} -\Delta u = \lambda u e^{u^2+|u|^p} \quad \text { in }\quad \Omega ,\quad u = 0 \quad \text { on }\partial \Omega , \end{aligned}$$ where $$\Omega \subseteq \mathbb {R}^2$$ is a bounded smooth domain and $$p\rightarrow 1^+$$ . If $$\Omega $$ is a ball, it is known that the case $$p=1$$ defines a critical threshold between the existence and the non-existence of radially symmetric sign-changing solutions. In this work we construct a blowing-up family of nodal solutions to such problem as $$p\rightarrow 1^+$$ , when $$\Omega $$ is an arbitrary domain and $$\lambda $$ is small enough. As far as we know, this is the first construction of sign-changing solutions for a Moser–Trudinger critical equation on a non-symmetric domain.

We consider the problem of prescribing Gaussian and geodesic curvatures for a conformal metric on the unit disk. This is equivalent to solving the following P.D.E. $$\begin{aligned} {\left\{ \begin{array}{ll}-\Delta u=2K(z)e^u&{}\hbox {in}\;\mathbb {D}^2,\\ \partial _\nu u+2=2h(z)e^\frac{u}{2}&{}\hbox {on}\;\partial \mathbb {D}^2,\end{array}\right. } \end{aligned}$$ where K, h are the prescribed curvatures. We construct a family of conformal metrics with curvatures $$K_\varepsilon ,h_\varepsilon $$ converging to K, h respectively as $$\varepsilon $$ goes to 0, which blows up at one boundary point under some generic assumptions.

We consider the nonlinear Schr\"odinger equation with focusing power-type nonlinearity on compact graphs with at least one terminal edge, i.e. an edge ending with a vertex of degree 1. On the one hand, we introduce the associated action functional and we provide a profile description of positive low action solutions at large frequencies, showing that they concentrate on one terminal edge, where they coincide with suitable rescaling of the unique solution to the corresponding problem on the real line. On the other hand, a Ljapunov-Schmidt reduction procedure is performed to construct one-peaked and multipeaked positive solutions with sufficiently large frequency, exploiting the presence of one or more terminal edges., Comment: 25 pages, 1 figure

We prove the non-degeneracy of the extremals of the Sobolev inequality \[ \int_{\mathbb R^N}|\nabla u|^p\,\rd x\ge \mathcal S_p\int_{\open R^N}|u|^\frac{Np}{N-p}\,\rd x,\quad u\in \mathcal D^{1,p}(\open R^N) \] when 1 < p < N, as solutions of a critical quasilinear equation involving the p-Laplacian.

Motivated by the statistical mechanics description of stationary 2D-turbulence, for a sinh-Poisson type equation with asymmetric nonlinearity, we construct a concentrating solution sequence in the form of a tower of singular Liouville bubbles, each of which has a different degeneracy exponent. The asymmetry parameter \begin{document} $γ∈(0, 1]$ \end{document} corresponds to the ratio between the intensity of the negatively rotating vortices and the intensity of the positively rotating vortices. Our solutions correspond to a superposition of highly concentrated vortex configurations of alternating orientation; they extend in a nontrivial way some known results for \begin{document} $\gamma=1$ \end{document} . Thus, by analyzing the case \begin{document} $\gamma≠1$ \end{document} we emphasize specific properties of the physically relevant parameter \begin{document} $\gamma$ \end{document} in the vortex concentration phenomena.

We consider the Gelfand problem on a planar domain. Under some conditions on the potential, we provide the first examples of multiplicity for blowing-up solutions at a given point in the domain. The argument is based on a refined Lyapunov-Schmidt reduction and the computation of the degree of a finite-dimensional map., 25 pages

We build blowing-up solutions for linear perturbation of the Yamabe problem on manifolds with umbilic boundary, provided the Weyl tensor is nonzero everywhere on the boundary and the dimension of the manifold is n ≥ 11 .

We study the existence of symmetric ground states to the supercritical problem $$\begin{aligned} -\Delta v=\lambda v+\left| v\right| ^{p-2}v\;\text {in}\;\Omega ,\quad v=0\;\text {on}\;\partial \Omega , \end{aligned}$$ in a domain of the form $$\begin{aligned} \Omega =\{(y,z)\in \mathbb {R}^{k+1}\times \mathbb {R}^{N-k-1}:\left( \left| y\right| ,z \right) \in \Theta \}, \end{aligned}$$ where \(\Theta \) is a bounded smooth domain such that \(\overline{\Theta } \subset \left( 0,\infty \right) \times \mathbb {R}^{N-k-1}\), \(1\le k\le N-3\), \(\lambda \in \mathbb {R}\), and \(p=\frac{2(N-k)}{N-k-2}\) is the \((k+1)\)-st critical exponent. We show that symmetric ground states exist for \(\lambda \) in some interval to the left of each symmetric eigenvalue and that no symmetric ground states exist in some interval \((-\infty ,\lambda _{*})\) with \(\lambda _{*}>0\) if \(k\ge 2\). Related to this question is the existence of ground states to the anisotropic critical problem $$\begin{aligned} -\text {div}(a(x)\nabla u)=\lambda b(x)u+c(x)\left| u\right| ^{2^{*}-2}u\;\text {in}\; \Theta ,\quad u=0\;\text {on}\; \partial \Theta , \end{aligned}$$ where a, b, c are positive continuous functions on \(\overline{\Theta }\). We give a minimax characterization for the ground states of this problem, study the ground state energy level as a function of \(\lambda \), and obtain a bifurcation result for ground states.

We consider the semilinear elliptic boundary value problem \[ -\Delta u=\left\vert u\right\vert ^{p-2}u\text{ in }\Omega,\text{\quad }u=0\text{ on }\partial\Omega, \] in a bounded smooth domain $\Omega$ of $\mathbb{R}^{N}$ for supercritical exponents $p>\frac{2N}{N-2}.$ Until recently, only few existence results were known. An approach which has been successfully applied to study this problem, consists in reducing it to a more general critical or subcritical problem, either by considering rotational symmetries, or by means of maps which preserve the Laplace operator, or by a combination of both. The aim of this paper is to illustrate this approach by presenting a selection of recent results where it is used to establish existence and multiplicity or to study the concentration behavior of solutions at supercritical exponents.

We consider the problem of finding pairs $(\lambda,\mathfrak{u})$, with $\lambda>0$ and $\mathfrak{u}$ a harmonic function in a three-dimensional torus-like domain $\mathcal{D}$, satisfying the nonlinear boundary condition $\partial_{\nu}\mathfrak{u}=\lambda\,\sinh\mathfrak{u}$ on $\partial\mathcal{D}$. This type of boundary condition arises in corrosion modeling (Butler--Volmer condition). We prove the existence of solutions which concentrate along some geodesics of the boundary $\partial\mathcal{D}$ as the parameter $\lambda$ goes to zero.

In this paper we consider the so-called Toda System in planar domains under Dirichlet boundary condition. We show the existence of continua of solutions for which one component is blowing up at a certain number of points. The proofs use singular perturbation methods.

We consider the following perturbed critical Dirichlet problem involving the Hardy-Schr\"odinger operator on a smooth bounded domain $\Omega \subset \mathbb{R}^N$, $N\geq 3$, with $0 \in \Omega$: $$ \left\{ \begin{array}{ll}-\Delta u-\gamma \frac{u}{|x|^2}-\epsilon u=|u|^{\frac{4}{N-2}}u &\hbox{in }\Omega u=0 & \hbox{on }\partial \Omega, \end{array}\right. $$ when $\epsilon>0$ is small and $\gamma< {(N-2)^2\over4}$. Setting $ \gamma_j= \frac{(N-2)^2}{4}\left(1-\frac{j(N-2+j)}{N-1}\right)\in(-\infty,0]$ for $j \in \mathbb{N},$ we show that if $\gamma\leq \frac{(N-2)^2}{4}-1$ and $\gamma \neq \gamma_j$ for any $j$, then for small $\epsilon$, the above equation has a positive --non variational-- solution that develops a bubble at the origin. If moreover $\gamma \frac{(N-2)^2}{4}-1$ and $\Omega$ is a ball $B$, then there is no radial positive solution for $\epsilon>0$ small. We complete the picture here by showing that, if $\gamma\geq \frac{(N-2)^2}{4}-4$, then the above problem has no radial sign-changing solutions for $\epsilon>0$ small. These results recover and improve what is known in the non-singular case, i.e., when $\gamma=0$., Comment: 41 pages, Updated version - if any - can be downloaded at http://www.birs.ca/~nassif/

In this paper we study the Lane-Emden-Fowler equation $$ (P)_ \epsilon \quad \left\{ \begin{aligned} &\Delta u+|u|^{q-2}u=0\ &\hbox{in}\ \mathcal D_ \epsilon,\\ & u=0\ &\hbox{on}\ \partial\mathcal D_ \epsilon.\\ \end{aligned}\right. $$ Here $\mathcal D_ \epsilon=\mathcal D\setminus \left\{x\in \mathcal D\ :\ \mathrm{dist}(x,\Gamma_l)\le \epsilon \right\}$, $\mathcal D$ is a smooth bounded domain in $\mathbb{R}^N$, $\Gamma_l$ is an $l-$dimensional closed manifold such that $\Gamma_l\subset\mathcal D$ with $1\le l\le N-3$ and $q={2(N-l)\over N-l-2} .$ We prove that, under some symmetry assumptions, the number of sign changing solutions to $ (P)_ \epsilon$ increases as $\epsilon$ goes to zero.

Let $M$ be a connected compact $C^\infty$ manifold of dimension $n\ge4$ without boundary. Let $ \mathcal{M}^k$ be the set of all $C^k$ Riemannian metrics on $M$. Any $g\in\mathcal{M}^k$ determines the Weyl tensor $$ \mathcal W^g\colon M\to \mathbb R^{4n},\qquad \mathcal W^g(\xi):=(W^g_{ijkl}(\xi))_{i,j,k,l=1,\dots,n}.$$ We prove that the set $$\mathcal{A}:=\big\{g\in \mathcal{M}^k : |\mathcal W^g(\xi)|+|D \mathcal W^g(\xi)|+|D^2 \mathcal W^g(\xi)|> 0\ \hbox{for any}\ \xi\in M\big\}$$ is an open dense subset of $\mathcal{M}^k$.

Given a 3-dimensional Riemannian manifold $(M,g)$, we investigate the existence of positive solutions of the Klein-Gordon-Maxwell system $$ \left\{ \begin{array}{cc} -\varepsilon^{2}\Delta_{g}u+au=u^{p-1}+\omega^{2}(qv-1)^{2}u & \text{in }M\\ -\Delta_{g}v+(1+q^{2}u^{2})v=qu^{2} & \text{in }M \end{array}\right. $$ and Schrodinger-Maxwell system $$ \left\{ \begin{array}{cc} -\varepsilon^{2}\Delta_{g}u+u+\omega uv=u^{p-1} & \text{in }M\\ -\Delta_{g}v+v=qu^{2} & \text{in }M \end{array}\right. $$ when $p\in(2,6). $ We prove that if $\varepsilon$ is small enough, any stable critical point $\xi_0$ of the scalar curvature of $g$ generates a positive solution $(u_\varepsilon,v_\varepsilon)$ to both the systems such that $u_\varepsilon$ concentrates at $\xi_0$ as $\varepsilon$ goes to zero.

We consider the supercritical problem −Δu=|u|p−1uin D,u=0on ∂D, where D is a bounded smooth domain in RN and p is smaller than the κ-th critical Sobolev exponent 2N,κ⁎:=N−κ+2N−κ−2 with 1⩽κ⩽N−3. We show that in some suitable torus-like domains D there exists an arbitrary large number of sign-changing solutions with alternate positive and negative layers which concentrate at different rates along a κ-dimensional submanifold of ∂D as p approaches 2N,κ⁎ from below.

We construct sign-changing concentrating solutions for a mean field equation describing turbulent Euler flows with variable vortex intensities and arbitrary orientation. We study the effect of variable intensities and orientation on the bubbling profile and on the location of the vortex points.

For a bounded domain $\Omega\subset\mathbb{R}^n$ let $H_\Omega:\Omega\times\Omega\to\mathbb{R}$ be the regular part of the Dirichlet Green function for the Laplace operator. Given a fixed arbitrary ${\mathcal C}^2$ function $f:{\mathcal D}\to\mathbb{R}$, defined on an open subset ${\mathcal D}\subset\mathbb{R}^{nN}$, and fixed coefficients $\lambda_1,\dots,\lambda_N\in\mathbb{R}\setminus\{0\}$ we consider the function $f_\Omega:{\mathcal D}\cap\Omega^N\to\mathbb{R}$ defined as \[ f_\Omega(x_1,\dots,x_N) = f(x_1,\dots,x_N) - \sum_{j,k=1}^N \lambda_j\lambda_k H_\Omega(x_j,x_k). \] We prove that $f_\Omega$ is a Morse function for most domains $\Omega$ of class ${\mathcal C}^{m+2,\alpha}$, any $m\ge0$, $0, Comment: 14 pages

Given ( M , g ) ${(M,g)}$ , a smooth compact Riemannian n-manifold, we prove that for a generic Riemannian metric g the critical points of the function 𝒲 g ⁢ ( ξ ) := | Weyl g ⁢ ( ξ ) | g 2 ${\mathcal{W}_{g}(\xi):=\lvert\mathrm{Weyl}_{g}(\xi)\rvert^{2}_{g}}$ with 𝒲 g ⁢ ( ξ ) ≠ 0 ${\mathcal{W}_{g}(\xi)\not=0}$ are nondegenerate.