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Monica Musso

Publications and source records attributed to Monica Musso.

67 records · Page 4Linked to original sources
arXiv2015

Interior bubbling solutions for the critical Lin-Ni-Takagi problem in dimension 3

We consider the problem of finding positive solutions of the problem $Δu - λu +u^5 = 0$ in a bounded, smooth domain $Ω$ in $\mathbb{R}^3$, under zero Neumann boundary conditions. Here $λ$ is a positive number. We analyze the role of Green's function of $-Δ+λ$ in the presence of solutions exhibiting single bubbling behavior at one point of the domain when $λ$ is regarded as a parameter. As a special case of our results, we find and characterize a positive value $λ_*$ such that if $λ-λ^*>0$ is sufficiently small, then this problem is solvable by a solution $u_λ$ which blows-up by bubbling at a certain interior point of $Ω$ as $λ\downarrow λ_*$.

math.AP↗
arXiv2015

A non-compactness result on the fractional Yamabe problem in large dimensions

Let $(X^{n+1}, g^+)$ be an $(n+1)$-dimensional asymptotically hyperbolic manifold with a conformal infinity $(M^n, [\hat{h}])$. The fractional Yamabe problem addresses to solve \[P^γ[g^+,\hat{h}] (u) = cu^{n+2γ\over n-2γ}, \quad u > 0 \quad \text{on } M\] where $c \in \mathbb{R}$ and $P^γ[g^+,\hat{h}]$ is the fractional conformal Laplacian whose principal symbol is $(-Δ)^γ$. In this paper, we construct a metric on the half space $X = \mathbb{R}^{n+1}_+$, which is conformally equivalent to the unit ball, for which the solution set of the fractional Yamabe equation is non-compact provided that $n \ge 24$ for $γ\in (0, γ^*)$ and $n \ge 25$ for $γ\in [γ^*,1)$ where $γ^* \in (0, 1)$ is a certain transition exponent. The value of $γ^*$ turns out to be approximately 0.940197.

math.AP↗
arXiv2015

Nondegeneracy of nonradial sign-changing solutions to the nonlinear Schrödinger equations

We prove that the non-radial sign-changing solutions to the nonlinear Schrödinger equation \begin{equation*} Δu-u+|u|^{p-1}u=0 \mbox{ in }\R^N, \quad u \in H^1 (\R^N ) \end{equation*} constructed by Musso, Pacard and Wei is non-degenerate. This provides the first example of non-degenerate sign-changing solution with finite energy to the above nonlinear Schrödinger equation.

math.AP↗
arXiv2015

Sign-changing blowing-up solutions for supercritical Bahri-Coron's problem

Let $Ω$ be a bounded domain in $\R^n$, $n\ge 3$ with smooth boundary $\partialΩ$ and a small hole. We give the first example of sign-changing {\it bubbling} solutions to the nonlinear elliptic problem $$ -Δu=|u|^{{n+2\over n-2} +\ve -1 } u \, \, \mbox{ in } Ω, \quad \quad u=0 \mbox{ on } \partial Ω, $$ where $\ve$ is a small positive parameter. The basic cell in the construction is the sign-changing nodal solution to the critical Yamabe problem $$ -Δw = |w|^{\frac{4}{n-2}} w, \ \ w \in {\mathcal D}^{1,2} (\R^n) $$ which has large number ($3n$) of kernels.

math.AP↗
arXiv2014

Concentration on minimal submanifolds for a Yamabe type problem

We construct solutions to a Yamabe type problem on a Riemannian manifold M without boundary and of dimension greater than 2, with nonlinearity close to higher critical Sobolev exponents. These solutions concentrate their mass around a non degenerate minimal submanifold of M, provided a certain geometric condition involving the sectional curvatures is satisfied. A connection with the solution of a class of P.D.E.'s on the submanifold with a singular term of attractive or repulsive type is established.

math.AP↗
arXiv2014

Non-topological condensates for the self-dual Chern-Simons-Higgs model

For the abelian self-dual Chern-Simons-Higgs model we address existence issues of periodic vortex configurations -- the so-called condensates-- of non-topological type as $k \to 0$, where $k>0$ is the Chern-Simons parameter. We provide a positive answer to the long-standing problem on the existence of non-topological condensates with magnetic field concentrated at some of the vortex points (as a sum of Dirac measures) as $k \to 0$, a question which is of definite physical interest.

math.AP↗
arXiv2014

Nondegeneracy of Nonradial Nodal Solutions to Yamabe Problem

We provide the first example of a sequence of {\em nondegenerate}, in the sense of Duyckaerts-Kenig-Merle \cite{DKM}, nodal nonradial solutions to the critical Yamabe problem $$ -ΔQ= |Q|^{\frac{2}{n-2}} Q, \ \ Q \in {\mathcal D}^{1,2} (\R^n). $$

math.AP↗
arXiv2013

Bubbling on Boundary Submanifolds for the Lin-Ni-Takagi Problem at Higher Critical Exponents

We consider the equation $d^2Δu - u+ u^{\frac{n-k+2}{n-k-2}} =0\,\hbox{in}Ω$, under zero Neumann boundary conditions, where $Ω$ is open, smooth and bounded and $d$ is a small positive parameter. We assume that there is a $k$-dimensional closed, embedded minimal submanifold $K$ of $\partialΩ$, which is non-degenerate, and certain weighted average of sectional curvatures of $\partialΩ$ is positive along $K$. Then we prove the existence of a sequence $d=d_j\to 0$ and a positive solution $u_d$ such that $$ d^2 |\nabla u_{d} |^2 \rightharpoonup S, δ_K \ass d \to 0 $$ in the sense of measures, where $δ_K$ stands for the Dirac measure supported on $K$ and $S$ is a positive constant.

math.AP↗
arXiv2007

Singular limits for the bi-laplacian operator with exponential nonlinearity in $\R^4$

Let $Ω$ be a bounded smooth domain in $\mathbb{R}^{4}$ such that for some integer $d\geq1$ its $d$-th singular cohomology group with coefficients in some field is not zero, then problem {Δ^{2}u-ρ^{4}k(x)e^{u}=0 & \hbox{in}Ω, u=Δu=0 & \hbox{on}\partialΩ, has a solution blowing-up, as $ρ\to0$, at $m$ points of $Ω$, for any given number $m$.

math.AP↗
arXiv2007

A Morse Index Theorem and bifurcation for perturbed geodesics on Semi-Riemannian Manifolds

Perturbed geodesics are trajectories of particles moving on a semi-Riemannian manifold in the presence of a potential. Our purpose here is to extend to perturbed geodesics on semi-Riemannian manifolds the well known Morse Index Theorem. When the metric is indefinite, the Morse index of the energy functional becomes infinite and hence, in order to obtain a meaningful statement, we substitute the Morse index by its relative form, given by the spectral flow of an associated family of index forms. We also introduce a new counting for conjugate points, which need not to be isolated in this context, and prove that the relative Morse index equals the total number of conjugate points. Finally we study the relation with the Maslov index of the flow induced on the Lagrangian Grassmannian.

math.DG↗
arXiv2007

Boundary singularities for weak solutions of semilinear elliptic problems

We construct positive weak solutions of a class of semilinear elliptic equation which vanish in suitable trace sense on the boundary of a given smooth bounded N-dimensional domain, but which are singular at prescribed isolated points of the boundary. Similar constructions are carried out for solutions which are singular on any given embedded submanifold of the boundary.

math.AP↗