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Fethi Mahmoudi

Publications and source records attributed to Fethi Mahmoudi.

14 recordsLinked to original sources
arXiv2017

Concentration at submanifolds for an elliptic Dirichlet problem near high critical exponents

Let $Ω$ be a open bounded domain in $\mathbb{R}^n $ with smooth boundary $\partialΩ$. We consider the equation $ Δu + u^{\frac{n-k+2}{n-k-2}-\varepsilon} =0\,\hbox{ in }\,Ω$, under zero Dirichlet boundary condition, where $\varepsilon$ 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 along which a certain weighted average of sectional curvatures of $\partialΩ$ is negative. Under these assumptions, we prove existence of a sequence $\varepsilon=\varepsilon_j$ and a solution $u_{\varepsilon}$ which concentrate along $K$, as $\varepsilon \to 0^+$, in the sense that $$ |\nabla u_{\varepsilon} |^2\,\rightharpoonup \, S_{n-k}^{\frac{n-k}{2}} \,δ_K \quad \mbox{as} \ \ \varepsilon \to 0 $$ where $δ_K $ stands for the Dirac measure supported on $K$ and $S_{n-k}$ is an explicit positive constant. This result generalizes the one obtained by del Pino-Musso-Pacard, where the case $k=1$ is considered.

math.AP↗
arXiv2015

Construction of a stable periodic solution to a semilinear heat equation with a prescribed profile

We construct a periodic solution to the semilinear heat equation with power nonlinearity, in one space dimension, which blows up in finite time $T$ only at one blow-up point. We also give a sharp description of its blow-up profile. The proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude. Thanks to the geometrical interpretation of the finite-dimensional parameters in terms of the blow-up time and blow-up point, we derive the stability of the constructed solution with respect to initial data.

math.AP↗
arXiv2015

Ground states and concentration phenomena for the fractional Schrödinger equation

We consider here solutions of the nonlinear fractional Schrödinger equation $$ε^{2s}(-Δ)^s u+V(x)u=u^p.$$ We show that concentration points must be critical points for $V$. We also prove that, if the potential $V$ is coercive and has a unique global minimum, then ground states concentrate suitably at such minimal point as $ε$ tends to zero. In addition, if the potential $V$ is radial, then the minimizer is unique provided $ε$ is small.

math.AP↗
arXiv2014

On the Ambrosetti-Malchiodi-Ni Conjecture for general submanifolds

We study positive solutions of the following semilinear equation $$\varepsilon^2Δ_{\bar g} u - V(z) u+ u^{p} =0\,\hbox{ on }\,M, $$ where $(M, \bar g )$ is a compact smooth $n$-dimensional Riemannian manifold without boundary or the Euclidean space $\mathbb R^n$, $\varepsilon$ is a small positive parameter, $p>1$ and $V$ is a uniformly positive smooth potential. Given $k=1,\dots,n-1$, and $1 < p < \frac{n+2-k}{n-2-k}$. Assuming that $K$ is a $k$-dimensional smooth, embedded compact submanifold of $M$, which is stationary and non-degenerate with respect to the functional $\int_K V^{\frac{p+1}{p-1}-\frac{n-k}{2}}dvol$, we prove the existence of a sequence $\varepsilon=\varepsilon_j\to 0$ and positive solutions $u_\varepsilon$ that concentrate along $K$. This result proves in particular the validity of a conjecture by Ambrosetti-Malchiodi-Ni, extending a recent result by Wang-Wei-Yang, where the one co-dimensional case has been considered. Furthermore, our approach explores a connection between solutions of the nonlinear Schrödinger equation and $f$-minimal submanifolds in manifolds with density.

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↗
arXiv2012

Weighted Hardy inequality with higher dimensional singularity on the boundary

Let $Ω$ be a smooth bounded domain in $\mahbb R^N$ with $N\ge 3$ and let $Σ_k$ be a closed smooth submanifold of $δΩ$ of dimension $1\le k\le N-2$. In this paper we study the weighted Hardy inequality with weight function singular on $Σ_k$. In particular we provide sufficient and necessary conditions for existence of minimizers.

math.AP↗
arXiv2007

Solutions to the nonlinear Schroedinger equation carrying momentum along a curve. Part II: proof of the existence result

We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroedinger Equation $- ε^2 Δψ+ V(x) ψ= |ψ|^{p-1} ψ$ on a manifold or in the Euclidean space. Here V represents the potential, p is an exponent greater than 1 and $ε$ a small parameter corresponding to the Planck constant. As $ε$ tends to zero (namely in the semiclassical limit) we prove existence of complex-valued solutions which concentrate along closed curves, and whose phase in highly oscillatory. Physically, these solutions carry quantum-mechanical momentum along the limit curves. In the first part of this work we identified the limit set and constructed approximate solutions, while here we give the complete proof of our main existence result.

math.AP↗
arXiv2007

Solutions to the nonlinear Schroedinger equation carrying momentum along a curve. Part I: study of the limit set and approximate solutions

We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroeodinger Equation $- ε^2 Δψ+ V(x) ψ= |ψ|^{p-1} ψ$, on a manifold or in the Euclidean space. Here V represents the potential, p an exponent greater than 1 and $ε$ a small parameter corresponding to the Planck constant. As $ε$ tends to zero (namely in the semiclassical limit) we prove existence of complex-valued solutions which concentrate along closed curves, and whose phase is highly oscillatory. Physically, these solutions carry quantum-mechanical momentum along the limit curves. In this first part we provide the characterization of the limit set, with natural stationarity and non-degeneracy conditions. We then construct an approximate solution up to order $ε^2$, showing that these conditions appear naturally in a Taylor expansion of the equation in powers of $ε$. Based on these, an existence result will be proved in the second part.

math.AP↗
arXiv2007

Transition Layer for the Heterogeneous Allen-Cahn Equation

We consider the equation $\e^{2}Δu=(u-a(x))(u^2-1)$ in $Ω$, $\frac{\partial u}{\partial ν} =0$ on $\partial Ω$, where $Ω$ is a smooth and bounded domain in $\R^n$, $ν$ the outer unit normal to $\paΩ$, and $a$ a smooth function satisfying $-1 0} and {a<0}. Assuming $\nabla a \neq 0$ on $K$ and $a\ne 0$ on $\partial Ω$, we show that there exists a sequence $\e_j \to 0$ such that the above equation has a solution $u_{\e_j}$ which converges uniformly to $\pm 1$ on the compact sets of $Ø_{\pm}$ as $j \to + \infty$.

math.AP↗
arXiv2006

Concentration on minimal submanifolds for a singularly perturbed Neumann problem

We consider the equation $- \e^2 \D u + u= u^p$ in $Ω\subseteq \R^N$, where $Ω$ is open, smooth and bounded, and we prove concentration of solutions along $k$-dimensional minimal submanifolds of $\partial Ø$, for $N \geq 3$ and for $k \in \{1, ..., N-2\}$. We impose Neumann boundary conditions, assuming $1<p <\frac{N-k+2}{N-k-2}$ and $\e \to 0^+$. This result settles in full generality a phenomenon previously considered only in the particular case $N = 3$ and $k = 1$.

math.AP↗
arXiv2006

Constant $k$-curvature hypersurfaces in Riemannian manifolds

Rugang Ye proved the existence of a family of constant mean curvature hypersurfaces in an $m+1$-dimensional Riemannian manifold $(M^{m+1},g)$, which concentrate at a point $p_0$ (which is required to be a nondegenerate critical point of the scalar curvature), moreover he proved that this family constitute a foliation of a neighborhood of $p_0$. In this paper we extend this result to the other curvatures (the $r$-th mean curvature for $1\le r\le m$).

math.DG↗
arXiv2006

Energy Quantization for Yamabe's problem in Conformal Dimension

T. Riviere proved an energy quantization for Yang-Mills fields defined on n-dimensional Riemannian manifolds, when $n$ is larger than the critical dimension 4. More precisely, he proved that the defect measure of a weakly converging sequence of Yang-Mills fields is quantized, provided the $W^{2,1}$ norm of their curvature is uniformly bounded. In the present paper, we prove a similar quantization phenomenon for the Yamabe problem in a bounded domain $Ω$ of $R^n$.

math.AP↗
arXiv2004

Constant mean curvature hypersurfaces condensing along a submanifold

Given any nondegenerate k-dimensional minimal submanifold K of codimension greater than 1, we prove the existence of families of constant mean curvature submanifolds, with mean curvature varying from one member of the family to another, which `condense' to K. In particular, our result proves the existence of constant mean curvature hypersurfaces with nontrivial topology in any Riemannian manifold.

math.DG↗