Hardy-Sobolev inequality with singularity a curve

TL;DR

This paper investigates the existence of positive solutions to a Hardy-Sobolev type PDE with a singularity along a curve in a bounded domain, highlighting geometric conditions that guarantee solutions in different dimensions.

Contribution

It establishes new existence criteria for solutions based on the local geometry of the curve and properties of the Green's function, extending previous results to singularities along curves.

Findings

For N ≥ 4, a geometric condition ensures existence of ground-state solutions.

For N = 3, positivity of the Green's function trace at the curve guarantees solutions.

The results connect geometric features with PDE solvability in singular settings.

Abstract

We consider a bounded domain $\Omega$ of $\mathbb{R}^N$, $N\geq 3$, and $h$ a continuous function on $\Omega$. Let $\Gamma$ be a closed curve contained in $\Omega$. We study existence of positive solutions $u\in H^1_0(\Omega)$ to the equation $$ -\Delta u+h u=\rho^{-\sigma}_\Gamma u^{2^*_\sigma-1} \qquad \textrm{ in } \Omega $$ where $2^*_\sigma:=\frac{2(N-\sigma)}{N-2}$, $\sigma\in (0,2)$, and $\rho_\Gamma$ is the distance function to $\Gamma$. For $N\geq 4$, we find a sufficient condition, given by the local geometry of the curve, for the existence of a ground-state solution. In the case $N=3$, we obtain existence of ground-state solution provided the trace of the regular part of the Green of $-\Delta+h$ is positive at a point of the curve.

Equations

$$-\Delta u+hu={\rho }_{\Gamma }^{-\sigma }{u}^{2_{\sigma }^{\ast }-1}\text{ in }\Omega$$

$$\int_{\mathbb{R}^{N}}|\nabla v|^{2}dx\geq C\biggl{(}\int_{\mathbb{R}^{N}}|z|^{-\sigma}|v|^{2^{*}_{\sigma}}dx\biggl{)}^{2/2^{*}_{\sigma}}\qquad\textrm{ for all $v\in\mathcal{D}^{1,2}({\mathbb{R}^{N}})$,}$$

$$S_{N,\sigma }:=\inf \left\{{\int }_{{\mathbb{R}}^N}|\nabla v{|}^{2}dx,v\in {\mathcal{D}}^{1,2}({\mathbb{R}}^N)\text{ and }{\int }_{{\mathbb{R}}^N}|z{|}^{-\sigma }|v{|}^{2_{\sigma }^{\ast }}dx=1\right\}.$$

$${\mu }_h(\Omega ,\Gamma ):=\underset{u\in H_0^1(\Omega )}{\inf }\frac{{\int }_{\Omega }|\nabla u{|}^{2}dx+{\int }_{\Omega }h{u}^{2}dx}{{\left({\int }_{\Omega }{\rho }_{\Gamma }^{-\sigma }|u{|}^{2_{\sigma }^{\ast }}dx\right)}^{\frac{2}{2_{\sigma }^{\ast }}}},$$

$$h(y_0)<-C_{N,\sigma }|\kappa (y_0){|}^2$$

$$\lambda <-\frac{C_{N,\sigma }}{R^2}.$$

$$\{\begin{array}{ll}-{\Delta }_{x}G(x,y)+h(x)G(x,y)=0& \text{ for every x∈Ω∖{y}}\\ G(x,y)=0& \text{ for every x∈∂Ω.}\end{array}$$

$$G(x,y)=\frac{c}{|x-y|}+c\text{m}(y)+o(1)\text{ as x→y.}$$

$$Q_r:=(-r,r)\times B_{{\mathbb{R}}^{N-1}}(0,r),$$

$$(t,z)\mapsto F_{y_0}(t,z):=\gamma (t)+\sum _{i=2}^N{z}_i{E}_i(t),$$

$${\rho }_{\Gamma }(y)=\underset{\overline{y}\in {\mathbb{R}}^N}{\min }|y-\overline{y}|.$$

$${\rho }_{\Gamma }\left(F_{y_0}(x)\right)=|z|\text{ for every x=(t,z)∈Qr.}$$

$$E_i^{\prime }(t)={\kappa }_i(t){\gamma }^{\prime }(t)+\sum _{j=2}^N{\tau }_i^j(t)E_j(t).$$

$${\tau }_i^j(t)=-{\tau }_j^i(t)\text{ for i,j=2,…,N. }$$

$$g_{11}(x)={\partial }_t{F}_{y_0}(x)\cdot {\partial }_t{F}_{y_0}(x),g_{1i}(x)={\partial }_t{F}_{y_0}(x)\cdot {\partial }_{z_i}{F}_{y_0}(x),g_{ij}(x)={\partial }_{z_j}{F}_{y_0}(x)\cdot {\partial }_{z_i}{F}_{y_0}(x).$$

$$\{\begin{array}{l}{g}_{11}(x)=1+2\sum _{i=2}^N{z}_i{\kappa }_i(0)+2t\sum _{i=2}^N{z}_i{\kappa }_i^{\prime }(0)+\sum _{ij=2}^N{z}_i{z}_j{\kappa }_i(0){\kappa }_j(0)+\sum _{ij=2}^N{z}_i{z}_j{\beta }_{ij}(0)+O\left(|x{|}^3\right)\\ g_{1i}(x)=\sum _{j=2}^N{z}_j{\tau }_j^i(0)+t\sum _{j=2}^N{z}_j{\left({\tau }_j^i\right)}^{\prime }(0)+O\left(|x{|}^3\right)\\ g_{ij}(x)={\delta }_{ij},\end{array}$$

$${\beta }_{ij}(t):=\sum _{l=2}^N{\tau }_i^l(t){\tau }_j^l(t).$$

$${\partial }_{t}F(x)={\gamma }^{\prime }(t)+\sum _{j=2}^N{z}_j{E}_j^{\prime }(t)\text{and}{\partial }_{z_i}F(x)=E_i(t).$$

$$g_{ij}(x)=E_i(t)\cdot E_j(t)={\delta }_{ij}.$$

$$g_{1i}(x)=\sum _{l=2}^N{z}_l{E}_l^{\prime }(t)\cdot E_i(t)=\sum _{j=2}^N{z}_j{\tau }_j^i(t)$$

$$g_{11}(x)={\partial }_{t}F(x)\cdot {\partial }_{t}F(x)=1+2\sum _{i=2}^N{z}_i{\kappa }_i(t)+\sum _{ij=2}^N{z}_i{z}_j{\kappa }_i(t){\kappa }_j(t)+\sum _{ij=2}^N{z}_i{z}_j\left(\sum _{l=2}^N{\tau }_i^l(t){\tau }_j^l(t)\right).$$

$${\kappa }_i(t)={\kappa }_i(0)+t{\kappa }_i^{\prime }(0)+O\left(t^2\right)\text{and}{\tau }_i^k(t)={\tau }_i^k(0)+t{\left({\tau }_i^k\right)}^{\prime }(0)+O\left(t^2\right).$$

$$\sqrt{|g|}(x)=1+\sum _{i=2}^N{z}_i{\kappa }_i(0)+t\sum _{i=2}^N{z}_i{\kappa }_i^{\prime }(0)+\frac{1}{2}\sum _{ij=2}^N{z}_i{z}_j{\kappa }_i(0){\kappa }_j(0)+O\left(|x{|}^3\right),$$

$$\{\begin{array}{l}{g}^{11}(x)=1-2\sum _{i=2}^N{z}_i{\kappa }_i(0)-2t\sum _{i=2}^N{z}_i{\kappa }_i^{\prime }(0)+3\sum _{ij=2}^N{z}_i{z}_j{\kappa }_i(0){\kappa }_j(0)+O\left(|x{|}^3\right)\\ g^{i1}(x)=-\sum _{j=2}^N{z}_j{\tau }_j^i(0)-t\sum _{j=2}^N{z}_j{\left({\tau }_j^i\right)}^{\prime }(0)+2\sum _{j=2}^N{z}_l{z}_j{\kappa }_l(0){\tau }_j^i(0)+O\left(|x{|}^3\right)\\ g^{ij}(x)={\delta }_{ij}+\sum _{lm=2}^N{z}_l{z}_m{\tau }_l^j(0){\tau }_m^i(0)+O\left(|x{|}^3\right).\end{array}$$

$$g(x)=id+H(x),$$

$$\{\begin{array}{l}{H}_{11}(x)=2\sum _{i=2}^N{z}_i{\kappa }_i(0)+2t\sum _{i=2}^N{z}_i{\kappa }_i^{\prime }(0)+\sum _{ij=2}^N{z}_i{z}_j{\kappa }_i(0){\kappa }_j(0)+\sum _{ij=2}^N{z}_i{z}_j{\beta }_{ij}(0)+O\left(|x{|}^3\right)\\ H_{1i}(x)=\sum _{j=2}^N{z}_i{\tau }_j^i(0)+O\left(|x{|}^2\right)\\ H_{ij}(x)=0.\end{array}$$

$$\sqrt{|g|}=\sqrt{\det \left(I+H\right)}=1+\frac{trH}{2}+\frac{{\left(trH\right)}^2}{4}-\frac{tr(H^2)}{4}+O\left(|H{|}^3\right).$$

$$\frac{trH}{2}=\sum _{i=2}^N{z}_i{\kappa }_i(0)+t\sum _{i=2}^N{z}_i{\kappa }_i^{\prime }(0)+\frac{1}{2}\sum _{ij=2}^N{z}_i{z}_j{\kappa }_i(0){\kappa }_j(0)+\frac{1}{2}\sum _{ij=2}^N{z}_i{z}_j{\beta }_{ij}(0)+O\left(|x{|}^3\right),$$

$$\frac{{\left(trH\right)}^2}{4}=\sum _{ij=2}^N{z}_i{z}_j{\kappa }_i(0){\kappa }_j(0)+O\left(|x{|}^3\right).$$

$$tr(H^2)(x)=\sum _{\alpha =1}^N{\left(H^2(x)\right)}_{\alpha \alpha }=\sum _{\alpha \beta =1}^N{H}_{\alpha \beta }(x)H_{\beta \alpha }(x)=\sum _{\alpha \beta =1}^N{H}_{\alpha \beta }^2(x)=H_{11}^2(x)+2\sum _{i=2}^N{H}_{i1}^2(x),$$

Full text

[TABLE]

where $2^{*}_{\sigma}:=\frac{2(N-\sigma)}{N-2}$, $\sigma\in(0,2)$, and $\rho_{\Gamma}$ is the distance function to $\Gamma$. For $N\geq 4$, we find a sufficient condition, given by the local geometry of the curve, for the existence of a ground-state solution. In the case $N=3$, we obtain existence of ground-state solution provided the trace of the regular part of the Green of $-\Delta+h$ is positive at a point of the curve.

1. Introduction

For $N\geq 3$, $0\leq k\leq N-1$ and $\sigma\in[0,2)$, we consider the Hardy-Sobolev inequality

[TABLE]

where $x=(t,z)\in\mathbb{R}^{k}\times\mathbb{R}^{N-k}$, $C=C(N,\sigma,k)>0$ and $2^{*}_{\sigma}:=\frac{2(N-\sigma)}{N-2}$. Here the Sobolev space $\mathcal{D}^{1,2}(\mathbb{R}^{N})$ is given by the completion of $C^{\infty}_{c}(\mathbb{R}^{N})$ with respect to the norm $v\longmapsto\left(\int_{\mathbb{R}^{N}}|\nabla v|^{2}dx\right)^{1/2}.$ Inequality (1.1) interpolates between cylindrical Hardy inequality, which corresponds to the case $\sigma=2$ and $k\not=N-2$, and the Sobolev inequality which is the case $\sigma=0$. Moreover it is invariant under scaling on $\mathbb{R}^{N}$ and by translations in the $t$-direction. It is well known that in the case of Hardy inequality, $\sigma=2$ and $k\not=N-2$, there is no positive constant $C$ and $v\in{\mathcal{D}}^{1,2}(\mathbb{R}^{N})$ for which equality holds in (1.1). For $\sigma\in[0,2)$, the best positive constant $C$ in (1.1) is

[TABLE]

In the case $\sigma=0$, $S_{N,0}$ is achieved by the standard bubble $(1+|x|^{2})^{\frac{2-N}{2}}$, which is unique up to scaling and translations, see e.g. Aubin [23] and Talenti [1]. For $k=0$, (1.1) is a particular case of the Caffarelli-Kohn-Nirenberg inequality, see [6]. In this case, Lieb showed in [20] that the function $(1+|x|^{2-\sigma})^{\frac{2-N}{2-\sigma}}$ achieves $S_{N,\sigma}$. When $k=N-1$, Musina proved in [21] that the support of the minimizer is contained in a half-space. Therefore (1.1) becomes the Hardy-Sobolev inequality with singularity all the boundary of the halfspace.

For $1\leq k\leq N-2$ and $\sigma\in(0,2)$, Badiale and Tarentello proved the existence of a minimizer $w$ for (1.2) in their paper [3], where they were motivated by questions from astrophysics. Moreover Mancini, Fabri and Sandeep shwoed decay and symmetry properties of $w$ in [10]. In particular, they prove that $w(t,z)=\theta(|t|,|z|)$, for some positive function $\theta$. An interesting classification result was also derived in [10] when $\sigma=1$, that every minimizer is of the form $((1+|z|)^{2}+|t|^{2})^{\frac{2-N}{2}}$, up to scaling in $\mathbb{R}^{N}$ and translations in the $t$-direction.

Since in this paper we are interested with Hardy-Sobolev inequality with weight singular at a given curve, our asymptotic energy level is given by $S_{N,\sigma}$ with $k=1$ and $\sigma\in(0,2)$.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$, $N\geq 3$, and $h$ a continuous function on $\Omega$. Let $\Gamma\subset\Omega$ be a smooth closed curve. In this paper, we are concerned with the existence of minimizers for the infinimum

[TABLE]

where $\sigma\in[0,2]$, $\displaystyle 2^{*}_{\sigma}:=\frac{2(N-\sigma)}{N-2}$ and $\rho_{\Gamma}(x):=\textrm{dist}(x,\Gamma)$. Here and in the following, we assume that $-\Delta+h$ defines a coercive bilinear form on $H^{1}_{0}(\Omega)$ —which is a necessary condition for the existence of minimmizer for $\mu_{h}(\Omega,\Gamma)$. We are interested with the effect of the geometry and/or the location of the curve $\Gamma$ on the existence of minimmizer for $\mu_{h}(\Omega,\Gamma)$.

We not that for $\sigma=0$, (1.3) reduces to the famous Brezis-Nirenberg problem [4]. In this case, for $N\geq 4$ it is enough that $h(y_{0})<0$ to get a minimizer, whereas for $N=3$, the problem is no more local and existence of minimizers is guaranteed by the positiveness of a certain mass —the trace of the regular part of the Green function of the operator $-\Delta+h$ with zero Dirichlet data, see Druet [9]. For $\sigma=2$, the problem reduces to a linear eigenvalue problem with Hardy potential, existence and nonexistence results were obtained by the second author in [25].

Here, we deal with the case $\sigma\in(0,2)$. Our results exhibit similar local/global phenomenon as in [4] and [9], with the additional property that for $N\geq 4$, the curvature of the curve at a point $y_{0}$ tells how much $h(y_{0})$ should be negative, while positive mass at a point $y_{0}\in\Gamma$ is enough in dimension $N=3$.

Our first main result is the following

Theorem 1.1**.**

Let $N\geq 4$, $\sigma\in(0,2)$ and $\Omega$ be a bounded domain of $\mathbb{R}^{N}$. Consider $\Gamma$ a smooth closed curve contained in $\Omega$. Let $h$ be a continuous function such that the linear operator $-\Delta+h$ is coercive. Then there exists a positive constant $C_{N,\sigma}$, only depending on $N$ and $\sigma$ with the property that if there exists $y_{0}\in\Gamma$ such that

[TABLE]

then $\mu_{h}\left(\Omega,\Gamma\right)<S_{N,\sigma},$ and $\mu_{h}\left(\Omega,\Gamma\right)$ is achieved by a positive function. Here $\kappa:\Gamma\to\mathbb{R}^{N}$ is the curvature vector of $\Gamma$.

Inequality (1.4) in Theorem 1.1 shows that the sign of the directional curvatures of $\Gamma$ is not important but the size of the curvature $\kappa$ at a point is.

For the explicit value of $C_{N,\sigma}$ appearing in (1.4), we refer the reader to Proposition 4.2 below. It is given by weighted integrals involving partial derivatives of $w$, a minimizer for $S_{N,\sigma}$. In the case $N=4$, we have $C_{N,\sigma}=\frac{3}{2}$.

We now give a consequence of Theorem 1.4 in the case where $h\equiv\lambda$ a constant function. We denote by $\lambda_{1}(\Omega)>0$ the first Dirichlet eigenvalue of $-\Delta$ in $\Omega$. It is easy to see that $-\Delta+\lambda$ is coercive for every $\lambda>-\lambda_{1}(\Omega)$. In our next result, we will consider a curve $\Gamma$ with curvature vanishing at a point. This is (trivially) the case when $\Gamma$ contains a segment.

Corollary 1.2**.**

Let $N\geq 4$, $\sigma\in(0,2)$ and $\Omega$ be a bounded domain of $\mathbb{R}^{N}$. Consider $\Gamma$ a smooth closed curve contained in $\Omega$. Suppose that the curvature $\kappa$ of $\Gamma$ vanishes at a point. Then for every $\lambda\in(-\lambda_{1}(\Omega),0)$, we have $\mu_{\lambda}\left(\Omega,\Gamma\right)<S_{N,\sigma},$ and $\mu_{\lambda}\left(\Omega,\Gamma\right)$ is achieved by a positive function.

We observe that if $\Gamma=S^{1}_{R}$ a circle of radius $R>0$ and $h\equiv\lambda\in\mathbb{R}$ then condition (1.4) translates into

[TABLE]

Therefore, provided $-\lambda_{1}(\Omega)<-\frac{C_{N,\sigma}}{R^{2}}$, we have that $\mu_{\lambda}(\Omega,S^{1}_{R})$ is achieved for every $\lambda\in(-\lambda_{1}(\Omega),-\frac{C_{N,\sigma}}{R^{2}})$. One is thus led to find domains for which $-\lambda_{1}(\Omega)<-\frac{C_{N,\sigma}}{R^{2}}$. A particular example is given by the annulus $\Omega_{\varepsilon}=B_{R+\varepsilon}\setminus B_{R-\varepsilon}$, which contains $S^{1}_{R}$ for $\varepsilon>0$. It is well known from e.g. the Faber-Krahn inequality that $\lambda_{1}(\Omega_{\varepsilon})\geq\frac{c(N)}{\varepsilon^{2}}$, so that for sufficiently small $\varepsilon$, one always has $-\lambda_{1}(\Omega_{\varepsilon})<-\frac{C_{N,\sigma}}{R^{2}}$.

We now turn to the $3$-dimensional case. We let $G(x,y)$ be the Dirichlet Green function of the operator $-\Delta+h$, with zero Dirichlet data. It satisfies

[TABLE]

In addition, for $N=3$, there exists a continuous function $\textbf{m}:\Omega\to\mathbb{R}$ and a positive constant $c>0$ such that

[TABLE]

We call the function $\textbf{m}:\Omega\to\mathbb{R}$ the mass of $-\Delta+h$ in $\Omega$. We note that $-\textbf{m}$ is occasionally called the Robin function of $-\Delta+h$ in the literature. We now state our second main result.

Theorem 1.3**.**

Let $\sigma\in(0,2)$ and $\Omega$ be a bounded domain of $\mathbb{R}^{3}$. Consider $\Gamma$ a smooth closed curve contained in $\Omega$. Let $h$ be a continuous function such that the linear operator $-\Delta+h$ is coercive. If $\textbf{m}(y_{0})>0$, for some $y_{0}\in\Gamma$, then $\mu_{h}\left(\Omega,\Gamma\right)<S_{3,\sigma},$ and $\mu_{h}\left(\Omega,\Gamma\right)$ is achieved by a positive function.

Since the mass m is independent on the curve, Theorem 1.3 shows that the location of the curve in the domain $\Omega$ — so to intersect the positive part of m— matters for the existence of solution in general. We note that there are situations in which the mass is a everywhere positive. This is the case four operator $-\Delta+\lambda$, provided $\lambda\in\left(-\lambda_{1}(B_{1}),-\frac{1}{4}\lambda_{1}(B_{1})\right)$, as observed in Brezis-Nirenberg [4]. We therefore have the

Corollary 1.4**.**

Let $B_{1}$ the unit ball of $\mathbb{R}^{3}$ and let $\Gamma$ be any smooth closed curve contained in $B_{1}$. If $\lambda\in\left(-\lambda_{1}(B_{1}),-\frac{1}{4}\lambda_{1}(B_{1})\right)$ then $\mu_{\lambda}\left(\Omega,\Gamma\right)<S_{3,\sigma},$ and $\mu_{\lambda}\left(\Omega,\Gamma\right)$ is achieved by a positive function.

The effect of curvatures in the study of Hardy-Sobolev inequalities have been intensively studied in the recent years. For each of these works, the sign of the curvatures at the point of singularity plays important roles for the existence a solution. The first paper, to our knowledge, being the one of Ghoussoub and Kang [17] who considered the Hardy-Sobolev inequality with singularity at the boundary. For more results in this direction, see the works of Ghoussoub and Robert in [12, 13, 14, 16], Demyanov and Nazarov [8], Chern and Lin [7], Lin and Li [19], the authors and Minlend in [11] and the references there in. We point out that in the pure Hardy-Sobolev case, $\sigma\in(0,2)$, with singularity at the boundary, one has existence of minimizers for every dimension $N\geq 3$ as long as the mean curvature of the boundary is negative at the point singularity, see [15].

The Hardy-Sobolev inequality with interior singularity on Riemannian manifolds have been studied by Jaber [18] and Thiam [24]. Here also the impact of the scalar curvature at the point singularity plays an important role for the existence of minimizers in higher dimensions $N\geq 4$. The paper [18] contains also existence result under positive mass condition for $N=3$.

We expect that the arguments in this paper can be generalized for the case $\Gamma\subset\Omega$ a $k$-dimensional closed submanifold, with $2\leq k\leq N-2$. Here we believe that the norm of the second fundamental from of $\Gamma$ will play a crucial role for the existence of minimizers. An other problem of interest would be the case $\Gamma\subset\partial\Omega$ is a $k$-dimensional submanifold of $\partial\Omega$ with, $1\leq k\leq N-1$. In this situation, we suspect that the sign if the mean curvature of $\partial\Omega$ at a point might influence on the the existence of minimizers. Finally we note that Ghoussoub and Robert in [14] obtained several results for the case $\Gamma$ a subspace of dimension $k\geq 2$, and among other results, if $\Gamma$ intersects $\partial\Omega$ transversely, they obtain existence results under some negativity assumptions on the mean curvature.

The proof of Theorem 1.1 and Theorem 1.3 rely on test function methods. Namely to build appropriate test functions allowing to compare $\mu_{h}(\Omega,\Gamma)$ and $S_{N,\sigma}$. While it always holds that $\mu_{h}(\Omega,\Gamma)\leq S_{N,\sigma}$, our main task is to find a function for which $\mu_{h}(\Omega,\Gamma)<S_{N,\sigma}$. This then allows to recover compactness and thus every minimizing sequence for $\mu_{h}(\Omega,\Gamma)$ converges to a minimizer. Building these approximates solutions requires to have sharp decay estimates of a minimizer $w$ for $S_{N,\sigma}$, see Section 3. In Section 4, we treat the case $N=4$ in the spirit of Aubin [1]. Here we find a continuous familly of test functions $(u_{\varepsilon})_{\varepsilon>0}$ concentrating at a point $y_{0}\in\Gamma$ which yields $\mu_{h}(\Omega,\Gamma)<S_{N,\sigma}$, as $\varepsilon\to 0$, provided (1.4) holds. In Section 5, we consider the case $N=3$, which is more difficult. Here we use the argument of Schoen [22] to build our test function. However we cannot adopt the method in [22] straightforwardly. In fact, in contrast to the case $N\geq 4$, we could only find a descrete family of test function $(\Psi_{\varepsilon_{n}})_{n\in\mathbb{N}}$ that leads to the inequality $\mu_{h}(\Omega,\Gamma)<S_{3,\sigma}$. This is due to the fact that the (flat) ground-state $w$ for $S_{3,\sigma}$, $\sigma\in(0,2)$, is not known explicitly, it is not radially symmetric, it is not smooth, and $S_{3,\sigma}$ is only invariant under translations in the $t-$direction. As in [22], we use some global test functions. These are similar to the test functions $(u_{\varepsilon_{n}})_{n\in\mathbb{N}}$ in dimension $N\geq 4$ near the concentration point $y_{0}$, but away from it is substituted with the regular part of the Green function $G(x,y_{0})$, which makes appear the mass $\textbf{m}(y_{0})$ in its first order Taylor expansion, see (1.6).

Acknowledgement: This work is supported by the Alexander von Humboldt Foundation and the German Academic Exchange Service (DAAD). Part of the paper was written while the authors visited the Institute of Mathematics of the Goethe-University Frankfurt. They wish to thank the institute for its hospitality and the DAAD for funding the visit of E.H.A.T. within the program 57060778. M.M.F. is partially supported by the ERC Advanced Grant 2013 n. 339958 “Complex Patterns for Strongly Interacting Dynamical Systems - COMPAT” and E.H.A.T is partially supported by the AIMS-NEI Small Research Grant.

2. Geometric Preliminaries

Let $\Gamma\subset\mathbb{R}^{N}$ be a smooth closed curve. Let $(E_{1};\dots;E_{N})$ be an orthonormal basis of $\mathbb{R}^{N}$. For $y_{0}\in\Gamma$ and $r>0$ small, we consider the curve $\gamma:\left(-r,r\right)\to\Gamma$, parameterized by arclength such that $\gamma(0)=y_{0}$. Up to a translation and a rotation, we may assume that $\gamma^{\prime}(0)=E_{1}$. We choose a smooth orthonormal frame field $\left(E_{2}(t);...;E_{N}(t)\right)$ on the normal bundle of $\Gamma$ such that $\left(\gamma^{\prime}(t);E_{2}(t);...;E_{N}(t)\right)$ is an oriented basis of $\mathbb{R}^{N}$ for every $t\in(-r,r)$, with $E_{i}(0)=E_{i}$.

We fix the following notation, that will be used a lot in the paper,

[TABLE]

where $B_{\mathbb{R}^{k}}(0,r)$ denotes the ball in $\mathbb{R}^{k}$ with radius $r$ centered at the origin. Provided $r>0$ small, the map $F_{y_{0}}:Q_{r}\to\Omega$, given by

[TABLE]

is smooth and parameterizes a neighborhood of $y_{0}=F_{y_{0}}(0,0)$. We consider $\rho_{\Gamma}:\Gamma\to\mathbb{R}$ the distance function to the curve given by

[TABLE]

In the above coordinates, we have

[TABLE]

Clearly, for every $t\in(-r,r)$ and $i=2,\dots N$, there are real numbers $\kappa_{i}(t)$ and $\tau^{j}_{i}(t)$ such that

[TABLE]

The quantity $\kappa_{i}(t)$ is the curvature in the $E_{i}(t)$-direction while $\tau_{i}^{j}(t)$ is the torsion from the osculating plane spanned by $\{\gamma^{\prime}(t);E_{j}(t)\}$ in the direction $E_{j}$. We note that provided $r>0$ small, $\kappa_{i}$ and $\tau_{i}^{j}$ are smooth functions on $(-r,r)$. Moreover, it is easy to see that

[TABLE]

Next, we derive the expansion of the metric induced by the parameterization $F_{y_{0}}$ defined above. For $x=(t,z)\in Q_{r}$, we define

[TABLE]

We have the following result.

Lemma 2.1**.**

There exits $r>0$, only depending on $\Gamma$ and $N$, such that for ever $x=(t,z)\in Q_{r}$

[TABLE]

where

[TABLE]

Proof.

To alleviate the notations, we will write $F=F_{y_{0}}$. We have

[TABLE]

Therefore

[TABLE]

By (2.2) and (2.5), we have

[TABLE]

and

[TABLE]

By Taylor expansions, we get

[TABLE]

Using these identities in (2.8) and (2.7), we get (2.4), thanks to (2.6). This ends the proof of the lemma. ∎

As a consequence we have the following result.

Lemma 2.2**.**

There exists $r>0$ only depending on $\Gamma$ and $N$, such that for every $x\in Q_{r}$, we have

[TABLE]

where $|g|$ stands for the determinant of $g$. Moreover $g^{-1}(x)$, the matrix inverse of $g(x)$, has components given by

[TABLE]

Proof.

We write

[TABLE]

where $id$ denotes the identity matrix on $\mathbb{R}^{N}$ and $H$ is a symmetric matrix with components $H_{\alpha\beta}$, for $\alpha,\beta=1,\dots,N$, given by

[TABLE]

We recall that as $|H|\to 0$,

[TABLE]

Now by (2.11), as $|x|\to 0$, we have

[TABLE]

so that

[TABLE]

Moreover, from (2.11), we deduce that

[TABLE]

so that

[TABLE]

Therefore plugging the expression from (2.13), (2.14) and (2.15) in (2.12), we get

[TABLE]

The proof of (2.9) is thus finished.

By Lemma 2.1 we can write

[TABLE]

where $A$ and $B$ are symmetric matrix with components $(A_{\alpha\beta})$ and $(A_{\alpha\beta})$, $\alpha,\beta=1,\dots,N$, given respectively by

[TABLE]

and

[TABLE]

We observe that, as $|x|\to 0$,

[TABLE]

We then deduce from (2.16) and (2.17) that

[TABLE]

[TABLE]

and

[TABLE]

This ends the proof. ∎

3. Some preliminary results

We consider the best constant for the cylindrical Hardy-Sobolev inequality

[TABLE]

As mentioned in the first section, it is attained by a positive function $w\in{\mathcal{D}}^{1,2}(\mathbb{R}^{N})$, satisfying

[TABLE]

see e.g. [3]. Moreover from [10], we have

[TABLE]

Next we prove further decay properties of $w$ involving its higher derivatives. We start with the following results.

Lemma 3.1**.**

Let $\theta$ be given by (3.2). Then we have the following properties.

• (i)

The function $t\mapsto\theta(t,\rho)$ is of class $C^{\infty}$ with all its derivatives uniformly bounded with respect to $\rho$.

• (ii)

There exists a constant $C>0$ such that for $|(t,\rho)|\leq 1$, we have

[TABLE]

Proof.

For the proof of $(i)$, see [10]. To prove $(ii)$, we first use polar coordinates to deduce that

[TABLE]

Integrating this identity in the $\rho$ variable, we therefore get, for every $\rho>0$,

[TABLE]

Moreover, we have

[TABLE]

By $(i)$ and the fact that $2^{*}_{\sigma}\geq 2$, we obtain

[TABLE]

Now using this in (3.3), we get $|\theta_{\rho\rho}|\leq C\rho^{-\sigma}$, for $|(t,\rho)|\leq 1$. The proof of $(ii)$ is completed. ∎

As a consequence we derive decay estimates of the derivatives of $w$ up to order two.

Corollary 3.2**.**

Let $w$ be a ground state for $S_{N,\sigma}$ then there exist positive constants $C_{1},C_{2}$, only depending on $N$ and $\sigma$, such that

For every $x\in\mathbb{R}^{N}$

[TABLE]

For $|x|=|(t,z)|\leq 1$

[TABLE]

For $|x|=|(t,z)|\geq 1$

[TABLE]

Proof.

For the proof of $(i)$, we refer to [10, Lemma 3.1]. The proof of $(ii)$ is an immediate consequence of Lemma 3.1(ii), recalling that $w(t,z)=\theta(|t|,|z|)$. Now $(iii)$ follows by Kelvin transform, using that the function $v:\mathbb{R}^{N}\to\mathbb{R}$, given by $v(t,z)=v(x)=\theta(|t||x|^{-2},|z||x|^{-2})|x|^{2-N}$ is also a ground state for $S_{N,\sigma}$, thus it satisfies $(ii)$. ∎

We close this section with the following result.

Lemma 3.3**.**

Let $v\in{\mathcal{D}}^{1,2}(\mathbb{R}^{N})$, $N\geq 3,$ satisfy $v(t,z)=\overline{\theta}(|t|,|z|)$, for some some function $\overline{\theta}:\mathbb{R}_{+}\times\mathbb{R}_{+}\to\mathbb{R}$. Then for $0<r<R$, we have

[TABLE]

Proof.

It is easy to see that

[TABLE]

We recall that

[TABLE]

It then follows that

[TABLE]

We first use Lemma 2.2 and (2.3), to get

[TABLE]

Next, we observe that

[TABLE]

where $\Upsilon(|t|,|z|)=\overline{\theta}_{t}(|t|,|z|)\overline{\theta}_{\rho}(|t|,|z|)\frac{1}{|t|}\frac{1}{|z|}.$ In addition, from (2.3), we see that

[TABLE]

Consequently, from (2.9) and (2.10), we obtain

[TABLE]

By (2.9) and (2.10), we have

[TABLE]

Using this, (3) and (3) in (3), we then deduce that

[TABLE]

Now by (2.9) and (2.10), we also have that

[TABLE]

This with (3.9) and (3.5) give the desired result.

∎

4. Existence of minimzers for $\mu_{h}(\Omega,\Gamma)$ in dimension $N\geq 4$

We consider $\Omega$ a bounded domain of $\mathbb{R}^{N}$, $N\geq 3$, and $\Gamma\subset\Omega$ be a smooth closed curve. For $u\in H^{1}_{0}(\Omega)\setminus\{0\}$, we define the ratio

[TABLE]

We will construct a family of test function $(u_{\varepsilon})_{\varepsilon}\in H^{1}_{0}(\Omega)$ and provide an expansion for $J(u_{\varepsilon})$ as $\varepsilon\to 0$. We let $\eta\in\mathcal{C}^{\infty}_{c}\left(F_{y_{0}}\left({Q}_{2r}\right)\right)$ be such that

[TABLE]

For $\varepsilon>0$, we consider $u_{\varepsilon}:\Omega\to\mathbb{R}$ given by

[TABLE]

In particular, for every $x=(t,z)\in\mathbb{R}\times\mathbb{R}^{N-1}$, we have

[TABLE]

It is clear that $u_{\varepsilon}\in H^{1}_{0}(\Omega).$ We have the following

Lemma 4.1**.**

For $J$ given by (4.1) and $u_{\varepsilon}$ given by (4.2), as $\varepsilon\to 0$, we have

[TABLE]

Proof.

To simplify the notations, we will write $F$ in the place of $F_{y_{0}}$. Recalling (4.2), we write

[TABLE]

where $W_{\varepsilon}(y)=w\left(\frac{F^{-1}(y)}{\varepsilon}\right)$. Then

[TABLE]

Integrating by parts, we have

[TABLE]

By the change of variable $y=\frac{F(x)}{\varepsilon}$ and (4.3), we can apply Lemma 3.3, to get

[TABLE]

Using Corollary 3.2, we find that

[TABLE]

By the change of variable $y=\frac{F(x)}{\varepsilon}$, (3.2), (2.1) and (2.9), we get

[TABLE]

Using (3.4), we have

[TABLE]

Hence by Taylor expanding, we get

[TABLE]

Finally, by (4), we conclude that

[TABLE]

We thus get the desired result.

∎

Proposition 4.2**.**

For $N\geq 5$, we define

[TABLE]

and

[TABLE]

Assume that, for some $y_{0}\in\Gamma$, there holds

[TABLE]

Then

[TABLE]

Proof.

We claim that

[TABLE]

To prove this claim, we let $\eta_{\varepsilon}(x)=\eta(\varepsilon x)$. We multiply (3.1) by $|z|^{2}\eta_{\varepsilon}w$ and integrate by parts to get

[TABLE]

We then deduce that

[TABLE]

Thanks to Corollary 3.2, we get (4.6) as claimed.

Next, by the continuity of $h$, for $\delta>0$, we can find $r_{\delta}>0$ such that

[TABLE]

**Case $N\geq 5.$

**Using (4.6) and (4.7) in (4.1), we obtain, for every $r\in(0,r_{\delta})$

[TABLE]

where we have used Corollary 3.2 to get the estimates

[TABLE]

It follows that, for every $r\in(0,r_{\delta})$,

[TABLE]

Suppose now that

[TABLE]

We can thus choose respectively $\delta>0$ small and $\varepsilon>0$ small so that $J(u_{\varepsilon})<S_{N,\sigma}$. Hence we get

[TABLE]

**Case $N=4.$

**From (4.1) and (4.7), we estimate, for every $r\in(0,r_{\delta})$

[TABLE]

This with (4.6) yield

[TABLE]

Since, by (3.4),

[TABLE]

we therefore have

[TABLE]

for some positive constant $C$ independent on $\varepsilon$. By (3.4), we have that

[TABLE]

so that

[TABLE]

Using polar coordinates and a change of variable, for $R>0$, we have

[TABLE]

Therefore, there exist numerical constants $c,\overline{c}>0$ such that for every $\varepsilon>0$ small, we have

[TABLE]

Now we assume that $\frac{3|\kappa(y_{0})|^{2}}{2}+h(y_{0})<0$. Therefore by Lemma 4.1 and (4.9), we get

[TABLE]

Then choosing $\delta>0$ small and $\varepsilon$ small, respectively, we deduce that $\mu_{h}\left(\Omega,\Gamma\right)\leq J(u_{\varepsilon})<S_{4,\sigma}$. This ends the proof of the proposition. ∎

Proof of Theorem 1.1 (completed).

Thanks to Proposition 4.2, we can apply Proposition 6.2 to get the result with $C_{N,\sigma}=\frac{A_{N,\sigma}}{B_{N,\sigma}}$ for $N\geq 5$ and $C_{4,\sigma}=\frac{3}{2}$.

∎

5. Existence of minimizer for $\mu_{h}(\Omega,\Gamma)$ in dimension three

We consider the function

[TABLE]

which satisfies

[TABLE]

We denote by $G$ the solution to the equation

[TABLE]

and satisfying

[TABLE]

We note that $G$ is proportional to the Green function of $-\Delta+h$ with zero Dirichlet data.

We let $\chi\in C^{\infty}_{c}(-2,2)$ with $\chi\equiv 1$ on $(-1,1)$ and $0\leq\chi<1$. For $r>0$, we consider the cylindrical symmetric cut-off function

[TABLE]

It is clear that

[TABLE]

For $y_{0}\in\Omega$, we let $r_{0}\in(0,1)$ such that

[TABLE]

We define the function $M_{y_{0}}:Q_{2r_{0}}\to\mathbb{R}$ given by

[TABLE]

It follows from (5.3) that $M_{y_{0}}\in L^{\infty}(Q_{r_{0}})$. By (5.2) and (5.1),

[TABLE]

whereas ${\mathcal{R}}\in L^{p}(Q_{r_{0}})$ for every $p\in(1,3)$. Hence by elliptic regularity theory, $M_{y_{0}}\in W^{2,p}(Q_{r_{0}/2})$ for every $p\in(1,3)$. Therefore by Morrey’s embdding theorem, we deduce that

[TABLE]

In view of (1.6), the mass of the operator $-\Delta+h$ in $\Omega$ at the point $y_{0}\in\Omega$ is given by

[TABLE]

We recall that the positive ground state solution $w$ satisfies

[TABLE]

where $x=(t,z)\in\mathbb{R}\times\mathbb{R}^{2}$. In addition by (3.4), we have

[TABLE]

The following result will be crucial in the following of this section.

Lemma 5.1**.**

Consider the function $v_{\varepsilon}:\mathbb{R}^{3}\setminus\{0\}\to\mathbb{R}$ given by

[TABLE]

Then there exists a constant $\textbf{c}>0$ and a sequence $(\varepsilon_{n})_{n\in\mathbb{N}}$ (still denoted by $\varepsilon$) such that

[TABLE]

and

[TABLE]

Proof.

By Corollary 3.2, we have that $(v_{\varepsilon})$ is bounded in $C^{2}_{loc}(\mathbb{R}^{3}\setminus\{z=0\})$. Therefore by Arzelá-Ascolli’s theorem $v_{\varepsilon}$ converges to $v$ in $C^{1}_{loc}(\mathbb{R}^{3}\setminus\{z=0\})$. In particular,

[TABLE]

It is plain, from (5.10), that

[TABLE]

By (5.9), we have

[TABLE]

where

[TABLE]

We let $\varphi\in C^{\infty}_{c}\left(\mathbb{R}^{3}\setminus\{0\}\right)$. We multiply (5.13) by $\varphi$ and integrate by parts to get

[TABLE]

By (5.12) and the dominated convergence theorem, we can pass to the limit in the above identity and deduce that

[TABLE]

In particular $v$ is equivalent to a function of class $C^{\infty}\left(\mathbb{R}^{3}\setminus\{0\}\right)$ which is still denoted by $v$. Thanks to (5.12), by Bôcher’s theorem, there exists a constant $\textbf{c}>0$ such that

[TABLE]

The proof of the lemma is thus finished. ∎

We start by recording some useful estimates.

Lemma 5.2**.**

There exists a constant $C>0$ such that for every $\varepsilon,r\in(0,r_{0}/2)$, we have

[TABLE]

[TABLE]

[TABLE]

*and *

[TABLE]

Proof.

The proof of this lemma is not difficult and uses only the estimates in Corollary 3.2. We therefore skip the details.

∎

5.1. Proof of Theorem 1.3

Given $y_{0}\in\Gamma\subset\Omega\subset\mathbb{R}^{3}$, we let $r_{0}$ as defined in (5.5). For $r\in(0,r_{0}/2)$, we consider $F_{y_{0}}:Q_{r}\to\Omega$ (see Section 2) parameterizing a neighborhood of $y_{0}$ in $\Omega$, with the property that $F_{y_{0}}(0)=y_{0}$. For $\varepsilon>0$, we consider $u_{\varepsilon}:\Omega\to\mathbb{R}$ given by

[TABLE]

We can now define the test function $\Psi_{\varepsilon}:\Omega\to\mathbb{R}$ by

[TABLE]

It is plain that $\Psi_{\varepsilon}\in H^{1}_{0}(\Omega)$ and

[TABLE]

The main result of this section is contained in the following

Proposition 5.3**.**

Let $(\varepsilon_{n})_{n\in\mathbb{N}}$ and c be the sequence and the number given by Lemma 5.1. Then there exists $r_{0},n_{0}>0$ such that for every $r\in(0,r_{0})$ and $n\geq n_{0}$

[TABLE]

for some numbers $\mathcal{O}_{r}(\varepsilon_{n})$ satisfying

[TABLE]

The proof of this proposition will be separated into two steps given by Lemma 5.4 and Lemma 5.5 below. To alleviate the notations, we will write $\varepsilon$ instead of $\varepsilon_{n}$ and we will remove the subscript $y_{0}$, by writing $M$ and $F$ in the place of ${M}_{y_{0}}$ and $F_{y_{0}}$ respectively. We define

[TABLE]

where $v_{\varepsilon}(x)=\varepsilon^{-1}w\left(\frac{x}{\varepsilon}\right).$ With these notations, (5.18) becomes

[TABLE]

We first consider the numerator in (5.3).

Lemma 5.4**.**

We have

[TABLE]

where $\nu$ is the unit outer normal of ${Q}_{r}$.

Proof.

Recalling (5.19), direct computations give

[TABLE]

By (5.4), $\eta_{r}v_{\varepsilon}=\eta_{r}\varepsilon^{-1}w(\cdot/\varepsilon)$ is cylindrically symmetric. Therefore by the change variable $y=F(x)$ and using Lemma 3.3, we get

[TABLE]

By computing, we find that

[TABLE]

From this and (5.14) and (5.15), we get

[TABLE]

We replace this in (5.1) to have

[TABLE]

We have the following estimates

[TABLE]

which easily follows from (5.10) and Corollary 3.2. By these estimates, Lemma 2.2 and (5.7) together with the change of variable $y=F(x)$, we have

[TABLE]

This with (5.22), (5.7) and (5.1) give

[TABLE]

Thanks to Lemma 5.1 and (5.23), we can thus use the dominated convergence theorem to deduce that, as $\varepsilon\to 0$,

[TABLE]

Similarly, we easily see that

[TABLE]

This and (5.24), then give

[TABLE]

Since the support of $\Psi_{\varepsilon}$ is contained in ${Q}_{4r}$ while the one of $\eta_{r}$ is in ${Q}_{2r}$, it is easy to deduce from (5.7) that

[TABLE]

and from Lemma 5.2, that

[TABLE]

Therefore by (5.25), we conclude that

[TABLE]

Recall that $G(x+y_{0},y_{0})=\eta_{r}(x){\mathcal{R}}(x)+M(x)$ for ever $x\in{Q}_{2r}$ and that by (5.2),

[TABLE]

Therefore, by integration by parts, we find that

[TABLE]

where $\overline{\nu}$ is the exterior normal vectorfield to ${Q}_{2r}\setminus{Q}_{r}$. Thanks to (5.7), we finally get

[TABLE]

where $\nu$ is the exterior normal vectorfield to ${Q}_{r}$.

Next we make the expansion of $\int_{F\left({Q}_{r}\right)}|\nabla\Psi_{\varepsilon}|^{2}dy$ for $r$ and $\varepsilon$ small. First, we observe that, by Lemma 5.2 and (5.7), we have

[TABLE]

By integration by parts and using (5.17), we deduce that

[TABLE]

Now (5.23), (5.11) and the dominated convergence theorem yield, for fixed $r>0$ and $\varepsilon\to 0$,

[TABLE]

Moreover (5.16) implies that

[TABLE]

From this together with (5.1) and (5.1), we obtain

[TABLE]

Combining this with (5.26), we then have

[TABLE]

Since (recalling (5.8)) $M(y)=M(0)+O(r)=\textbf{m}(y_{0})+O(r)$ in ${Q}_{2r}$, we get the claimed result in the statement of the lemma. ∎

The following result together with the previous lemma provides the proof of Proposition5.3.

Lemma 5.5**.**

We have

[TABLE]

Proof.

Since $2^{*}_{\sigma}>2$, there exists a positive constant $C(\sigma)$ such that

[TABLE]

As a consequence, we obtain

[TABLE]

By Hölder’s inequality and (2.9), we have

[TABLE]

recalling that $\|w\|_{L^{2^{*}_{\sigma}}(\mathbb{R}^{3};|z|^{-\sigma})}=1$. Furthermore, since $2^{*}_{\sigma}>2$, by (5.7), we easily get

[TABLE]

Moreover by change of variables and (5.17), we also have

[TABLE]

By this, (5.1), (5.32) and (5.1), it results

[TABLE]

We define $B_{\varepsilon}(x):=M(\varepsilon x)\sqrt{|g_{\varepsilon}|}(x)=M(\varepsilon x)\sqrt{|g|}(\varepsilon x)$. Then by the change of variable $y=\frac{F(x)}{\varepsilon}$ in the above identity and recalling (2.9), then by oddness, we have

[TABLE]

Therefore by (5.17) we then have

[TABLE]

Multiply (5.9) by $B_{\varepsilon}\in{\mathcal{C}}^{1}(\overline{Q_{r}})$ and integrate by parts to get

[TABLE]

Since $|\nabla B_{\varepsilon}|\leq C\varepsilon$, by Lemma 5.1 and (5.7), we then have

[TABLE]

Consequently

[TABLE]

on the one hand. On the other hand by Lemma 5.1, (5.7) and the dominated convergence theorem, we get

[TABLE]

so that

[TABLE]

It then follows from (5.33) that

[TABLE]

Since ${M}(0)=\textbf{m}(y_{0})$, see (5.8), the proof of the lemma is thus finished. ∎

Proof of Proposition 5.3 (completed).

By Lemma 5.4 and Lemma 5.5, we have

[TABLE]

Finally, recalling that ${\mathcal{R}}(x)=\frac{1}{|x|}$, we can compute

[TABLE]

From this and (5.34), we then have

[TABLE]

which finishes the proof. ∎

As a consequence of Proposition 5.3, we can now complete the

Proof of Theorem 1.3 (completed).

The proof follows from Lemma 5.3 that if $\textbf{m}(y_{0})>0$ for some $y_{0}\in\Gamma$ then $\mu_{h}(\Omega,\Gamma)<S_{3,\sigma}$. The fact that $\mu_{h}(\Omega,\Gamma)$ is attained by a positive function is an immediate consequence of Proposition 6.2 belwo. ∎

6. Appendix: Existence of minimizer for $\mu_{h}\left(\Omega,\Gamma\right)$

Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$, $N\geq 3$, and $h$ a continuous function on $\Omega$. Let $\Gamma$ be a smooth closed curve $\Gamma$ contained in $\Omega$. We consider

[TABLE]

We also recall that

[TABLE]

with $x=(t,z)\in\mathbb{R}\times\mathbb{R}^{N-1}$. Our aim in this section is to show that if $\mu_{h}\left(\Omega,\Gamma\right)<S_{N,\sigma}$ then the best constant $\mu_{h}\left(\Omega,\Gamma\right)$ is achieved. The argument of proof is standard. However, for sake of completeness, we add the proof. We start with the following

Lemma 6.1**.**

Let $\Omega$ be an open subset of $\mathbb{R}^{N}$, with $N\geq 3$, and let $\Gamma\subset\Omega$ be a closed smooth curve. Then for every $r>0$, there exist positive constants $c_{r}>0$, only depending on $\Omega,\Gamma,N,\sigma$ and $r$, such that for every $u\in H^{1}_{0}(\Omega)$

[TABLE]

where $2^{*}_{\sigma}=\frac{2(N-\sigma)}{N-2}$ and $\sigma\in(0,2)$.

Proof.

We let $r>0$ small. We can cover a tubular neighborhood of $\Gamma$ by a finite number of sets $\left(T^{y_{i}}_{r}\right)_{1\leq i\leq m}$ given by

[TABLE]

We refer to Section 2 for the parameterization $F_{y_{i}}:Q_{r}\to\Omega$. See e.g. [2, Section 2.27], there exists $\left(\varphi_{i}\right)_{1\leq i\leq m}$ a partition of unity subordinated to this covering such that

[TABLE]

for some constant $K>0$. We define

[TABLE]

Recall that that $\rho_{\Gamma}\geq C>0$ on $\Omega\setminus U$, for some positive constant $C>0$. Therefore, since $\frac{2}{2^{*}_{\sigma}}<1$, by (6.4) we get

[TABLE]

By change of variables and Lemma 2.2, we have

[TABLE]

In addition the Hardy-Sobolev inequality (1.2) yields

[TABLE]

Therefore by change of variables and Lemma 2.2, we get

[TABLE]

Applying Young’s inequality using (6.3) and (6.4), we find that

[TABLE]

Summing for $i$ equal $1$ to $m$, we get

[TABLE]

This together with (6) give

[TABLE]

Since $\varepsilon$ and $r$ can be chosen arbitrarily small, we get the desired result. ∎

We can now prove the following existence result.

Proposition 6.2**.**

Consider $\mu_{h}(\Omega,\Gamma)$ and $S_{N,\sigma}$ given by (6.1) and (6.2) respectively. Suppose that

[TABLE]

Then $\mu_{h}\left(\Omega,\Gamma\right)$ is achieved by a positive function.

Proof.

Let $\left(u_{n}\right)_{n\in\mathbb{N}}$ be a minimizing sequence for $\mu_{h}\left(\Omega,\Gamma\right)$ normalized so that

[TABLE]

By coercivity of $-\Delta+h$, the sequence $\left(u_{n}\right)_{n\in\mathbb{N}}$ is bounded in $H^{1}_{0}(\Omega)$ and thus , up to a subsequence,

[TABLE]

and

[TABLE]

The weak convergence in $H^{1}_{0}(\Omega)$ implies that

[TABLE]

By Brezis-Lieb lemma [5] and the strong convergence in the Lebesgue spaces $L^{p}(\Omega)$, we have

[TABLE]

By Lemma 6.1, (6.8) —note that $2^{*}_{\sigma}<2^{*}_{0}$ , we then deduce that

[TABLE]

Using (6.9), (6.10) and (6.11), we have

[TABLE]

By the concavity of the map $t\mapsto t^{2/2^{*}_{\sigma}}$ on $[0,1]$, we have

[TABLE]

From this, then taking the limits respectively as $n\to+\infty$ and as $r\to 0$ in (6), we find that

[TABLE]

Thanks to (6.6), we then get

[TABLE]

Since by (6.7) and Fatou’s lemma,

[TABLE]

we conclude that

[TABLE]

It then follows from (6.7) that $u_{n}\to u$ in $L^{2^{*}_{\sigma}}(\Omega;\rho_{\Gamma}^{-\sigma})$ and thus $u_{n}\to u$ in $H^{1}_{0}(\Omega)$. Therefore $u$ is a minimizer for $\mu_{h}(\Omega,\Gamma)$. Since $|u|$ is also a minimizer for $\mu_{h}(\Omega,\Gamma)$, we may assume that $u\gneqq 0$. Therefore $u>0$ by the maximum principle. ∎