Abstract
We introduce a new notion that we call a metric potential capacity associated to a positive potential V and a weak-star topology of a suitable Banach space. We show among other things that if the metric potential capacity of a closed set F is zero and the square root of the potential is locally q-integrable for some \(q\in [1,{+\,\infty }[\), then the usual Sobolev capacity in \(W^{1,q}\) of the set F is zero. Properties of potential capacities are presented and applications to Schrödinger type equations are also made with very singular potentials including the modified Pösh–Teller potential \(V_\alpha (x)=\Big |\sin |x|\Big |^{-\alpha },\ \alpha >0\). We discuss about existence and non existence of a solution.
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This paper is dedicated to my friend Ildefonso DIAZ for his seventieth birthday.
Thank you Ilde for these 30 years of friendships.
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Appendix
Appendix
1.1 Definitions of an Assouad dimension, and Lemma of Aikawa
Let us consider a set \(E\subset \Omega ,\ r>0\) and denote by N(E, r) the minimal number of open balls of radius r centered at a point of E necessary to cover E. Then, we have
Definition 5.1
Assouad dimension of a set K [2, 22] Let \(K\subset \Omega \). The (upper) Assouad dimension of K is:
To fix an idea of such dimension, if K is a smooth sub-manifold whose Hausdorff dimension is \(\lambda \in [0,n]\), then \(\dim _A(K)=\lambda \).
The link between the Assouad dimension and the integrability of the function \(\textrm{dist}\,(y;K)\dot{=}d_K(y)\) is made via Aïkawa condition, which reads as:
Definition 5.2
Aïkawa’s condition Let K be a closed set of \(\mathrm{I\!R}^n\) and define:
Lemma 5.1
(See [21, 22]) Let K be a closed non void set. Then
-
(1)
\(\dim _A(K)=\inf \mathcal A(K)\le n.\)
-
(2)
If s is such that \(\dim _A(K)<s\) or \(s\ge n\) then \(s\in \mathcal A(K)\).
-
(3)
If \(s\in \mathcal A(K)\) with \(0<s<n\) then there is \(0<s'<s\) such that \(s'\in \mathcal A(K)\) in particular \(\dim _A(K)<s\).
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(4)
Assume that K is a compact subset of \(\mathrm{I\!R}^n\) and let \(s\ge 0\). Then \(s\in \mathcal A(K)\) if and only if for some \(0<R\le {+\,\infty }\), there exists a constant \(c>0\) such that the inequality in \(\mathcal A(K) \) holds for \(x\in K,\ 0<r<R\).
1.2 Definition of an Hausdorff measure
Definition 5.3
(See [26, 32]) Let \(0\le s<{+\,\infty },\ 0<\delta \le {+\,\infty }\), a compact \(K\subset \mathrm{I\!R}^n\).
Then
where \(\omega _s=\pi ^{\frac{s}{2}}\left[ \Gamma \left( \dfrac{s}{2}+1\right) \right] ^{-1}.\)
The Hausdorff measure of dimension \(s\in \mathrm{I\!R}_+\) of K is
It is shown (see for instance [26]) that \(\mathcal H^s(K)=0\) if and only if \(\mathcal H^s_\infty (K)=0\).
So the Hausdorff dimension of a compact \(K\subset \mathrm{I\!R}^n\) is
If \(s=n\) the \(meas(K)=\mathcal H^n(K)\).
1.3 Approximation Lemma of a distance function
We need the following Lemma whose proof is given in [20, 29, 32].
Lemma 5.2
Let \(A\subset \mathrm{I\!R}^n\) be closed and for \(x\in \mathrm{I\!R}^n\) let \(d(x)=d(x;A)\) denote the distance from x to A. Let
Then there is a function \(\rho \in C^\infty (U-A)\) and a positive number \(M=M(n)\) such that
In particular, the result holds if \(A=\partial \Omega \) boundary of an open bounded set \(\Omega \) , in this case
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Rakotoson, J.M. A metric potential capacity: some qualitative properties of Schrödinger’s equations with a non negative potential. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116, 112 (2022). https://doi.org/10.1007/s13398-022-01254-0
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DOI: https://doi.org/10.1007/s13398-022-01254-0