Some quasilinear minimization problems involving Hardy potentials

Abstract: In this note, we consider several quasiliniear minimization problems involving various Hardy type potentials for functions in $W<sup>{1,p}(Ω)$ with weighted mean zero. We prove that the strict inequality for the infimum yields the existence of a minimizer.

• The paper proves positivity of weighted mean-zero Hardy constants and attainment below the universal Hardy threshold for point and submanifold singularities, as well as critical logarithmic weights.
• The authors combine Cherrier-type Hardy refinements with the Valeriola–Willem gradient Brezis–Lieb theorem to recover compactness and establish strong convergence of minimizing sequences.
• The strict gap condition is verified for selected balls using coordinate-function test cases, while the method remains limited to subcritical dimensions, admissible geometries, and domains without a general gap criterion.

Setting and motivation

The paper studies quasilinear minimization problems of Hardy type on a smooth bounded domain $\Omega \subset \mathbb{R}^N$ with $0 \in \Omega$. The classical $W^{1,p}_0$-Hardy inequality asserts that the best constant

$H_p = \left(\frac{N-p}{p}\right)^p$

is positive, independent of $\Omega$, and never attained in $W^{1,p}_0(\Omega)$ for $1 < p < N$. This non-attainability is a manifestation of the lack of compactness in the embedding $$W^{1,p}_0(\Omega) \hookrightarrow L^p(\Omega, |x|^{-p}\,dx)$$. The authors' strategy, following Chabrowski–Peral–Ruf for the case $p=2$ (2606.15849), is to restore compactness by restricting to the closed subspace

$$X_p = \left\{ u \in W^{1,p}(\Omega) : \int_\Omega \frac{u}{|x|^p}\,dx = 0 \right\},$$

i.e., functions with weighted mean zero, and minimizing the Rayleigh quotient over $0 \in \Omega$0:

$0 \in \Omega$1

The same program is carried out in two further settings: the geometric Hardy inequality of Barbatis–Filippas–Tertikas with potential $0 \in \Omega$2 attached to a submanifold $0 \in \Omega$3 of codimension $0 \in \Omega$4 satisfying the condition $0 \in \Omega$5 weakly, and the critical logarithmic Hardy inequality for $0 \in \Omega$6, where the weight is $0 \in \Omega$7.

Main results

The three theorems share a common structure. For $0 \in \Omega$8 (with $0 \in \Omega$9 covering the point singularity), $W^{1,p}_0$0; moreover, if the strict inequality

$W^{1,p}_0$1

holds, then the infimum is attained by a non-zero function in $W^{1,p}_0$2. Analogously, for the critical exponent,

$W^{1,p}_0$3

implies attainability of $W^{1,p}_0$4 on $W^{1,p}_0$5. The positivity statements hold unconditionally; only the attainment requires the strict gap below the universal constant. Theorem 2 extends the earlier work of Sano–Takahashi from $W^{1,p}_0$6 to all $W^{1,p}_0$7.

The strictness assumption is essential: it is used precisely at the points where the contradiction arguments would otherwise yield only the trivial inequality $W^{1,p}_0$8. Whether the gap condition holds depends on the geometry of $W^{1,p}_0$9; the paper does not characterize when it fails.

Analytic framework

Two ingredients drive the proofs. First, the authors establish Cherrier-type refinements of the Hardy inequalities valid on all of $H_p = \left(\frac{N-p}{p}\right)^p$0 (not merely $H_p = \left(\frac{N-p}{p}\right)^p$1): for every $H_p = \left(\frac{N-p}{p}\right)^p$2 there exists $H_p = \left(\frac{N-p}{p}\right)^p$3 such that

$H_p = \left(\frac{N-p}{p}\right)^p$4

with an analogous inequality for the critical logarithmic weight. These follow by cutting off with a test function $H_p = \left(\frac{N-p}{p}\right)^p$5 near $H_p = \left(\frac{N-p}{p}\right)^p$6 and applying the elementary inequality $H_p = \left(\frac{N-p}{p}\right)^p$7; the lower-order term absorbs the error away from the singularity.

Second, since the Brezis–Lieb identity for gradients generally fails for weakly convergent sequences in $H_p = \left(\frac{N-p}{p}\right)^p$8 when $H_p = \left(\frac{N-p}{p}\right)^p$9, the authors invoke a theorem of Valeriola–Willem: if a weakly convergent sequence satisfies the scalar condition

$\Omega$0

where $\Omega$1 for $\Omega$2 and $\Omega$3 otherwise, then $\Omega$4 a.e., the Brezis–Lieb identity holds for the gradient, and $\Omega$5 strongly in $\Omega$6 for $\Omega$7.

The core of each proof is verifying this scalar condition for a minimizing Palais–Smale sequence produced by the Ekeland variational principle on the constraint manifold $\Omega$8. Testing the approximate Euler–Lagrange equation against $\Omega$9, where $W^{1,p}_0(\Omega)$0 normalizes $W^{1,p}_0(\Omega)$1 into the constraint subspace, decomposes the expression into four terms; these vanish respectively by strong convergence of the Lagrange multipliers $W^{1,p}_0(\Omega)$2 in the dual space, dominated convergence (using that $W^{1,p}_0(\Omega)$3 when $W^{1,p}_0(\Omega)$4), boundedness of the weighted $W^{1,p}_0(\Omega)$5 norms, and weak convergence of $W^{1,p}_0(\Omega)$6 to zero in $W^{1,p}_0(\Omega)$7. With the Brezis–Lieb identity in hand, two contradiction steps show that the weak limit $W^{1,p}_0(\Omega)$8 is non-trivial and saturates the constraint $W^{1,p}_0(\Omega)$9, hence attains the infimum.

An implication worth noting: because the argument relies on $1 < p < N$0, the supercritical regime $1 < p < N$1 is outside the scope of the method as presented.

Verification of the gap condition

The paper provides concrete sufficient conditions under which the strict inequality holds, using the coordinate function $1 < p < N$2 as a test function.

For balls $1 < p < N$3 and $1 < p < N$4, symmetry gives $1 < p < N$5, and Hölder's inequality yields $1 < p < N$6. Hence the gap condition holds whenever $1 < p < N$7. For $1 < p < N$8, the estimate $1 < p < N$9 suffices, which holds for fixed $$W^{1,p}_0(\Omega) \hookrightarrow L^p(\Omega, |x|^{-p}\,dx)$$0 once $$W^{1,p}_0(\Omega) \hookrightarrow L^p(\Omega, |x|^{-p}\,dx)$$1 — true for large $$W^{1,p}_0(\Omega) \hookrightarrow L^p(\Omega, |x|^{-p}\,dx)$$2 since the ratio diverges as $$W^{1,p}_0(\Omega) \hookrightarrow L^p(\Omega, |x|^{-p}\,dx)$$3.

In the critical case, testing with $$W^{1,p}_0(\Omega) \hookrightarrow L^p(\Omega, |x|^{-p}\,dx)$$4 gives $$W^{1,p}_0(\Omega) \hookrightarrow L^p(\Omega, |x|^{-p}\,dx)$$5 as $$W^{1,p}_0(\Omega) \hookrightarrow L^p(\Omega, |x|^{-p}\,dx)$$6, via an asymptotic expansion of $$W^{1,p}_0(\Omega) \hookrightarrow L^p(\Omega, |x|^{-p}\,dx)$$7 obtained through integration by parts and l'Hôpital's rule. Thus the gap condition is satisfied for "fat" balls with $$W^{1,p}_0(\Omega) \hookrightarrow L^p(\Omega, |x|^{-p}\,dx)$$8 close to $$W^{1,p}_0(\Omega) \hookrightarrow L^p(\Omega, |x|^{-p}\,dx)$$9. These verifications are restricted to balls; no general geometric criterion on $p=2$0 is given.

Limitations and open questions

Several restrictions are inherent to the treatment. The attainment results are conditional on the strict gap $p=2$1, and the paper offers no characterization of domains or manifolds for which the gap fails — nor examples showing non-attainment when equality holds. The sufficient conditions for the gap are established only for balls, leaving the question open for general domains. The manifold hypothesis (K) excludes configurations where $p=2$2 changes sign, and the supercritical range $p=2$3 is not addressed. Finally, whether the minimizers exhibit specific qualitative behavior (regularity up to the singularity, asymptotic profiles analogous to those known in the linear case) is not investigated here.

Conclusion

The paper establishes positivity and conditional attainability of weighted-mean-zero Hardy constants in three quasilinear settings — point singularities, submanifold singularities, and the critical logarithmic weight — by combining Cherrier-type global Hardy inequalities with the Valeriola–Willem Brezis–Lieb machinery for gradients. The contribution is a systematic extension of techniques previously available for $p=2$4 to the full quasilinear range $p=2$5, together with explicit ball-domain regimes in which the required spectral gap provably holds.