arXiv subjects

Monideep Ghosh

Publications and source records attributed to Monideep Ghosh.

3 recordsLinked to original sources

arXiv2025

Quantitative stability for the conformally invariant Chang-Gui inequality on the exponentiation of functions on the sphere

In this work, we focus on a recent variant of the Trudinger-Moser-Onofri inequality introduced by S. Y. Alice Chang and Changfeng Gui \cite{CG-2023}: \begin{align*} α\int_{\mathbb{S}^2}|\nabla_{\mathbb{S}^2}u|^2 {\rm d}ω+2 \int_{\mathbb{S}^2} u {\rm d}ω-\frac{1}{2}\ln\left[\left(\int_{\mathbb{s}^2}e^{2u}{\rm d}ω\right)^2-\sum_{i=1}^3\left(\int_{\mathbb{s}^2}ω_i e^{2u}{\rm d} ω\right)^2\right] \geq 0 \end{align*} holds on $H^1(\mathbb{S}^2)$ if and only if $α\geq \frac{2}{3}$. In this regime, the infimum is attained only by trivial functions when $α> \frac{2}{3},$ whereas for the critical value $α= \frac{2}{3}$ nontrivial extremals exist, and Chang-Gui further provided a complete classification of such solutions. Building upon their result, we found a nice conformal invariance of the associated functional. Exploiting this invariance, we were able to characterize the full family of extremals in terms of conformal maps of $\mathbb{S}^2$ and, moreover, establish a sharp quantitative stability result in the gradient norm.

math.AP↗

arXiv2025

Existence of Bianchi-Egnell stability extremizer for the Hardy-Sobolev inequality

In this article, we prove the best Bianchi-Egnell constant for the Hardy-Sobolev (HS) inequality \begin{align*} C_{\tiny\mbox{BE}}(γ) := \inf_{u \ \small \mbox{not an optimizer}} \frac{\int_{\mathbb{R}^n} \left(|\nabla u|^2 - \fracγ{|x|^2}u^2\right) \ {\rm d}x - S_γ\|u\|_{L^{2^{\star}}}^2}{\mbox{dist} (u, \ \mbox{set of optimizers})^2}, \end{align*} is attained, extending the result of König [arXiv:2211.14185] for the classical Sobolev inequality (that corresponds to $γ= 0$). One of the main difficulties is that the third eigenspace of the linearized operator may contain only spherical harmonics of degree $1$, and hence, an essential non-vanishing criterion fails [arXiv:2210.08482]. This non-vanishing criterion is indispensable for proving the best Bianchi-Egnell constant $C_{\tiny\mbox{BE}}(γ) < C_{\tiny\mbox{BE}}^{\tiny\mbox{loc}}(γ)$ that prevents a minimizing sequence converging to one of the optimizers. In addition, not being translation invariant, extracting a non-zero weak limit from a minimizing sequence presents difficulties. We found another hidden critical level $C_{\tiny\mbox{BE}}(γ) <1 - \frac{S_γ}{S},$ where $S$ is the best Sobolev constant that plays a significant role in proving the existence of an extremizer. In particular, we show that there exists a $γ_0>0$ such that for $γ\geq γ_0,\ C_{\tiny\mbox{BE}}(γ)$ is attained. Moreover, we remark that there is a region $γ_0 \leq γ< γ_c^{\star},$ where the third eigenspace of the linearized operator contains only spherical harmonics of degree $1.$ Our result improves some of the results in Wei-Wu [arXiv:2308.04667] corresponding to the HS inequality.

math.AP↗

arXiv2025

A note on the log-perturbed Brézis-Nirenberg problem on the hyperbolic space

We consider the log-perturbed Brézis-Nirenberg problem on the hyperbolic space \begin{align*} Δ_{\mathbb{B}^N}u+λu +|u|^{p-1}u+θu \ln u^2 =0, \ \ \ \ u \in H^1(\mathbb{B}^N), \ u > 0 \ \mbox{in} \ \mathbb{B}^N, \end{align*} and study the existence vs non-existence results. We show that whenever $θ>0,$ there exists an $H^1$-solution, while for $θ<0$, there does not exist a positive solution in a reasonably general class. Since the perturbation $ u \ln u^2$ changes sign, Pohozaev type identities do not yield any non-existence results. The main contribution of this article is obtaining an "almost" precise lower asymptotic decay estimate on the positive solutions for $θ<0,$ culminating in proving their non-existence assertion.

math.AP↗