arXiv subjects

Asmita Rai

Publications and source records attributed to Asmita Rai.

4 recordsLinked to original sources

arXiv2026

Existence and Asymptotic Behavior of Normalized Ground States for a Weighted Elliptic System with Caffarelli--Kohn--Nirenberg Critical Exponent

In this paper, we study the coupled singular weighted elliptic system $$\left\{ \begin{aligned} -\operatorname{div}(|x|^{-2a}\nabla u)+\lambda_1 \frac{u}{|x|^{2a}}&=\beta p\frac{|v|^q|u|^{p-2}u}{|x|^{b(p+q)}}+\frac{|u|^{2^{\sharp}-2}u}{|x|^{b2^{\sharp}}},\quad\text{in}~\mathbb{R}^N, -\operatorname{div}(|x|^{-2a}\nabla v)+\lambda_2\frac{v}{|x|^{2a}}&=\beta q\frac{|u|^p|v|^{q-2}v}{|x|^{b(p+q)}}+\frac{|v|^{2^\sharp-2}v}{|x|^{b{2^\sharp}}},\quad\text{in}~\mathbb{R}^N, \displaystyle\int_{\mathbb{R}^N}\frac{|u|^2}{|x|^{2a}} dx=\rho_1^2,\quad &\int_{\mathbb{R}^N}\frac{|v|^2}{|x|^{2a}}dx=\rho_2^2. \end{aligned} \right. $$ where $N\ge3$, $\beta\in\mathbb{R}$, $\rho_1,\rho_2>0$, $\max\{0,\tfrac{N-4}{2}\}\le a<\tfrac{N-2}{2}$, $a 1$ and $2 0$ we obtain positive normalized ground states, with positive Lagrange multipliers, in the mass subcritical, mass critical, and mass supercritical regimes, for explicit ranges of $\beta$. In the mass supercritical regime, a threshold $\beta_0\ge0$ appears: a ground state exists for every $\beta>\beta_0$ and does not exist for $0<\beta<\beta_0$. We give conditions on $p,q$ under which $\beta_0=0$, and we prove that $\beta_0>0$ when $p,q\ge2$. Finally, we describe the ground states as $\beta\to0^+$ and as $\beta\to+\infty$: after a dilation, they converge to the ground states of the limit system without critical terms, or one component vanishes and the other concentrates on an extremal of the Caffarelli--Kohn--Nirenberg inequality.

math.AP↗

arXiv2026

Normalized Solutions for a Weighted Laplacian Problem with the Caffarelli-Kohn-Nirenberg Critical Exponent

This article establishes the existence and multiplicity of normalized solutions to the weighted nonlinear Schr\"odinger-type equation governed by the Caffarelli-Kohn-Nirenberg operator, $$ -\text{div}(|x|^{-2a}\nabla u)=\lambda \frac{u}{|x|^{2a}}+\beta\frac{|u|^{q-2}u}{|x|^{bq}} +\frac{|u|^{2^{\sharp}-2}u}{|x|^{b{2^{\sharp}}}}\quad \text{in}~\mathbb{R}^N,$$ $$\int_{\mathbb{R}^N}\frac{|u|^2}{|x|^{2a}}dx=\rho^2,$$ where $\lambda\in \mathbb{R}$, $\beta,~\rho>0$, $0< a<\frac{N-2}{2}$, $a<b<a+1$, $2^{\sharp}:=\frac{2N}{N-2(1+a-b)}$ and $2<q<{2^{\sharp}}$. Through constrained variational techniques, refined estimates on the best constants in the Caffarelli-Kohn-Nirenberg inequalities, and a bespoke concentration-compactness lemma, the study secures mass-subcritical ground states alongside multiple constrained critical points, together with high-energy ground state solutions in the mass-critical and supercritical regimes -- notwithstanding the noncompactness arising from the critical Caffarelli-Kohn-Nirenberg nonlinearity over the unbounded domain.

math.AP↗

arXiv2025

Normalized solution to Kirchhoff-fractional system involving critical Choquard nonlinearity

In this article, we explore the fractional Kirchhoff-Choquard system given by $$ \left\{ \begin{array}{lr} (a+b\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}} u|^2\;dx)(-\Delta)^su=\lambda_1u+(I_{\mu}*|v|^{{2^*_{\mu,s}}})|u|^{{2^*_{\mu,s}}-2}u +\alpha p (I_{\mu}*|v|^{q})|u|^{p-2}u \;\text{in}\;\mathbb{R}^N,\\ (a+b\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}} v|^2\;dx)(-\Delta)^sv=\lambda_2v+ (I_{\mu}*|u|^{{2^*_{\mu,s}}})|v|^{{2^*_{\mu,s}}-2}u +\alpha q(I_{\mu}*|u|^{p})|v|^{q-2}v \;\;\text{in}\;\mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u|^2=d_1^2,\;\;\int_{\mathbb{R}^N}|v|^2=d_2^2. \end{array} \right. $$ where $N> 2s$, $s \in (0,1)$, $\mu \in (0, N)$, $\alpha \in\mathbb{R}$. Here, $I_{\mu}:\mathbb{R}^N \to \mathbb{R}$ denotes the Riesz potential. We denote by $2_{\mu,*}:=\frac{2N-\mu}{N}$ and $\frac{2N-\mu}{N-2s}:={2^*_{\mu,s}}$, the lower and upper Hardy-Littlewood-Sobolev critical exponents, repectively, and assume that $2_{\mu,*} < p,q< {2^*_{\mu,s}}$. Our primary focus is on the existence of normalized solutions for the case $\alpha>0$ in two scenarios: the $L^2$ subcritical case characterized by $22_{\mu,*}<p + q < 4 + \frac{4s-2\mu}{N}$ and $L^2$ supercritical associated with $4+\frac{8s-2\mu}{N}< p + q < 2{2^*_{\mu,s}}$.

math.AP↗

arXiv2025

Normalized solutions for fractional Choquard equation with critical growth on bounded domain

In this work, we establish the multiplicity of positive solutions for the following critical fractional Choquard equation with a perturbation on the star-shaped bounded domain $$ \left\{ \begin{array}{lr} (-\Delta)^s u = \lambda u +\alpha|u|^{p-2}u+ \left( \int\limits_{\Omega} \frac{|u(y)|^{2^{*}_{\mu ,s}}}{|x-y|^ \mu}\, dy\right) |u|^{2^{*}_{\mu ,s}-2}u\; \text{in} \; \Omega,\\ u>0\; \text{in}\; \Omega,\; \\ u = 0\; \text{in} \; \mathbb{R}^{N}\backslash\Omega, \\ \int_{\Omega}|u|^2 dx=d, \end{array} \right. $$ where, $s\in(0,1), N>2s$, $\alpha\in \mathbb{R}$, $d>0$, $2<p<2^*_s:=\frac{2N}{N-2s}$ and $2^{*}_{\mu ,s}:=\frac{2N-\mu}{N-2s}$ represents fractional Hardy-Littlewood-Sobolev critical exponent. Using the minimization technique over an appropriate set and the uniform mountain pass theorem, we prove the existence of first and second solutions, respectively.

math.AP↗