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Divya Goel

Publications and source records attributed to Divya Goel.

At least 19 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.

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arXiv2026

Critical Quasilinear Schr\"odinger Equations on the Heisenberg Group: Existence and Nonexistence

We study the quasilinear Schr\"odinger equation \begin{align*} -\Delta_{\mathbb{H}} u +V(\xi)u-\Delta_{\mathbb{H}} (\left|u\right|^{2\alpha})\left|u\right|^{2\alpha-2} u= \lambda \left|u\right|^{q-2}u + \left|u\right|^{p-2}u \quad \text{ in } \mathbb{H}^N, \end{align*} where $\Delta_{\mathbb{H}}$ is the Kohn Laplacian on the Heisenberg group $\mathbb{H}^N$, $4\alpha \frac12$, and $2\alpha Q^{*}$ is the critical exponent, $Q=2N+2$ being the homogeneous dimension and $Q^{*}=\frac{2Q}{Q-2}$. For $p=2\alpha Q^{*}$ and $\lambda>0$, we obtain a nontrivial solution, assuming that the potential is bounded below by a positive constant and is either asymptotically constant from above or invariant under a discrete subgroup of $\mathbb{H}^N$. In the opposite direction, we prove a Poho\v{z}aev identity for $\Delta_{\mathbb{H}}$ and combine it with the Nehari identity, obtaining a family of identities from which the quasilinear energy disappears exactly at the exponent $2\alpha Q^{*}$. This yields a nonexistence theorem under a monotonicity condition on the potential with respect to anisotropic dilations and shows that no nontrivial solution exists for $\lambda \leq 0$; the sign of the subcritical perturbation determines solvability. Along the way, we show that every weak solution is bounded and decays exponentially in the Kor\'anyi gauge.

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arXiv2026

Normalized Solutions of the $L^2$-Supercritical NLS Equation on Noncompact Metric Graphs with a Short-Range Potential

We are concerned with the existence of normalized solutions to the $L^2$-supercritical nonlinear Schr\"odinger equation on a noncompact metric graph $G$, \[ \begin{cases} -u''+W(x)u+\lambda u=\chi(x)|u|^{p-2}u, & \text{on every edge } e \text{ of } G,\\[2mm] \displaystyle\sum_{e\succ v}u'_e(v)=0, & \text{at every vertex } v\in V, \end{cases} \] under the mass constraint $\int_G |u|^2\,dx=\mu>0$, where $\lambda$ arises as a Lagrange multiplier. Here $p>6$, $\chi$ is the characteristic function of the compact core $\mathcal K$, so that the nonlinearity is localized, and the potential $W$ is bounded, nonnegative and vanishing at infinity along every unbounded edge, with $\int_{1}^{\infty}xW(x)\, dx<\infty$ on at least one of them. For every $\mu>0$ we obtain a positive solution with $\lambda>0$, arising as a constrained critical point at a strictly positive energy level. Our approach combines a uniform mountain-pass geometry for a family of approximating functionals, the monotonicity trick with Morse index type information, and a blow-up analysis ruling out the divergence of the Lagrange multipliers. A key point is the strict positivity of the multiplier, for which we exhibit a potential of inverse-square decay admitting a zero-energy $L^2$ solution on a half-line. To the best of our knowledge, this is the first existence result for normalized solutions of the $L^2$-supercritical NLS equation on a noncompact metric graph in the presence of an external potential.

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arXiv2026

Positive and nodal solutions for a parametric quasilinear $Q$-sub-Laplacian problem with critical exponential growth on the Heisenberg group

In this article, we investigate the following modified quasilinear equation with parameter driven by the $Q$-subLaplacian: \begin{align*} \begin{cases} -\Delta_Q u - \Delta_Q\bigl(|u|^{2\alpha}\bigr)\,|u|^{2\alpha-2} u = \lambda f(\xi,u) & \text{in } \Omega, \\[2mm] u = 0 & \text{on } \partial\Omega, \end{cases} \end{align*} where $\Delta_Q(\cdot):= \mathrm{div}_{\mathbb{H}}\bigl(|\nabla_{\mathbb{H}}(\cdot)|^{Q-2}\nabla_{\mathbb{H}}(\cdot)\bigr)$ denotes the $Q$-subLaplacian on the Heisenberg group $\mathbb{H}^N$, $Q=2N+2$ is the homogeneous dimension, $\Omega \subset \mathbb{H}^N$ is a smooth bounded domain, $\lambda>0$, $\alpha > \frac{1}{2}$, and $f$ has critical or subcritical exponential growth of order $\exp\bigl(\beta|t|^{2\alpha Q/(Q-1)}\bigr)$. We prove three results: the existence of a nontrivial positive weak solution in the critical case for all large $\lambda$, and the existence of a least-energy nodal solution with exactly two nodal domains, under subcritical and critical exponential growth. A change of variables $u=g(v)$ reduces the problem to a quasilinear problem whose energy functional is of class $C^1$; the exponent $2\alpha Q/(Q-1)$ arises from the growth of $g$. We handled the exponential growth using the sharp Moser-Trudinger inequality of Cohn and Lu. The positive solution is obtained by the mountain pass theorem and the nodal solutions by minimization on a nodal Nehari set.

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arXiv2026

Quasilinear Schr\"odinger Critical Problem on the Heisenberg group \(\mathbb{H}^N\)

We study the existence of standing wave solutions for the following quasilinear Schr\"odinger equations with critical growth on the Heisenberg group $$ -\Delta_{\mathbb{H}} u +V(\xi)u-\Delta_{\mathbb{H}} (\left|u\right|^{2\alpha})\left|u\right|^{2\alpha-2} u= \lambda \left|u\right|^{q-2}u + \left|u\right|^{p-2}u \text{ in }\mathbb{H}^N $$ where $\mathbb{H}^N$ is Heisenberg group, $\Delta_{\mathbb{H}}$ is Kohn Laplacian operator, $4\alpha \frac{1}{2}.$ By a suitable nonlinear change of variables, the quasilinear equation is transformed into a semilinear one, allowing the use of variational methods in the Folland--Stein Sobolev space $S^{1,2}(\mathbb{H}^N)$. Applying the mountain pass theorem together with a concentration--compactness argument adapted to the sub-Riemannian framework, we establish the existence of a nontrivial solution.

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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.

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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 $4+\frac{8s-2\mu}{N}< p + q < 2{2^*_{\mu,s}}$.

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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.

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arXiv2025

$p$-biharmonic Kirchhoff equations with critical Choquard nonlinearity

In this article, we deal with the following involving $p$-biharmonic critical Choquard-Kirchhoff equation $$ \left(a+b\left(\int_{\mathbb R^N}|\Delta u|^p dx\right)^{\theta-1}\right) \Delta_{p}^{2}u = \alpha \left(|x|^{-\mu}*u^{p^*_\mu}\right)|u|^{p^*_\mu-2}u+ \lambda f(x) |u|^{r-2} u \; \text{in}\; \mathbb R^N, $$ where $a\geq 0$, $b> 0$, $0<\mu 2p$, $p\geq 2$, $\theta\geq1$, $\alpha$ and $\lambda$ are positive real parameters, $p_{\mu}^{*}= \frac{p(2N-\mu)}{2(N-2p)}$ is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. The function $f \in L^{t}(\mathbb R^N)$ with $t= \frac{p^{*}}{(p^* -r)}$ if $p<r<p^*:=\frac{Np}{N-2p}$ and $t=\infty$ if $r\geq p^{*}$. We first prove the concentration compactness principle for the $p$-biharmonic Choquard-type equation. Then using the variational method together with the concentration-compactness, we established the existence and multiplicity of solutions to the above problem with respect to parameters $\lambda$ and \(\alpha\) for different values of $r$. The results obtained here are new even for $p-$Laplacian.

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arXiv2025

Spectral statistics of preferred orientation quantum graphs

We study the spectral statistics of quantum (metric) graphs whose vertices are equipped with preferred orientation vertex conditions. When comparing their spectral statistics to those predicted by suitable random matrix theory ensembles, one encounters some deviations. We point out these discrepancies and demonstrate that they occur in various graphs and even for Neumann-Kirchhoff vertex conditions, which was overlooked so far. Detailed explanations and computations are provided for this phenomena. To achieve this, we explore the combinatorics of periodic orbits, with a particular emphasis on counting Eulerian cycles.

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arXiv2024

Normalized Solutions to the Kirchhoff-Choquard Equations with Combined Growth

This paper is devoted to the study of the following nonlocal equation: \begin{equation*} -\left(a+b\|\nabla u\|_{2}^{2(\theta-1)}\right) \Delta u =\lambda u+\alpha (I_{\mu}\ast|u|^{q})|u|^{q-2}u+(I_{\mu}\ast|u|^{p})|u|^{p-2}u \ \hbox{in} \ \mathbb{R}^{N}, \end{equation*} with the prescribed norm $ \int_{\mathbb{R}^{N}} |u|^{2}= c^2,$ where $N\geq 3$, $0<\mu 0$, $1<\theta<\frac{2N-\mu}{N-2}$, $\frac{2N-\mu}{N} 0$ is a suitably small real parameter, $\lambda\in\mathbb{R}$ is the unknown parameter which appears as the Lagrange's multiplier and $I_{\mu}$ is the Riesz potential. We establish existence and multiplicity results and further demonstrate the existence of ground state solutions under the suitable range of $\alpha$. We demonstrate the existence of solution in the case of $q$ is $L^2-$supercritical and $p= \frac{2N-\mu}{N-2}$, which is not investigated in the literature till now. In addition, we present certain asymptotic properties of the solutions. To establish the existence results, we rely on variational methods, with a particular focus on the mountain pass theorem, the min-max principle, and Ekeland's variational principle.

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arXiv2023

High energy solutions for $p$-Kirchhoff elliptic problems with Hardy-Littlewood-Sobolev nonlinearity

This article deals with the study of the following Kirchhoff-Choquard problem: \begin{equation*} \begin{array}{cc} \displaystyle M\left(\, \int\limits_{\mathbb{R}^N}|\nabla u|^p\right) (-\Delta_p) u + V(x)|u|^{p-2}u = \left(\, \int\limits_{\mathbb{R}^N}\frac{F(u)(y)}{|x-y|^{\mu}}\,dy \right) f(u), \;\;\text{in} \; \mathbb{R}^N, u > 0, \;\; \text{in} \; \mathbb{R}^N, \end{array} \end{equation*} where $M$ models Kirchhoff-type nonlinear term of the form $M(t) = a + bt^{\theta-1}$, where $a, b > 0$ are given constants; $1<p<N$, $\Delta_p = \text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian operator; potential $V \in C^2(\mathbb{R}^N)$; $f$ is monotonic function with suitable growth conditions. We obtain the existence of a positive high energy solution for $\theta \in \left[1, \frac{2N-\mu}{N-p}\right) $ via the Poho\v{z}aev manifold and linking theorem. Apart from this, we also studied the radial symmetry of solutions of the associated limit problem.

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arXiv2022

Critical growth fractional Kirchhoff elliptic problems

This article is concerned with the existence and multiplicity of positive weak solutions for the following fractional Kirchhoff-Choquard problem: \begin{equation*} \begin{array}{cc} \displaystyle M\left( \|u\|^2\right) (-\Delta)^s u = \ds\lambda f(x)|u|^{q-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\quad \text{in} \; \Omega, \,\, u = 0\quad \text{in} \; \mathbb{R}^{N}\backslash\Omega, \end{array} \end{equation*} where $\Omega$ is open bounded domain of $\mathbb{R}^{N}$ with $C^2$ boundary, $N > 2s$ and $s \in (0,1)$, here $M$ models Kirchhoff-type coefficient of the form $M(t) = a + bt^{\te-1}$, where $a, b > 0$ are given constants. $(-\Delta)^s$ is fractional Laplace operator, $\lambda > 0$ is a real parameter. We explore using the variational methods, the existence of solution for ${q} \in (1,2^*_s)$ and $\te \geq 1$. % and we also consider the case when $\te > 2^*_{\mu,s}$ for $2< q < 2^*_{s}$. Here $2^*_s = \frac{2N}{N-2s}$ and $2^{*}_{\mu ,s} = \frac{2N-\mu}{N-2s}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality.

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arXiv2020

On weighted $L^p$-Hardy inequality on domains in $\mathbb{R}^n$

We consider weighted $L^p$-Hardy inequalities involving the distance to the boundary of a domain in the $n$-dimensional Euclidean space with nonempty boundary. Using criticality theory, we give an alternative proof of the following result of F.~G.~Avkhadiev (2006) Theorem: Let $\Omega \subsetneqq \mathbb{R}^n$, $n\geq 2$, be an arbitrary domain, $1 n$. Let $\mathrm{d}_\Omega(x) =\mathrm{dist}(x,\partial \Omega )$ denote the distance of a point $x\in \Omega$ to $\partial \Omega$. Then the following Hardy-type inequality holds $$ \int_{\Omega }\frac{|\nabla \varphi |^p}{\mathrm{d}_\Omega^{\alpha}}\,\mathrm{d}x \geq \left( \frac{\alpha +p-n}{p}\right)^p \int_{\Omega }\frac{|\varphi|^p}{\mathrm{d}_\Omega^{p+\alpha}}\,\mathrm{d}x \qquad \forall \varphi\in C^{\infty }_c(\Omega),$$ and the lower bound constant $\left( \frac{\alpha +p-n}{p}\right)^p$ is sharp.

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arXiv2020

Singular doubly nonlocal elliptic problems with Choquard type critical growth nonlinearities

The theory of elliptic equations involving singular nonlinearities is well studied topic but the interaction of singular type nonlinearity with nonlocal nonlinearity in elliptic problems has not been investigated so far. In this article, we study the very singular and doubly nonlocal singular problem $(P_\lambda)$(See below). Firstly, we establish a very weak comparison principle and the optimal Sobolev regularity. Next using the critical point theory of non-smooth analysis and the geometry of the energy functional, we establish the global multiplicity of positive weak solutions.

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arXiv2019

Regularity results on a class of doubly nonlocal problems

The purpose of this article is twofold. First, an issue of regularity of weak solution to the problem $(P)$ (See below) is addressed. Secondly, we investigate the question of $H^s$ versus $C^0$- weighted minimizers of the functional associated to problem $(P)$ and then give applications to existence and multiplicity results.

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arXiv2019

Brezis-Nirenberg type result for Kohn Laplacian with critical Choquard Nonlinearity

In this article, we are study the following Dirichlet problem with Choquard type non linearity \[ -\Delta_{\mathbb{H}} u = a u+ \left(\int_{\Omega}\frac{|u(\eta)|^{Q^*_\lambda}}{|\eta^{-1}\xi|^{\lambda}}d\eta\right)|u|^{Q^*_\lambda-2}u \; \text{in}\; \Omega,\quad u = 0 \; \text{ on } \partial \Omega , \] where $\Omega$ is a smooth bounded subset of the Heisenberg group $\mathbb{H}^N, N\in \mathbb N$ with $C^2$ boundary and $\Delta_{\mathbb{H}}$ is the Kohn Laplacian on the Heisenberg group $\mathbb{H}^N$. Here, $Q^*_\lambda=\frac{2Q-\lambda}{Q-2},\; Q= 2N+2$ and $a$ is a positive real parameter. We derive the Brezis-Nirenberg type result for the above problem. Moreover, we also prove the regularity of solutions and nonexistence of solutions depending on the range of $a$.

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