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Sarika Goyal

Publications and source records attributed to Sarika Goyal.

At least 19 recordsLinked to original sources
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.

math.AP↗
arXiv2026

On the existence results for $m$-Harmonic equation with critical Choquard Nonlinearity

This article established the existence results for the $m$-harmonic equation involving critical Choquard nonlinearity and subcritical perturbation. We first explore the minimizers of the $m$-harmonic operator with the critical Choquard equation. Then, using these minimizers, we establish delicate estimates to show the energy below the threshold level, which helps to recover the compactness. Further, we prove the existence of a nontrivial solution for our problem with different kinds of local and nonlocal subcritical perturbations. To the best of our knowledge, this is the first article dealing with the polyharmonic equation and critical Choquard type nonlinearity. The results obtained are even new for $m\geq 2$.

math.AP↗
arXiv2025

Asymptotic behaviour and existence of positive solutions for mixed local nonlocal elliptic equations with Hardy potential

We investigate the existence and multiplicity of positive solutions to the following problem driven by the superposition of the Laplacian and the fractional Laplacian with Hardy potential \begin{equation*} \left\{ \begin{aligned} -\Delta u + (-\Delta)^s u - \mu \frac{u}{|x|^2} &= \lambda |u|^{p-2} u + |u|^{2^*-2} u \quad \text{in } \Omega \subset \mathbb{R}^N, u &= 0 \quad \text{in } \mathbb{R}^N \setminus \Omega, \end{aligned} \right. \end{equation*} where $ \Omega \subset \mathbb{R}^N $ is a bounded domain with smooth boundary, $ 0 < s < 1 $, $ 1 < p < 2^* $, with $ 2^* = \frac{2N}{N-2} $, $ \lambda > 0 $, and $ \mu \in (0, \bar{\mu}) $ where $\bar \mu = \left( \frac{N-2}{2} \right)^2$. The aim of this paper is twofold. First, we establish uniform asymptotic estimates for solutions of the problem by means of a suitable transformation. Then, according to the value of the exponent $p$, we analyze three distinct cases and prove the existence of a positive solution. Moreover, in the sublinear regime $1 < p < 2$, we demonstrate the existence of multiple positive solutions for small perturbations of the fractional Laplacian.

math.AP↗
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.

math.AP↗
arXiv2025

On the eigenvalues and Fu\v{c}\'{\i}k spectrum of $p$-Laplace local and nonlocal operator with mixed interpolated Hardy term

In this article, we are concerned with the eigenvalue problem driven by the mixed local and nonlocal $p$-Laplacian operator having the interpolated Hardy term \begin{equation*} \mathcal{T}(u) :=- \Delta_p u + (- \Delta_p)^s u - \mu \frac{|u|^{p-2}u}{|x|^{p \theta}}, \end{equation*} where $0<s<1<p<N$, $\theta \in [s,1]$, and $\mu \in (0,\mu_0(\theta))$. First, we establish a mixed interpolated Hardy inequality and then show the existence of eigenvalues and their properties. We also investigate the Fu\v{c}\'{\i}k spectrum, the existence of the first nontrivial curve in the Fu\v{c}\'{\i}k spectrum, and prove some of its properties. Moreover, we study the shape optimization of the domain with respect to the first two eigenvalues, the regularity of the eigenfunctions, the Faber-Krahn inequality, and a variational characterization of the second eigenvalue.

math.AP↗
arXiv2024

Quasilinear Schr\"{o}dinger Equation involving Critical Hardy Potential and Choquard type Exponential nonlinearity

In this article, we study the following quasilinear Schr\"{o}dinger equation involving Hardy potential and Choquard type exponential nonlinearity with a parameter $\alpha$ \begin{equation*} \left\{ \begin{array}{l} - \Delta_N w - \Delta_N(|w|^{2\alpha}) |w|^{2\alpha - 2} w - \lambda \frac{|w|^{2\alpha N-2}w}{\left( |x| \log\left(\frac{R}{|x|} \right) \right)^N} = \left(\int_{\Omega} \frac{H(y,w(y))}{|x-y|^{\mu}}dy\right) h(x,w(x))\; \mbox{in }\; \Omega, w > 0 \mbox{ in } \Omega \setminus \{ 0\}, \quad \quad w = 0 \mbox{ on } \partial \Omega, \end{array} \right. \end{equation*} where $N\geq 2$, $\alpha>\frac12$, $0\leq \lambda< \left(\frac{N-1}{N}\right)^N$, $0 < \mu < N$, $h : \mathbb R^N \times \mathbb R \rightarrow \mathbb R$ is a continuous function with critical exponential growth in the sense of the Trudinger-Moser inequality and $H(x,t)= \int_{0}^{t} h(x,s) ds$ is the primitive of $h$. With the help of Mountain Pass Theorem and critical level which is obtained by the sequence of Moser functions, we establish the existence of a positive solution for a small range of $\lambda$. Moreover, we also investigate the existence of a positive solution for a non-homogeneous problem for every $0\leq \lambda <\left(\frac{N-1}{N}\right)^N.$ To the best of our knowledge, the results obtained here are new even in case of $N$-Laplace equation with Hardy potential.

math.AP↗
arXiv2023

Quasilinear Schrödinger equations with Stein-Weiss type convolution and critical exponential nonlinearity in $\mathbb R^N$

In this article, we investigate the existence of the positive solutions to the following class of quasilinear {Schrödinger} equations involving Stein-Weiss type convolution \begin{align*} -Δ_N u -Δ_N (u^{2})u +V(x)|u|^{N-2}u= \left(\int_{\mathbb R^N}\frac{F(y,u)}{|y|^β|x-y|^μ}~dy\right)\frac{f(x,u)}{|x|^β} \;\; \text{ in}\; \mathbb R^N, \end{align*} where $N\geq 2,\,$ $0<μ<N,\, β\geq 0,$ and $2β+μ\leq N.$ The potential $V:\mathbb R^N\to \mathbb R$ is a continuous function satisfying $0<V_0\leq V(x)$ for all $x\in \mathbb R^N$ and some appropriate assumptions. The nonlinearity $f:\mathbb R^N\times \mathbb R\to \mathbb R$ is a continuous function with critical exponential growth in the sense of the Trudinger-Moser inequality and $F(x,s)=\int_{0}^s f(x,t)dt$ is the primitive of $f$.

math.AP↗
arXiv2019

Kirchhoff equations with Choquard exponential type nonlinearity involving the fractional Laplacian

In this article, we deal with the existence of non-negative solutions of the class of following non local problem $$ \left\{ \begin{array}{lr} \quad - M\left(\displaystyle\int_{\mathbb R^n}\int_{\mathbb R^{n}} \frac{|u(x)-u(y)|^{\frac{n}{s}}}{|x-y|^{2n}}~dxdy\right) (-Δ)^{s}_{n/s} u=\left(\displaystyle\int_Ω\frac{G(y,u)}{|x-y|^μ}~dy\right)g(x,u) \; \text{in}\; Ω,\\ \quad \quad u =0\quad\text{in} \quad \mathbb R^n \setminus Ω, \end{array} \right. $$ where $(-Δ)^{s}_{n/s}$ is the $n/s$-fractional Laplace operator, $n\geq 1$, $s\in(0,1)$ such that $n/s\geq 2$, $Ω\subset \mathbb R^n$ is a bounded domain with Lipschitz boundary, $M:\mathbb R^+\rightarrow \mathbb R^+$ and $g:Ω\times\mathbb R\rightarrow \mathbb R$ are continuous functions, where $g$ behaves like $\exp({|u|^{\frac{n}{n-s}}})$ as $|u|\rightarrow\infty$.

math.AP↗
arXiv2017

On the first curve of Fučik Spectrum Of $p$-fractional Laplacian Operator with nonlocal normal boundary conditions

In this article, we study the Fučik spectrum of the $p$-fractional Laplace operator with nonlocal normal derivative conditions which is defined as the set of all $(a,b)\in \mb R^2$ such that $$ \mc (F_p)\left\{ \begin{array}{lr} Λ_{n,p}(1-\al)(-Δ)_{p}^{\al} u + |u|^{p-2}u = \frac{χ_{Ω_\e}}{\e} (a (u^{+})^{p-1} - b (u^{-})^{p-1}) \;\quad \text{in}\; Ω,\quad \\ \mc{N}_{\al,p} u = 0 \; \quad \mbox{in}\; \mb R^n \setminus \overlineΩ, \end{array} \right. $$ has a non-trivial solution $u$, where $Ω$ is a bounded domain in $\mb R^n$ with Lipschitz boundary, $p \geq 2$, $n>p \al $, $\e, \al \in(0,1)$ and $Ω{_\e}:=\{x \in Ω: d(x,\pa Ω)\leq \e \}$. We showed existence of the first non-trivial curve $\mc C$ of this spectrum which is used to obtain the variational characterization of a second eigenvalue of the problem $\mc (F_p)$. We also discuss some properties of this curve $\mc C$, e.g. Lipschitz continuous, strictly decreasing and asymptotic behaviour and nonresonance with respect to the Fučik spectrum.

math.AP↗
arXiv2016

A note on the eigenvalues of fractional Hardy-Sobolev operator with indefinite weight

In this article, we study the eigenvalue of nonlinear $p-$fractional Hardy operator \begin{align*} (-Δ)_p^αu - μ\frac{|u|^{p-2}u}{|x|^{pα}} = λV(x) |u|^{p-2}u \; \text{in}\; Ω, \quad u = 0 \; \mbox{in}\; \mathbb{R}^n \setminusΩ, \end{align*} where $n>pα$, $p\geq2$, $α\in(0,1)$, $0\leq μ<C_{n,α,p}$ and $Ω$ is a domain in $\mathbb{R}^n$ with Lipschitz boundary containing $0$. In particular, $Ω=\mathbb{R}^n$ is admitted. The weight function $V$ may change sign and may have singular points. We also show that the least positive eigenvalue is simple and it is unique associated to a non-negative eigenfunction. Moreover, we proved that there exists a sequence of eigenvalues $λ_k \to \infty$ as $k\to\infty$.

math.AP↗
arXiv2016

On the solvability of resonance problems for nonlocal elliptic equations

In this article, we consider the following problem: $$ \quad \left\{ \begin{array}{lr} \quad (-Δ)^s u = αu^+ -βu^{-} + f(u) + h \; \text{in}\;Ω\quad \quad \quad \quad u =0 \; \text{on}\; \mathbb{R}^n\setminus Ω, \end{array} \right. $$ where $Ω\subset \mathbb{R}^n$ is a bounded domain with Lipschitz boundary, $n> 2s$, $0<s<1$, $(α, β) \in \mathbb{R}^2$, $f: \mathbb{R}\to \mathbb{R}$ is a bounded and continuous function and $h\in L^2(Ω)$. We prove the existence results in two cases: First, the nonresonance case, where $(α,β)$ is not an element of the Fučik spectrum. Second, the resonance case, where $(α,β)$ is an element of the Fučik spectrum. Our existence results follows as an application of the Saddle point Theorem. It extends some results, well known for Laplace operator, to the nonlocal operator.

math.AP↗
arXiv2016

Multiplicity results of fractional-Laplace system with sign-changing and singular nonlinearity

In this article, we study the following fractional-Laplacian system with singular nonlinearity \begin{equation*} (P_{λ,μ}) \left\{ \begin{array}{lr} (-Δ)^s u = λf(x) u^{-q}+ \fracα{α+β}b(x) u^{α-1} w^β\; \text{in}\;Ω\\ (-Δ)^s w = μg(x) w^{-q}+ \fracβ{α+β} b(x) u^α w^{β-1}\; \text{in}\;Ω\\ \quad \quad u, w>0\;\text{in}\;Ω, \quad u = w = 0 \; \mbox{in}\; \mathbb{R}^n \setminusΩ, \end{array} \quad \right. \end{equation*} where $Ω$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary $\partial Ω$, $n>2s$, $s\in(0,1)$, $0 1$, $β>1$ satisfy $2<α+β< 2_{s}^*-1$ with $2_{s}^*=\frac{2n}{n-2s}$, the pair of parameters $(λ,μ) \in \mathbb{R}^2\setminus\{(0,0)\}$. The weight functions $f,g: Ω\subset\mathbb{R}^n \to \mathbb{R}$ such that $0<f$, $g\in L^{\frac{α+β}{α+β-1+q}}(Ω)$, and $b:Ω\subset \mathbb{R}^n \to \mathbb{R}$ is a sign-changing function such that $b(x)\in L^{\infty}(Ω)$. Using variational methods, we show existence and multiplicity of positive solutions of $(P_{λ,μ})$ with respect to the pair of parameters $(λ,μ)$.

math.AP↗
arXiv2016

Multiplicity results of fractional $p$-Laplace equations with sign-changing and singular nonlinearity

In this article, we study the following fractional $p$-Laplacian equation with singular nonlinearity \begin{equation*} (P_{\la}) \left\{ \begin{array}{lr} - 2\int_{\mb R^n}\frac{|w(y)-w(x)|^{p-2}(w(y)-w(x))}{|x-y|^{n+ps}}dy = a(x) w^{-q}+ \la b(x) w^r\; \text{in}\; \Om \quad \quad w>0\;\text{in}\;\Om, \quad w = 0 \; \mbox{in}\; \mb R^n \setminus\Om, \end{array} \quad \right. \end{equation*} where $\Om$ is a bounded domain in $\mb R^n$ with smooth boundary $\partial \Om$, $n> ps$,$s\in(0,1)$, $\la>0$, $0<q<1$, $q<p-1<r< p_{s}^*-1$ with $p_{s}^*=\frac{np}{n-ps}$, $a: \Om\subset\mb R^n \ra \mb R$ such that $0< a(x)\in L^{\frac{p^{*}_{s}}{p^{*}_{s}-1+q}}(\Om)$, and $b:\Om\subset\mb R^n \ra \mb R$ is a sign-changing function such that $b(x)\in L^{\frac{p^{*}_{s}}{p^{*}_{s}-1-r}}(\Om)$. Using variational methods, we show existence and multiplicity of positive solutions of $(P_{\la})$ with respect to the parameter $\la$.

math.AP↗
arXiv2016

Polyharmonic Kirchhoff type equations with singular exponential nonlinearities

\noi In this article, we study the existence of non-negative solutions of the following polyharmonic Kirchhoff type problem with critical singular exponential nolinearity $$ \quad \left\{ \begin{array}{lr} \quad -M\left(\displaystyle\int_Ω|\nabla^m u|^{\frac{n}{m}}dx\right)Δ_{\frac{n}{m}}^{m} u = \frac{f(x,u)}{|x|^α} \; \text{in}\; \Om{,} \quad \quad u = \nabla u=\cdot\cdot\cdot= {\nabla}^{m-1} u=0 \quad \text{on} \quad \partial \Om{,} \end{array} \right. $$ where $\Om\subset \mb R^n$ is a bounded domain with smooth boundary, $n\geq 2m\geq 2$ and $f(x,u)$ behaves like $e^{|u|^{\frac{n}{n-m}}}$ as $|u|\ra\infty$. Using mountain pass structure and {the} concentration compactness principle, we show the existence of a nontrivial solution. %{OR}\\ In the later part of the paper, we also discuss the above problem with convex-concave type sign changing nonlinearity. Using {the} Nehari manifold technique, we show the existence and multiplicity of non-negative solutions. \medskip

math.AP↗
arXiv2015

On The Fučik Spectrum Of Non-Local Elliptic Operators

In this article, we study the Fučik spectrum of fractional Laplace operator which is defined as the set of all $(\al,\ba)\in \mb R^2$ such that \begin{equation*} \quad \left. \begin{array}{lr} \quad (-\De)^s u = \al u^{+} - \ba u^{-} \; \text{in}\; \Om \quad \quad \quad \quad u = 0 \; \mbox{in}\; \mb R^n \setminus\Om.\\ \end{array} \quad \right\} \end{equation*} has a non-trivial solution $u$, where $\Om$ is a bounded domain in $\mb R^n$ with Lipschitz boundary, $n>2s$, $s\in(0,1)$. The existence of a first nontrivial curve $\mc C$ of this spectrum, some properties of this curve $\mc C$, e.g. Lipschitz continuous, strictly decreasing and asymptotic behavior are studied in this article. A variational characterization of second eigenvalue of the fractional eigenvalue problem is also obtained. At the end, we study a nonresonance problem with respect to Fučik spectrum.

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