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Tuhina Mukherjee

Publications and source records attributed to Tuhina Mukherjee.

At least 37 records · Page 2Linked to original sources
arXiv2023

On the extreme value of Nehari manifold for nonlocal singular Schr{ö}dinger-Kirchhoff equations in $\mathbb{R}^N$

This article investigates the existence, non-existence, and multiplicity of weak solutions for a parameter-dependent nonlocal Schrödinger-Kirchhoff type problem on $\mathbb R^N$ involving singular non-linearity. By performing fine analysis based on Nehari submanifolds and fibre maps, our goal is to show the problem has at least two positive solutions even if $λ$ lies beyond the extremal parameter $λ_\ast$.

math.AP↗
arXiv2023

On critically coupled (s_1, s_2)-fractional system of Schrödinger equations with Hardy potential

In this article, our main concern is to study the existence of bound and ground state solutions for the following fractional system of Schrödinger equations with Hardy potentials: \begin{equation*} \left\{ \begin{aligned} (-Δ)^{s_{1}} u - λ_{1} \frac{u~~}{|x|^{2s_{1}}} - u^{2_{s_{1}}^{*}-1} = ναh(x) u^{α-1}v^β & \quad \mbox{in} ~ \mathbb{R}^{N}, (-Δ)^{s_{2}} v - λ_{2} \frac{v~~}{|x|^{2s_{2}}} - v^{2_{s_{2}}^{*}-1} = νβh(x) u^αv^{β-1} & \quad \mbox{in} ~ \mathbb{R}^{N}, u,v >0 \quad \mbox{in} ~ \mathbb{R}^{N} \setminus \{0\}, \end{aligned} \right. \end{equation*} where $s_{1},s_{2} \in (0,1)~\text{and}~λ_{i}\in (0, Λ_{N,s_{i}})$ with $Λ_{N,s_{i}} = 2 π^{N/2} \frac{Γ^{2}(\frac{N+2s_i}{4}) Γ(\frac{N+2s_i}{2})}{Γ^{2}(\frac{N-2s_i}{4}) ~|Γ(-s_{i})|}, (i=1,2)$. By imposing certain assumptions on the parameters and on the function h, we obtain ground-state solutions using the concentration-compactness principle and the mountain-pass theorem.

math.AP↗
arXiv2022

Existence of ground state solutions for a Choquard double phase problem

In this paper we study quasilinear elliptic equations driven by the double phase operator involving a Choquard term of the form \begin{align*} -\mathcal{L}_{p,q}^{a}(u) + |u|^{p-2}u+ a(x) |u|^{q-2}u = \left( \int_{\mathbb{R}^N} \frac{F(y, u)}{|x-y|^μ}\,\mathrm{d} y\right)f(x,u) \quad\text{in } \mathbb{R}^N, \end{align*} where $\mathcal{L}_{p,q}^{a}$ is the double phase operator given by \begin{align*} \mathcal{L}_{p,q}^{a}(u):= \operatorname{div}\big(|\nabla u|^{p-2}\nabla u + a(x) |\nabla u|^{q-2}\nabla u \big), \quad u\in W^{1,\mathcal{H}}(\mathbb{R}^N), \end{align*} $0<μ<N$, $1<p<N$, $p<q<p+ \frac{αp}{N}$, $0 \leq a(\cdot)\in C^{0,α}(\mathbb{R}^N)$ with $α\in (0,1]$ and $f\colon\mathbb{R}^N\times\mathbb{R}\to\mathbb{R}$ is a continuous function that satisfies a subcritical growth. Based on the Hardy-Littlewood-Sobolev inequality, the Nehari manifold and variational tools, we prove the existence of ground state solutions of such problems under different assumptions on the data.

math.AP↗
arXiv2022

On an anisotropic double phase problem with singular and sign changing nonlinearity

This article consists of study of anisotropic double phase problems with singular term and sign changing subcritical as well as critical nonlinearity. Seeking the help of well known Nehari manifold technique, we establish existence of at least two opposite sign energy solutions in the subcritical case and one negative energy solution in the critical case. The results in the critical case is even new in the classical $p$-Laplacian case.

math.AP↗
arXiv2021

On double phase Kirchhoff problems with singular nonlinearity

In this paper, we study multiplicity results for double phase problems of Kirchhoff type with right-hand sides that include a parametric singular term and a nonlinear term of subcritical growth. Under very general assumptions on the data, we prove the existence of at least two weak solutions that have different energy sign. Our treatment is based on the fibering method in form of the Nehari manifold. We point out that we cover both the non-degenerate as well as the degenerate Kirchhoff case in our setting.

math.AP↗
arXiv2021

On an anisotropic p-Laplace equation with variable singular exponent

In this article, we study the following anisotropic p-Laplacian equation with variable exponent given by \begin{equation*} (P)\left\{\begin{split} -Δ_{H,p}u&=\frac{\la f(x)}{u^{q(x)}}+g(u)\text{ in }Ω,\\ u&>0\text{ in }Ω,\,u=0\text{ on }\partialΩ, \end{split}\right. \end{equation*} under the assumption $Ω$ is a bounded smooth domain in $\mathbb{R}^N$ with $p,N\geq 2$, $\la>0$ and $0<q \in C(\bar \Om)$. For the purely singular case that is $g\equiv 0$, we proved existence and uniqueness of solution. We also demonstrate the existence of multiple solution to $(P)$ provided $f\equiv 1$ and $g(u)=u^r$ for $r\in (p-1,p^*-1)$.

math.AP↗
arXiv2021

Non-homogeneous $(p_1,p_2)$-fractional Laplacian systems with lack of compactness

The present paper studies the existence of weak solutions for the following type of non-homogeneous system of equations \begin{equation*} (S) \left\{\begin{aligned} (-Δ)^{s_1}_{p_1} u &=u|u|^{α-1}|v|^{β+1}+f_1(x) \,\mbox{ in }\, Ω, \\ (-Δ)^{s_2}_{p_2} v &=|u|^{α+1}v|v|^{β-1}+f_2(x) \,\mbox{ in }\, Ω, \\ u=v &= 0 \,\mbox{ in }\, \mathbb{R}^N \setminus Ω, \\ \end{aligned} \right. \end{equation*} where $Ω\subset \mathbb{R}^N$ is smooth bounded domain, $s_1,s_2 \in (0,1)$, $1 \max\{p_1s_1,p_2s_2\}$, $α>-1$ and $β>-1$. We employ the variational techniques where the associated energy functional is minimized over Nehari manifold set while imposing appropriate bound on dual norms of $f_1,f_2$.

math.AP↗
arXiv2020

On the existence of three non-negative solutions for a $(p,q)$-Laplacian system

The present paper studies the existence of weak solutions for \begin{equation*} (\mathcal{P}) \left\{\begin{aligned} (-Δ)^{s_1}_{p_1} u &=\la f_1\,(x,u,v) +g_1(x,u) \,\mbox{ in }\, \Om, \\ (-Δ)^{s_2}_{p_2} v &=\la f_2\,(x,u,v) +g_2(x,v) \,\mbox{ in }\, \Om, \\ u=v &= 0 \,\mbox{in }\, \Rn \setminus \Om, \\ \end{aligned} \right. \end{equation*} where $\Om \subset \Rn$ is a smooth bounded domain with smooth boundary, $s_1,s_2 \in (0,1)$, $1<p_i<\frac{N}{s_i}$, $i=1,2$, $f_i$ and $g_i$ has certain growth assumptions for $i=1,2$. We prove existence of at least three non negative solutions of $(\mathcal P)$ under restrictive range of $λ$ using variational methods. As a consequence, we also conclude that a similar result can be obtained when we consider a more general non local operator $\mathcal L_{ϕ_i}$ instead of $(-Δ)^{s_i}_{p_i}$ in $(\mathcal P)$.

math.AP↗
arXiv2020

The Nehari manifold method for Fractional Kirchhoff problem involving singular and exponential nonlinearity

In this paper we establish the existence of at least two weak solutions for the following fractional Kirchhoff problem involving singular and exponential nonlinearity \begin{equation*} \left\{\begin{split} M\left(\|u\|^{\frac{n}{s}}\right)(-Δ)^s_{n/s}u & = μu^{-q}+ u^{r-1}\exp( u^β)\;\text{in}\;\Om, u&>0,\;\text{in}\; \Om, u &= 0,\;\text{in}\; \mb R^n \setminus{\Om}, \end{split} \right. \end{equation*} where $\Om$ is smooth bounded domain in $\mb R^n$, {$n\geq 1$}, $s\in (0,1)$, $μ>0$ is a real parameter, $β<\frac{n}{n-s}$ and $q\in (0,1)$. We have considered the degenerate Kirchhoff case here and used the Nehari manifold techniques to obtain the results.

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↗
arXiv2018

Critical growth elliptic problems with Choquard type nonlinearity:A survey

This article deals with a survey of recent developments and results on Choquard equations where we focus on the existence and multiplicity of solutions of the partial differential equations which involve the nonlinearity of convolution type. Because of its nature, these equations are categorized under the nonlocal problems. We give a brief survey on the work already done in this regard following which we illustrate the problems we have addressed. Seeking the help of variational methods and asymptotic estimates, we prove our main results.

math.AP↗
arXiv2018

n-Kirchhoff Choquard equations with exponenetial nonlinearity

This article deals with the study of the following Kirchhoff equation with exponential nonlinearity of Choquard type (see $(KC)$ below). We use the variational method in the light of Moser-Trudinger inequality to show the existence of weak solutions to $(KC)$. Moreover, analyzing the fibering maps and minimizing the energy functional over suitable subsets of the Nehari manifold, we prove existence and multiplicity of weak solutions to convex-concave problem $(\mathcal{P}_{\la,M})$ below.

math.AP↗
arXiv2018

A Global multiplicity result for a very singular critical nonlocal equation

In this article, we show the global multiplicity result for the following nonlocal singular problem \begin{equation*} (P_\la):\;\quad (-\De)^s u = u^{-q} + \la u^{{2^*_s}-1}, \quad u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{in}\; \mb R^n \setminus\Om, \end{equation*} where $\Om$ is a bounded domain in $\mb{R}^n$ with smooth boundary $\partial \Om$, $n > 2s,\; s \in (0,1),\; \la >0,\; q>0$ satisfies $q(2s-1)<(2s+1)$ and $2^*_s=\frac{2n}{n-2s}$. Employing the variational method, we show the existence of at least two distinct weak positive solutions for $(P_\la)$ in $X_0$ when $\la \in (0,\La)$ and no solution when $\la>\La$, where $\La>0$ is appropriately chosen. We also prove a result of independent interest that any weak solution to $(P_λ)$ is in $C^α(\R^n)$ with $α=α(s,q)\in (0,1)$. The asymptotic behaviour of weak solutions reveals that this result is sharp.

math.AP↗
arXiv2018

On Concentration of least energy solutions for magnetic critical Choquard equations

In the present paper, we consider the following magnetic nonlinear Choquard equation $$ \left\{ \begin{array}{ll} & (-i \nabla+A(x))^2u + μg(x)u = λu + (|x|^{-α} * |u|^{2^*_α})|u|^{2^*_α-2}u ,\; u>0 \;\text{in} \; \mathbb{ R}^n , & u \in H^1(\mathbb{R}^n, \mathbb{ C}) \end{array} \right\}.$$ where $n \geq 4$, $2^*_α= \frac{2n-α}{n-2}$, $λ>0$, $μ\in \mathbb{ R}$ is a parameter, $α\in (0,n)$, $A(x): \mathbb{R}^n \rightarrow \mathbb{ R}^n$ is a magnetic vector potential and $g(x)$ is a real valued potential function on $\mathbb{R}^n$. Using variational methods, we establish the existence of least energy solution under some suitable conditions. Moreover, the concentration behavior of solutions is also studied as $μ\rightarrow +\infty$.

math.AP↗
arXiv2018

Existence of three positive solutions for a nonlocal singular dirichlet boundary problem

In this article, we prove the existence of at least three positive solutions for the following nonlocal singular problem \begin{equation*} (P_\la)\left\{ \begin{split} (-\De)^su &= \la\frac{f(u)}{u^q}, \; \; u>0 \;\; \text{in}\;\; \Om,\\ u &= 0\;\; \text{in}\;\; \mb R^n \setminus \Om \end{split} \right. \end{equation*} where $(-\De)^s$ denotes the fractional Laplace operator for $s\in (0,1)$, $n>2s$, $q \in (0,1)$, $\la>0$ and $\Om$ is smooth bounded domain in $\mb R^n$. Here $f :[0,\infty) \to [0,\infty)$ is a continuous nondecreasing map satisfying $\lim\limits_{u\to \infty}\frac{f(u)}{u^{q+1}}=0$. We show that under certain additional assumptions on $f$, $(P_\la)$ possesses at least three distinct solutions for a certain range of $\la$. We use the method of sub-supersolutions and a critical point theorem by Amann \cite{amann} to prove our results. Moreover, we prove a new existence result for a suitable infinite semipositone nonlocal problem which played a crucial role to obtain our main result and is of independent interest. \medskip

math.AP↗
arXiv2017

Doubly nonlocal system with Hardy-Littlewood-Sobolev critical nonlinearity

This article concerns about the existence and multiplicity of weak solutions for the following nonlinear doubly nonlocal problem with critical nonlinearity in the sense of Hardy-Littlewood-Sobolev inequality \begin{equation*} \left\{ \begin{split} (-Δ)^su &= λ|u|^{q-2}u + \left(\int_Ω\frac{|v(y)|^{2^*_μ}}{|x-y|^μ}~\mathrm{d}y\right) |u|^{2^*_μ-2}u\; \text{in}\; Ω (-Δ)^sv &= δ|v|^{q-2}v + \left(\int_{\Om}\frac{|u(y)|^{2^*_μ}}{|x-y|^μ}~\mathrm{d}y \right) |v|^{2^*_μ-2}v \; \text{in}\; Ω u &=v=0\; \text{in}\; \mb R^n\setminusΩ, \end{split} \right. \end{equation*} where $Ω$ is a smooth bounded domain in $\mb R^n$, $n >2s$, $s \in (0,1)$, $(-Δ)^s$ is the well known fractional Laplacian, $μ\in (0,n)$, $2^*_μ= \displaystyle\frac{2n-μ}{n-2s}$ is the upper critical exponent in the Hardy-Littlewood-Sobolev inequality, $1 0$ are real parameters. We study the fibering maps corresponding to the functional associated with $(P_{λ,δ})$ and show that minimization over suitable subsets of Nehari manifold renders the existence of atleast two non trivial solutions of $(P_{\la,δ})$ for suitable range of $\la$ and $δ$.

math.AP↗
arXiv2017

Existence and stabilization results for a singular parabolic equation involving the fractional Laplacian

In this article, we study the following parabolic equation involving the fractional Laplacian with singular nonlinearity \begin{equation*} \quad (P_{t}^s) \left\{ \begin{split} \quad u_t + (-Δ)^s u &= u^{-q} + f(x,u), \;u >0\; \text{in}\; (0,T) \times Ω, u &= 0 \; \mbox{in}\; (0,T) \times (\mb R^n \setminusΩ), \quad \quad \quad \quad u(0,x)&=u_0(x) \; \mbox{in} \; {\mb R^n}, \end{split} \quad \right. \end{equation*} where $Ω$ is a bounded domain in $\mb{R}^n$ with smooth boundary $\partial Ω$, $n> 2s, \;s \in (0,1)$, $q>0$, ${q(2s-1)<(2s+1)}$, $u_0 \in L^\infty(Ω)\cap X_0(Ω)$ and $T>0$. We suppose that the map $(x,y)\in Ω\times \mb R^+ \mapsto f(x,y)$ is a bounded below Carathéodary function, locally Lipschitz with respect to second variable and uniformly for $x \in Ω$ it satisfies \begin{equation}\label{cond_on_f} { \limsup_{y \to +\infty} \frac{f(x,y)}{y}<λ_1^s(Ω)}, \end{equation} where $\la_1^s(Ω)$ is the first eigenvalue of $(-Δ)^s$ in $Ω$ with homogeneous Dirichlet boundary condition in $\mathbb{R}^n \setminus Ω$. We prove the existence and uniqueness of weak solution to $(P_t^s)$ on assuming $u_0$ satisfies an appropriate cone condition. We use the semi-discretization in time with implicit Euler method and study the stationary problem to prove our results. We also show additional regularity on the solution of $(P_t^s)$ when we regularize our initial function $u_0$.

math.AP↗