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Tuhina Mukherjee

Publications and source records attributed to Tuhina Mukherjee.

At least 19 recordsLinked to original sources
arXiv2026

On polyharmonic Kirchhoff double phase problems without AR-conditions

In this paper, we study a class of polyharmonic Kirchhoff problems driven by a double phase operator. The reaction term has subcritical growth but does not satisfy the Ambrosetti--Rabinowitz condition. Motivated by the work of Harrabi-Hamdani-Fiscella \cite{Harrabi-Hamdani-Fiscella-2024} on m-polyharmonic Kirchhoff problems without Ambrosetti--Rabinowitz conditions, we extend their analysis to a nonhomogeneous double phase setting. We study the problem in the natural Musielak--Orlicz--Sobolev framework associated with the double phase structure. The main novelty of the paper lies in combining the nonlocal Kirchhoff term with a higher-order double phase operator under assumptions weaker than the classical Ambrosetti--Rabinowitz condition. By developing suitable modular estimates and compactness arguments, we establish the variational setting and obtain existence and multiplicity results by minimax methods.

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arXiv2026

On doubly critical polyharmonic double phase problems: Existence and non-existence of solutions

In this article, we investigate the existence and nonexistence of weak solutions to higher-order doubly critical elliptic problems with weights, driven by a polyharmonic double phase operator. More precisely, we deal with the following problem \begin{equation} \begin{cases} \mathcal{L}^m_{p,q}(u) = f(x,u) ~&\text{in } Ω,\\[6pt] u=\nabla u=\cdots\nabla^{m-1} u=0 &\text{on }{\partialΩ}, \end{cases} \end{equation} where $Ω\subset \mathbb{R}^N$ with $N \geq 2$ is a smooth bounded domain with Lipschitz boundary $\partialΩ$, $m \in \mathbb{N}$, $1 < p < q < \frac{N}{m}$ with $(N-1)q\leq Np$, the nonlinear term $f\colonΩ\times\mathbb{R}\to \mathbb{R}$ is a Carathéodory function, which has doubly critical growth, and $\mathcal{L}^m_{p,q}$ represents a polyharmonic double phase operator. By establishing new compactness results within a suitable Musielak--Orlicz--Sobolev framework and applying variational methods, we prove the existence of nontrivial weak solutions. In addition, we derive nonexistence results under appropriate assumptions by establishing a Pohozaev-type identity for higher--order derivatives. Our approach extends classical techniques to capture the intricate features of the double-phase operator for higher--order derivatives, and addresses the difficulties arising from critical nonlinearities, in particular extending the results of [F. Colasuonno, K. Perera, J. Differ. Equ., 422 (2025), 426--488] in a polyharmonic double phase setup overcoming the non-closedness of truncations in higher-order Sobolev spaces.

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arXiv2026

A Note on Sharpened Singular Adams-Type Inequalities

We establish a sharp Adams-type inequality in higher-order function spaces with singular weights on $\mathbb{R}^n$. A sharp singular concentration-compactness principle, improving Lions' result, is also proved. The study distinguishes between critical and subcritical sharp singular Adams-type inequalities and shows their equivalence. Furthermore, we analyze the asymptotic behavior of the associated bounds and relate the suprema of the critical and subcritical cases. A new compact embedding, crucial to our analysis, is also derived. Moreover, as an application of these results, by employing the mountain pass theorem, we study the existence of nontrivial solutions to a class of nonhomogeneous quasilinear elliptic equations involving the $(p,\frac{n}{2})$-biharmonic operator with singular exponential growth.

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arXiv2026

Spectral Properties of the Logarithmic Laplacian with Indefinite Weights

In this paper, we investigate a weighted eigenvalue problem driven by the Logarithmic Laplacian with indefinite weights. We prove the existence of an unbounded sequence of Lusternik-Schnirelman eigenvalues and show that the first eigenvalue is simple, with the associated eigenfunction having constant sign in the domain. In contrast, eigenfunctions corresponding to higher eigenvalues necessarily change sign. We further establish a nodal domain type inequality relating the higher eigenvalues to the measure of the positive and negative parts of the corresponding eigenfunctions, which is of independent interest. As an application, we prove that the first eigenvalue is isolated. In addition, we obtain alternative variational characterizations of the first and second eigenvalues and establish monotonicity properties of the eigenvalues with respect to both the weight function and the domain.

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arXiv2026

On Coron problems with Choquard term and mixed operator

In this article, we study a Coron-type problem involving a critical Choquard nonlinearity driven by a mixed operator combining the Laplacian and fractional Laplacian. In annular-type domains, we prove the existence of nontrivial positive solutions when the inner hole is sufficiently small. Using variational methods and concentration compactness arguments, we establish a global compactness result for Palais- Smale sequences and obtain high-energy solutions using topological methods. We also derive regularity results for weak solutions.

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arXiv2026

On the Fučík spectrum of the Logarithmic Laplacian

In this paper, we investigate the Fučík spectrum $Σ_L$ associated with the logarithmic Laplacian. This spectrum is defined as the set of all pairs $(α,β) \in \mathbb{R}^2$ for which the problem \[ L_Δu = αu^+-βu^- ~\text{in} ~ Ω\quad \text{and} \quad u=0 ~\text{in} ~\mathbb{R}^N\setminus Ω\] admits a nontrivial solution $u$. Here, $Ω\subset \mathbb{R}^N$ is a bounded domain with $C^{1,1}$ boundary, $u^\pm = \max\{\pm u,0\}$, and $u = u^+ - u^-$. We show that the lines $λ_1^L \times \mathbb{R}$ and $\mathbb{R} \times λ_1^L$, where $λ_1^L$ denotes the first eigenvalue of $L_Δ$, lies in the spectrum $Σ_L$ and are isolated within the spectrum. Furthermore, we establish the existence of the first nontrivial curve in $Σ_L$ and analyze its qualitative properties, including Lipschitz continuity, strict monotonicity, and asymptotic behavior. In addition, we obtain a variational characterization of the second eigenvalue of the logarithmic Laplacian and show that all eigenfunctions corresponding to eigenvalues $λ> λ_1^L$ are sign-changing. Finally, we address a nonresonance problem with respect to the Fučík spectrum $Σ_L$, employing variational methods and carefully overcoming the difficulties arising from the contrasting features of the first eigenvalue $λ_1^L$.

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arXiv2025

On singularly perturbed $(p, N )$-Laplace Schrödinger equation with logarithmic nonlinearity

This article focuses on the study of the existence, multiplicity and concentration behavior of ground states as well as the qualitative aspects of positive solutions for a $(p, N)$-Laplace Schrödinger equation with logarithmic nonlinearity and critical exponential nonlinearity in the sense of Trudinger-Moser in the whole Euclidean space $\mathbb{R}^N$. Through the use of smooth variational methods, penalization techniques, and the application of the Lusternik-Schnirelmann category theory, we establish a connection between the number of positive solutions and the topological properties of the set in which the potential function achieves its minimum values.

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arXiv2025

Non variational type critical growth nonlocal system

This study investigates the existence, uniqueness, and multiplicity of positive solutions for a system of fractional differential equations given by: \begin{equation*} (-Δ)^{s_i} u_{i}+λ_{i} u_{i}=\sum_{j=1}^{n} α_{i j}\left|u_{j}\right|^{q_{i j}}\left|u_{i}\right|^{p_{i j}-2} u_{i} , u_i\in {\mathscr{D}^{s_i,2}\left(\mathbb{R}^{N}\right)}, i=1,2,\cdots,n, \end{equation*} where $N>2s=\max\{2s_i\}$, $s_i\in(0,1)$, $n\geq 2$, $λ_{i} \geq 0$, $α_{ij}>0$, $p_{ij}<2^{*}_{s}$, and $p_{ij}+q_{ij}=2^{*}_{s}=\min\{{\frac{2N}{N-2s_i}\}}$ for $i\neq j \in \{1,2,...,n\}$. $2^{*}_s$ called the fractional critical sobolev exponent and $2^{*}_s=2 N /(N-2s)$ for $N > 2s$ and $2^{*}_s=+\infty$ for $N=2s$ or $N<2s$. Our work establishes novel uniqueness and multiplicity results for positive solutions, applicable whether the system possesses a variational structure or not. We provide a comprehensive characterization of the exact number of positive solutions under specific parameter configurations. Our analysis shows that the positive solution set behaves differently across three distinct regimes: $p_{ij}<2$, $p_{ij}=2$, and $2<p_{ij}<2^{*}_{s}$.

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arXiv2025

On singular problems associated with mixed operators under mixed boundary conditions

In this paper, we study the following singular problem associated with mixed operators (the combination of the classical Laplace operator and the fractional Laplace operator) under mixed boundary conditions \begin{equation*} \label{1} \left\{ \begin{aligned} \mathcal{L}u &= g(u), \quad u > 0 \quad \text{in} \quad Ω, u &= 0 \quad \text{in} \quad U^c, \mathcal{N}_s(u) &= 0 \quad \text{in} \quad \mathcal{N}, \frac{\partial u}{\partial ν} &= 0 \quad \text{in} \quad \partial Ω\cap \overline{\mathcal{N}}, \end{aligned} \right. \tag{$P_λ$} \end{equation*} where $U= (Ω\cup {\mathcal{N}} \cup (\partialΩ\cap\overline{\mathcal{N}}))$, $Ω\subseteq \mathbb{R}^N$ is a non empty open set, $\mathcal{D}$, $\mathcal{N}$ are open subsets of $\mathbb{R}^N\setminus{\bar{Ω}}$ such that ${\mathcal{D}} \cup {\mathcal{N}}= \mathbb{R}^N\setminus{\barΩ}$, $\mathcal{D} \cap {\mathcal{N}}= \emptyset $ and $Ω\cup \mathcal{N}$ is a bounded set with smooth boundary, $λ>0$ is a real parameter and $\mathcal{L}= -Δ+(-Δ)^{s},~ \text{for}~s \in (0, 1).$ Here $g(u)=u^{-q}$ or $g(u)= λu^{-q}+ u^p$ with $0<q<1<p\leq 2^*-1$. We study $(P_λ)$ to derive the existence of weak solutions along with its $L^\infty$-regularity. Moreover, some Sobolev-type variational inequalities associated with these weak solutions are established.

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arXiv2024

On elliptic problems with mixed operators and Dirichlet-Neumann boundary conditions

In this paper, we study the existence, nonexistence and multiplicity of positive solutions to the problem given by \begin{equation*} \label{1} \left\{\begin{split} \mathcal{L}u\: &= λu^{q} + u^{p}, \quad u>0 ~~ \text{in} ~Ω, u&=0~~\text{in} ~~{D^c}, \mathcal{N}_s(u)&=0 ~~\text{in} ~~{Π_2}, \frac{\partial u}{\partial ν}&=0 ~~\text{in}~~ \partial Ω\cap \overline{Π_2}. \end{split} \right.\tag{$P_λ$} \end{equation*} {where $D= \left(Ω\cup {Π_2} \cup (\partialΩ\cap\overline{Π_2})\right)$ and $D^c$ is the complement of $D$, $Ω\subseteq \mathbb{R}^n$ is a non empty open set, $Π_{1}$, $Π_{2}$ are open subsets of $\mathbb{R}^n\setminus{\bar Ω}$ such that $\overline{Π_{1} \cup {Π_{2}}}= \mathbb{R}^n\setminusΩ$, $Π_{1} \cap Π_{2}= \emptyset$ and $Ω\cup Π_2$ is a bounded set with smooth boundary}, $λ>0$ is a real parameter, $ 0 < q < 1 2$ and $\mathcal{L}= -Δ+(-Δ)^{s},~ \text{for}~s \in (0, 1).$ We first present a functional setting to study any problem involving $\mathcal L$ under mixed boundary conditions in the presence of concave-convex power nonlinearity, {for a suitable range of $λ$, $q$ and $p$}. Our article also contains results related to Picone's identity, strong maximum principles and comparison principles.

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arXiv2024

On an eigenvalue problem associated with mixed operators under mixed boundary conditions

In this paper, we study a class of eigenvalue problems involving both local as well as nonlocal operators, precisely the classical Laplace operator and the fractional Laplace operator in the presence of mixed boundary conditions, that is \begin{equation} \label{1} \left\{\begin{split} \mathcal{L}u\: &= λu,~~u>0~ \text{in} ~Ω, u&=0~~\text{in} ~~{U^c}, \mathcal{N}_s(u)&=0 ~~\text{in} ~~{\mathcal{N}}, \frac{\partial u}{\partial ν}&=0 ~~\text{in}~~ \partial Ω\cap \overline{\mathcal{N}}, \end{split} \right.\tag{$P_λ$} \end{equation} where $U= (Ω\cup {\mathcal{N}} \cup (\partialΩ\cap\overline{\mathcal{N}}))$, $Ω\subseteq \mathbb{R}^n$ is a non empty open set, $\mathcal{D}$, $\mathcal{N}$ are open subsets of $\mathbb{R}^n\setminus{\bar{Ω}}$ such that $\overline{\mathcal{D} \cup {\mathcal{N}}}= \mathbb{R}^n\setminusΩ$, $\mathcal{D} \cap {\mathcal{N}}= \emptyset $ and $Ω\cup \mathcal{N}$ is a bounded set with smooth boundary, $λ>0$ is a real parameter and $$\mathcal{L}= -Δ+(-Δ)^{s},~ \text{for}~s \in (0, 1).$$ We establish the existence and some characteristics of the first eigenvalue and associated eigenfunctions to the above problem, based on the topology of the sets $\mathcal{D}$ and $\mathcal{N}$. Next, we apply these results to establish bifurcation type results, both from zero and infinity for the problem \eqref{ql} which is an asymptotically linear problem inclined with $(P_λ)$.

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arXiv2024

Existence of normalized ground state solution to a mixed Schrödinger system in a plane

In this paper, we establish the existence of positive ground state solutions for a class of mixed Schrödinger systems with concave-convex nonlinearities in $\mathbb{R}^2$, subject to $L^2$-norm constraints; that is, \[ \left\{ \begin{aligned} -\partial_{xx} u + (-Δ)_y^s u + λ_1 u &= μ_1 u^{p-1} + βr_1 u^{r_1-1} v^{r_2}, && -\partial_{xx} v + (-Δ)_y^s v + λ_2 v &= μ_2 v^{q-1} + βr_2 u^{r_1} v^{r_2-1}, && \end{aligned} \right. \] subject to the $L^2$-norm constraints: \[ \int_{\mathbb{R}^2} u^2 \,\mathrm{d}x\mathrm{d}y = a \quad \text{and} \quad \int_{\mathbb{R}^2} v^2 \,\mathrm{d}x\mathrm{d}y = b, \] where $(x,y)\in \mathbb{R}^2$, $u, v \geq 0$, $s \in \left(1/2, 1 \right)$, $μ_1, μ_2, β> 0$, $r_1, r_2 > 1$, the prescribed masses $a, b > 0$, and the parameters $λ_1, λ_2$ appear as Lagrange multipliers. Moreover, the exponents $p, q, r_1 + r_2$ satisfy: \[ \frac{2(1+3s)}{1+s} < p, q, r_1 + r_2 < 2_s, \] where $2_s = \frac{2(1+s)}{1-s}$. To obtain our main existence results, we employ variational techniques such as the Mountain Pass Theorem, the Pohozaev manifold, Steiner rearrangement, and others, consolidating the works of Louis Jeanjean et al. \cite{jeanjean2024normalized}.

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arXiv2024

On the study of $(p, Q)$-Laplace Choquard equations with critical Trudinger-Moser nonlinearity in $\mathbb{H}^N$

This paper deals with the existence and multiplicity of nontrivial solutions for $(p, Q)$-Laplace equations with the Stein-Weiss reaction under critical exponential nonlinearity in the Heisenberg group $\mathbb{H}^N$. In addition, a weight function and two positive parameters have also been included in the nonlinearity. The developed analysis is significantly influenced by these two parameters. Further, the mountain pass theorem, the Ekeland variational principle, the Trudinger-Moser inequality, the doubly weighted Hardy-Littlewood-Sobolev inequality and a completely new Brézis-Lieb type lemma for Choquard nonlinearity play key roles in our proofs.

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arXiv2023

Degenerate Schr{ö}dinger-Kirchhoff $(p, N)$-Laplacian problem With singular Trudinger-Moser nonlinearity in $\mathbb{R}^N$

In this paper, we deal with the existence of nontrivial nonnegative solutions for a $(p, N)$-Laplacian Schr{ö}dinger-Kirchhoff problem in $\mathbb{R}^N$ with singular exponential nonlinearity. The main features of the paper are the $(p, N)$ growth of the elliptic operators, the double lack of compactness, and the fact that the Kirchhoff function is of degenerate type. To establish the existence results, we use the mountain pass theorem, the Ekeland variational principle, the singular Trudinger-Moser inequality, and a completely new Brézis-Lieb type lemma for singular exponential nonlinearity.

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arXiv2023

Existence and non-existence results to a mixed Schrodinger system in a plane

This article focuses on the existence and non-existence of solutions for the following system of local and nonlocal type \begin{equation*} \left\{ \begin{aligned} -\partial_{xx}u + (-Δ)_{y}^{s_{1}} u + u - u^{2_{s_{1}}^{}-1} = καh(x,y) u^{α-1}v^β & \quad \mbox{in} ~ \mathbb{R}^{2}, -\partial_{xx}v + (-Δ)_{y}^{s_{2}} v + v- v^{2_{s_{2}}^{}-1} = κβh(x,y) u^αv^{β-1} & \quad \mbox{in} ~ \mathbb{R}^{2}, u,v ~ \geq ~0 \quad \mbox{in} ~ \mathbb{R}^{2}, \end{aligned} \right. \end{equation*} where $s_{1},s_{2} \in (0,1),~α,β>1,~α+β\leq \min \{ 2_{s_{1}}^{},2_{s_{2}}^{}\}$, and $2_{s_i}^{} = \frac{2(1+s_i)}{1-s_i}, i=1,2$. The existence of a ground state solution entirely depends on the behaviour of the parameter $κ>0$ and on the function $h$. In this article, we prove that a ground state solution exists in the subcritical case if $κ$ is large enough and $h$ satisfies (1.3). Further, if $κ$ becomes very small in this case then there does not exist any solution to our system. The study in the critical case, i.e. $s_1=s_2=s, α+β=2_s$, is more complex and the solution exists only for large $κ$ and radial $h$ satisfying (H1). Finally, we establish a Pohozaev identity which enables us to prove the non-existence results under some smooth assumptions on $h$.

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arXiv2023

Nonlocal critical exponent singular problems under mixed Dirichlet-Neumann boundary conditions

In this paper, we study the following singular problem, under mixed Dirichlet-Neumann boundary conditions, and involving the fractional Laplacian \begin{equation*} \label{1} \begin{cases} (-Δ)^{s}u = λu^{-q} + u^{2^*_s-1}, \quad u>0 \quad \text{in }Ω, \mathcal A(u) = 0 \quad \text{on}~ \partialΩ= \sum_{D} \cup \sum_{\mathcal{N}}, \end{cases} \tag{$P_λ$} \end{equation*} where $Ω\subset \mathbb{R}^N$ is a bounded domain with smooth boundary $\partialΩ$, $1/2 0$ is a real parameter, $ 0 < q < 1 $, $N>2s$, $2^*_s=2N/(N-2s)$ and $$\mathcal{A}(u)= u \mathcal{X}_{\sum_{D}} + {\partial_νu}\mathcal{X}_{ \sum_{\mathcal{N}}}, \quad{\partial_ν=\frac{\partial }{\partialν}}.$$ Here $\sum_{D}$, $\sum_{\mathcal{N}}$ are smooth $(N-1)$ dimensional submanifolds of $\partial Ω$ such that $\sum_{D} \cup \sum_{\mathcal{N}}= \partialΩ$, $\sum_{D} \cap \sum_{\mathcal{N}}= \emptyset $ and $\sum_{D} \cap \overline{\sum_{\mathcal{N}}} = τ'$ is a smooth $(N-2)$ dimensional submanifold of $\partialΩ$. Within a suitable range of $λ$, we establish existence of at least two opposite energy solutions for \eqref{1} using the standard Nehari manifold technique.

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arXiv2023

On a fractional system of NLS-KDV equations with Hardy potentials

In this article, our main concern is to study the existence of bound and ground state solutions for the following fractional system of nonlinear Schrödinger-Korteweg-De Vries (NLS-KdV, in short) 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} &= 2νh(x) u^{}v^{} & \quad \mbox{in} ~ \mathbb{R}^{N}, (-Δ)^{s_{2}} v - λ_{2} \frac{v}{|x|^{2s_{2}}} - v^{2_{s_{2}}^{*}-1} &= νh(x) u^{2} & \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 parameter $ν$ and on the function $h$, we obtain ground-state solutions using the concentration-compactness principle and the mountain-pass theorem.

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