arXiv subjects

Sekhar Ghosh

Publications and source records attributed to Sekhar Ghosh.

At least 19 recordsLinked to original sources

arXiv2026

Optimization for the first weighted eigenvalue of local-nonlocal operators with a potential

In this work, we study the optimization problem for the first eigenvalue of mixed local nonlocal operators plus a potential $V$. We begin by investigating the existence and fundamental properties of the eigenvalues, with special emphasis on the first eigenvalue. Finally, we discuss the dependence of the first eigenvalue on the potential function and establish the existence of optimal potentials within certain admissible classes. In particular, we show the existence of a unique maximizer and a minimizer of the first eigenvalue on any bounded, closed, and convex subset of $L^q(\Omega)$. Moreover, these results enable us to characterize the maximizers and minimizers in the closed unit ball of $L^q(\Omega)$, as well as in the class of rearrangements of any $V\in L^q(\Omega)$.

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arXiv2026

A positive ground state for a planar Choquard equation with mixed diffusion and critical exponential growth

We study a two-dimensional Choquard equation driven by the mixed local and nonlocal operator $L:=-\Delta+(-\Delta)^s$, where the nonlinearity has critical exponential growth of Trudinger--Moser type. Under a coercive assumption on the potential and suitable one-sided assumptions on the nonlinearity, we prove the existence of a least energy positive solution. The proof combines Nehari manifold minimization, compactness below the critical Trudinger--Moser threshold, local regularity, and a strong maximum principle.

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arXiv2026

Mixed local-nonlocal quasilinear problems with mixed interpolated Hardy potential

This paper addresses the existence of nontrivial solutions to a class of mixed local-nonlocal problems involving a mixed interpolated Hardy potential. We first establish a concentration-compactness principle for mixed local and nonlocal operators. This result is combined with Ricceri's variational principle to obtain an existence result for quasilinear elliptic problems under different growth assumptions on the nonlinearity. Furthermore, we apply the classical mountain pass theorem to obtain a second existence result in the superlinear case.

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arXiv2026

On weak and viscosity solutions to a nonhomogeneous mixed local-nonlocal equation

This paper explores the relationship between weak and viscosity solutions to a nonhomogeneous mixed local and non-local $p$-Laplace equation in a bounded Lipschitz domain in $\mathbb{R}^N$. Under certain conditions, we derive the comparison principle for weak subsolutions and weak supersolutions to the problem. For $1<p<\infty$, we establish that continuous weak supersolutions to the problem are viscosity supersolutions, using the comparison principle. Furthermore, we show that bounded viscosity supersolutions are weak supersolutions for $p \geq 2$.

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arXiv2026

Harnack inequality for superposition operators of mixed fractional order

The main aim of this paper is to establish the H\"older continuity and the Harnack inequality for weak solutions to Dirichlet problems associated with superposition operators of mixed fractional order, thereby complementing our previous work \cite{BGKL2026}. To achieve this, we extend the De Giorgi--Nash--Moser theory to the framework of superposition operators by introducing a novel {\it nonlocal superposition tail}, which appears to be the first contribution of its kind in the literature. The obtained results are new even in the classical linear case $p=2$, thereby illustrating the broader applicability of the analytical techniques developed in this work. As intermediate steps toward the proof of the main results, we also establish a logarithmic estimate for weak supersolutions, local boundedness for weak subsolutions, a weak Harnack inequality for weak supersolutions, an expansion of positivity for weak supersolutions, and tail estimates for weak solutions.

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arXiv2026

Regularity of superposition operators of mixed fractional order

We extend the De Giorgi--Nash--Moser theory to superposition operators of mixed fractional operators. In particular, we investigate several regularity properties for this class of operators. We establish the Caccioppoli-type inequality with tail for weak subsolutions, local boundedness of weak subsolutions, local H\"older continuity of weak solutions, the weak Harnack inequality for weak supersolutions, and the lower semicontinuity of weak supersolutions. Furthermore, we prove the expansion of positivity, a preliminary Harnack inequality, and the upper semicontinuity of weak subsolutions. Our results apply to both fixed-sign and sign-changing solutions involving mixed local--nonlocal superposition fractional operators. Notably, the results are new even in the classical linear case $p=2$, demonstrating the broader applicability of the techniques developed in this work.

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arXiv2026

Multiplicity of Solutions to the Brezis-Nirenberg Problem on Hyperbolic Spaces

This article investigates the multiplicity of solutions to the Brezis-Nirenberg problem on smooth bounded domains in the hyperbolic space $\mathbb{B}^N$ for $N \ge 4$. Specifically, we study the critical semilinear equation $-\Delta_{\mathbb{B}^N} u = \lambda u + |u|^{2^*-2}u$ under Dirichlet boundary conditions for $\lambda > \frac{N(N-2)}{4}$. Overcoming the analytic challenges induced by the hyperbolic geometry and the intricate concentration profiles of Palais-Smale sequences, we establish the existence of multiple pairs of nontrivial solutions. Using the equivariant Ljusternik-Schnirelmann category, we obtain lower bounds on the number of solutions depending on the position of the parameter $\lambda$ relative to the Dirichlet spectrum of the Laplace-Beltrami operator.

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arXiv2025

Spectral analysis, maximum principles and shape optimization for nonlinear superposition operators of mixed fractional order

The main objective of this paper is to investigate the spectral properties, maximum principles, and shape optimization problems for a broad class of nonlinear ``superposition operators" defined as continuous superpositions of operators of mixed fractional order, modulated by a signed finite Borel measure on the unit interval. This framework encompasses, as particular cases, mixed local and nonlocal operators such as $-\Delta_p+(-\Delta_p)^s$, finite (possibly infinite) sums of fractional $p$-Laplacians with different orders, as well as operators involving fractional Laplacians with ``wrong" signs. The main findings, obtained through variational techniques, concern the spectral analysis of the Dirichlet eigenvalue problem associated with general superposition operators with special emphasis on various properties of the first eigenvalue and its corresponding eigenfunction. We establish weak and strong maximum principles for positive superposition operators by introducing an appropriate notion of the {\it nonlocal tail} for this class of superposition operators and deriving a logarithmic estimate, both of which are of independent interest. Utilizing these newly developed tools, we further investigate the spectral properties of such superposition operators and prove that the first eigenvalue is isolated and simple. Moreover, we show that the eigenfunctions corresponding to positive eigenvalues are globally bounded and that they change sign when associated with higher eigenvalues. In addition, we demonstrate that the second eigenvalue is well-defined and provide the mountain pass characterization. Finally, we address shape optimization problems, in particular, the Faber--Krahn inequality associated with the principal frequency associated with the superposition operators.

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arXiv2025

Fractional Morrey-Sobolev type embeddings and nonlocal subelliptic problems with oscillating nonlinearities on stratified Lie groups

In this paper, we establish the fractional Morrey-Sobolev type embeddings on stratified Lie groups. This extends and complements the Sobolev type embeddings derived in \cite{GKR}. As an application of the results, we study the following nonlocal subelliptic problem, \begin{equation} \begin{cases} (-\Delta_{\mathbb{G}, p})^s u= \lambda \beta(x) g(u) & \text{in} \quad \Omega, \\ u=0\quad & \text{in}\quad \mathbb{G}\backslash \Omega, \end{cases} \end{equation} where $0 0})$ and $g \in C(\mathbb{R}, \R) $ oscillates near the origin or at infinity. By using the variational principle of Ricceri, we prove the existence and asymptotic behaviors of infinitely many solutions to the problem under consideration. We emphasize that the results obtained here are also novel for $\mathbb{G}$ being the Heisenberg group and $p=2$.

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arXiv2025

Superlinear problems involving nonlinear superposition operators of mixed fractional order

In this work, we study a class of elliptic problems involving nonlinear superpositions of fractional operators of the form \[ A_{\mu,p}u := \int_{[0,1]} (-\Delta)_{p}^{s} u \, d\mu(s), \] where $\mu$ is a signed measure on $[0,1]$, coupled with nonlinearities of superlinear type. Our analysis covers a variety of superlinear growth assumptions, beginning with the classical Ambrosetti--Rabinowitz condition. Within this framework, we construct a suitable variational setting and apply the Fountain Theorem to establish the existence of infinitely many weak solutions. The results obtained are novel even in the special cases of superpositions of fractional $p$-Laplacians, or combinations of the fractional $p$-Laplacian with the $p$-Laplacian. More generally, our approach applies to finite sums of fractional $p$-Laplacians with different orders, as well as to operators in which fractional Laplacians appear with ``wrong'' signs. A distinctive contribution of the paper lies in providing a unified variational framework that systematically accommodates this broad class of operators.

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arXiv2025

Infinitely many solutions for nonlinear superposition operators of mixed fractional order involving critical exponent

This paper addresses a class of elliptic problems involving the superposition of nonlinear fractional operators with the critical Sobolev exponent in the sublinear regimes. We establish the existence of infinitely many nontrivial weak solutions using a variational framework combining a truncation argument with the notion of genus. A central part of our analysis is the verification of the Palais--Smale (PS) condition for the associated energy functional for every $q \in (1, p_{s_\sharp}^*)$, despite the challenges posed by the lack of compactness due to the critical exponent. The results obtained in the paper are new even in the classical case $p = 2$, highlighting the broader applicability of the methods developed here.

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arXiv2025

On equivalence of weak and viscosity solutions to nonlocal double phase problems with nonhomogeneous data

This work focuses on the nonhomogeneous nonlocal double phase problem \begin{align*} L_au(x)=f(x,u,D_s^p u, D_{a,t}^q u) \text{ in } \Omega, \end{align*} where $\Omega\subset\mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $0<s,t<1<p\leq q<\infty$ with $tq\leq sp$ and the operator $L_a$ is defined as \begin{align*} L_a u(x)&=2\operatorname{P.V.}\int_{\mathbb{R}^N}|u(x)-u(y)|^{p-2}(u(x)-u(y))K_{s,p}(x,y) &\ \ \ +2\operatorname{P.V.}\int_{\mathbb{R}^N}a(x,y)|u(x)-u(y)|^{q-2}(u(x)-u(y))K_{t,q}(x,y)dy. \end{align*} We establish the equivalence between weak and viscosity solutions under boundedness and continuity assumptions. In addition, the local boundedness of weak solutions in some special cases on $f$ is also obtained using the notion of De Giorgi classes.

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arXiv2025

Brezis-Nirenberg type problems associated with nonlinear superposition operators of mixed fractional order

This paper aims to study the Brezis-Nirenberg type problem driven by the nonlinear superposition of operators of the form $$A_{\mu, p}u:=\int_{[0,1]}(-\Delta)_{p}^{s} u\,\, d \mu(s),$$ where $\mu$ denotes the signed measure over $[0, 1]$. We consider nonlinear nonlocal equations associated with $A_{\mu, p}$, involving critical nonlinearity and lower-order perturbation. Using variational techniques, we establish existence results for the critical problem by employing weak lower semicontinuity arguments under general assumptions on the perturbation term. We discuss the multiplicity results when the perturbation term vanishes at the origin. Additionally, when the lower-order term is a pure power function, we examine the Brezis-Nirenberg-type problem using the mountain pass technique. Furthermore, we address the existence of solutions to subcritical problems associated with $A_{\mu, p}.$ Our findings are novel, even in the case of the sum of two distinct fractional $p$-Laplacians or a combination of a fractional $p$-Laplacian with a classical $p$-Laplacian. More generally, our framework is sufficiently broad to accommodate finite sums of different fractional $p$-Laplacians as well as cases involving fractional Laplacians with ``wrong" signs. A key contribution of this study is the development of a unified approach that systematically addresses these problems by incorporating a broad class of operators and lower-order perturbation terms within a common theoretical framework. The results remain new even in the case of linear superposition of fractional operators of different orders.

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arXiv2025

On sign-changing solutions for mixed local and nonlocal $p$-Laplace operator

In this paper, we use the method of invariant sets of descending flows to demonstrate the existence of multiple sign-changing solutions for a class of elliptic problems with zero Dirichlet boundary conditions. By combining Nehari manifold techniques with a constrained variational approach and Brouwer degree theory, we establish the existence of a least-energy sign-changing solution. Furthermore, we prove that the energy of the least energy sign-changing solution is strictly greater than twice the ground state energy. This work extends the celebrated results of Bartsch $et~al.$ [Proc. Lond. Math. Soc. (3), 91(1): 129-152, 2005] and Chang $et~al.$ [Adv. Nonlinear Stud., 19(1): 29-53, 2019] to the mixed local and nonlocal $p$-Laplace operator, providing a novel contribution even in the case when $p=2$.

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arXiv2025

Critical Equations Involving Nonlocal Subelliptic Operators on Stratified Lie Groups: Spectrum, Bifurcation and Multiplicity

In this paper, we explore the bifurcation phenomena and establish the existence of multiple solutions for the nonlocal subelliptic Brezis-Nirenberg problem: \begin{equation*} \begin{cases} (-\Delta_{\mathbb{G}})^s u= |u|^{2_s^*-2}u+\lambda u \quad &\text{in}\quad \Omega, \\ u=0\quad & \text{in}\quad \mathbb{G}\backslash \Omega, \end{cases} \end{equation*} where $(-\Delta_{\mathbb{G}})^s$ is the fractional sub-Laplacian on the stratified Lie group $\mathbb{G}$ with homogeneous dimension $Q,$ $\Omega$ is a open bounded subset of $\mathbb{G},$ $s \in (0,1)$, $Q> 2s,$ $2_s^*:=\frac{2Q}{Q-2s}$ is subelliptic fractional Sobolev critical exponent, $\lambda>0$ is a real parameter. This work extends the seminal contributions of Cerami, Fortunato, and Struwe to nonlocal subelliptic operators on stratified Lie groups. A key component of our study involves analyzing the subelliptic $(s, p)$-eigenvalue problem for the (nonlinear) fractional $p$-sub-Laplacian $(-\Delta_{p,{\mathbb{G}}})^s$ \begin{align*} (-\Delta_{p,{\mathbb{G}}})^s u&=\lambda |u|^{p-2}u,~\text{in}~\Omega,\nonumber u&=0~\text{ in }~{\mathbb{G}}\setminus\Omega, \end{align*} with $0 ps$, over the fractional Folland-Stein-Sobolev spaces on stratified Lie groups applying variational methods. Particularly, we prove that the $(s, p)$-spectrum of $(-\Delta_{p,{\mathbb{G}}})^s$ is closed and the second eigenvalue $\lambda_2(\Omega)$ with $\lambda_2(\Omega)>\lambda_1(\Omega)$ is well-defined and provides a variational characterization of $\lambda_2(\Omega)$. We emphasize that the results obtained here are also novel for $\mathbb{G}$ being the Heisenberg group.

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arXiv2024

Existence results for mixed local and nonlocal elliptic equations involving singularity and nonregular data

In this paper, we prove the existence of weak, veryweak and duality solutions to a class of elliptic problems involving singularity and measure data which is given by: $-\Delta u+(-\Delta)^s u = \frac{f(x)}{u^\gamma} +\mu$ in $\Omega$ with the zero Dirichlet boundary data $u=0$ in $\mathbb R^N \setminus \Omega$. The existence of weak solutions is obtained by approximating a sequence of problems for $0<\gamma\leq1$ and $\gamma>1$. We employ Schauder's fixed point theorem and embeddings of Marcinkiewicz spaces. The novelty of our work is that we prove the existence of a duality solution and its equivalence with weak solutions to the problem $\mathcal{L}u=\mu$. Moreover, we prove a veryweak maximum principle and a Kato-type inequality for the mixed local-nonlocal operator $\mathcal{L}=-\Delta +(-\Delta)^s$, which are crucial tools to guarantee the existence of veryweak solutions to the problem. Using a Kato-type inequality, maximum principle together with sub-super solution method, we prove the existence of veryweak solution for $0<\gamma<1$. Our work extends the studies due to Oliva and Petitta [ESAIM Control Optim. Calc. Var., 22(1):289--308, 2016.] and Petitta [Adv. Nonlinear Stud., 16(1):115--124, 2016.] for the mixed local-nonlocal operator.

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arXiv2024

Subelliptic Nonlocal Brezis-Nirenberg Problems on Stratified Lie Groups

In this paper, we investigate the subelliptic nonlocal Brezis-Nirenberg problem on stratified Lie groups involving critical nonlinearities, namely, \begin{align*} (-\Delta_{\mathbb{G}, p})^s u&= \mu |u|^{p_s^*-2}u+\lambda h(x, u) \quad \text{in}\quad \Omega, \\ u&=0\quad \text{in}\quad \mathbb{G}\backslash \Omega, \end{align*} where $(-\Delta_{\mathbb{G}, p})^s$ is the fractional $p$-sub-Laplacian on a stratified Lie group $\mathbb{G}$ with homogeneous dimension $Q,$ $\Omega$ is an open bounded subset of $\mathbb{G},$ $s \in (0,1)$, $\frac{Q}{s}>p\geq2,$ $p_s^*:=\frac{pQ}{Q-ps}$ is subelliptic fractional Sobolev critical exponent, $\mu, \lambda>0$ are real parameters and $h$ is a lower order perturbation of the critical power $|u|^{p_s^*-2}u$. Utilising direct methods of the calculus of variation, we establish the existence of at least one weak solution for the above problem under the condition that the real parameter $\lambda$ is sufficiently small. Additionally, we examine the problem for $\mu = 0$, representing subelliptic nonlocal equations on stratified Lie groups depending on one real positive parameter and involving a subcritical nonlinearity. We demonstrate the existence of at least one solution in this scenario as well. We emphasize that the results obtained here are also novel for $p=2$ even for the Heisenberg group.

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arXiv2024

A weighted eigenvalue problem for mixed local and nonlocal operators with potential

We study an {\it indefinite weighted eigenvalue problem} for an operator of {\it mixed-type} (that includes both the classical {\it $p$-Laplacian} and the {\it fractional $p$-Laplacian}) in a bounded open subset $\Omega\subset \mathbb{R}^N \,(N\geq2)$ with {\it Lipschitz boundary} $\partial \Omega$, which is given by \begin{align*} -\Delta_p u + (-\Delta_p)^su+V(x)|u|^{p-2}u&=\lambda g(x)|u|^{p-2}u~\text{in}~\Omega, u&=0~\text{in}~\mathbb{R}^N\setminus\Omega, \end{align*} where $\lambda >0$ is a parameter, exponents $0 0$ a.e. in $\Omega$. Using the variational tools together with a {\it weak comparison} and {\it strong maximum principles}, we investigate the existence and uniqueness of {\it principal eigenvalue} and discuss its qualitative properties. Moreover, with the help of {\it Ljusternik-Schnirelman category theory}, it is proved that there exists a {\it nondecreasing sequence of positive eigenvalues} which goes to infinity. Further, we show that {\it the set of all positive eigenvalues is closed}, and {\it eigenfunctions} associated with every {\it positive eigenvalue} are bounded.

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