Author

# Abhishek Sarkar

Other affiliations: Tata Institute of Fundamental Research, TIFR Centre for Applicable Mathematics

Bio: Abhishek Sarkar is an academic researcher from University of West Bohemia. The author has contributed to research in topics: p-Laplacian & Mathematics. The author has an hindex of 3, co-authored 12 publications receiving 18 citations. Previous affiliations of Abhishek Sarkar include Tata Institute of Fundamental Research & TIFR Centre for Applicable Mathematics.

##### Papers

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TL;DR: In this paper, the authors look for the weight functions that admit the generalized Hardy-Rellich type inequality into certain Lorentz-Zygmund spaces proved by Hansson and later by Brezis and Wainger.

Abstract: In this paper, we look for the weight functions (say g) that admit the following generalized Hardy-Rellich type inequality: into certain Lorentz-Zygmund spaces proved by Hansson and later by Brezis and Wainger.

6 citations

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TL;DR: In this paper, the authors investigated the Fredholm alternative for the p -Laplacian in an exterior domain which is the complement of the closed unit ball in R N (N ≥ 2 ).

Abstract: We investigate the Fredholm alternative for the p -Laplacian in an exterior domain which is the complement of the closed unit ball in R N ( N ≥ 2 ). By employing techniques of Calculus of Variations we obtain the multiplicity of solutions. The striking difference between our case and the entire space case is also discussed.

5 citations

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TL;DR: In this paper, the authors studied the fourth order elliptic problem with exponential nonlinearity and proved existence results under assumptions only on the shape of Q near its critical points, which is more general than the non-degeneracy conditions assumed so far.

5 citations

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TL;DR: In this paper, the weight functions that admit the generalized Hardy-Rellich type inequality were studied and the Muckenhoupt condition for the one dimensional weighted Hardy inequalities and a symmetrization inequality were used to obtain admissible weights in certain Lorentz-Zygmund spaces.

Abstract: In this article, we look for the weight functions (say $g$) that admits the following generalized Hardy-Rellich type inequality: $ \int_{\Omega} g(x) u^2 dx \leq C \int_{\Omega} |\Delta u|^2 dx, \forall u \in \mathcal{D}^{2,2}_0(\Omega), $ for some constant $C>0$, where $\Omega$ is an open set in $\mathbb{R}^N$ with $N\ge 1$. We find various classes of such weight functions, depending on the dimension $N$ and the geometry of $\Omega.$ Firstly, we use the Muckenhoupt condition for the one dimensional weighted Hardy inequalities and a symmetrization inequality to obtain admissible weights in certain Lorentz-Zygmund spaces. Secondly, using the fundamental theorem of integration we obtain the weight functions in certain weighted Lebesgue spaces. As a consequence of our results, we obtain simple proofs for the embeddings of $\mathcal{D}^{2,2}_0(\Omega)$ into certain Lorentz-Zygmund spaces proved by Hansson and later by Brezis and Wainger.

3 citations

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TL;DR: In this article, the eigenvalue problem is investigated and the existence of the first eigenpair and the asymptotic estimates for u (x ) and ∇ u ( x ) as | x | → R 1 + or R 2 − are also investigated.

2 citations

##### Cited by

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14 Mar 2020

TL;DR: In this article, the authors consider nonlinear Robin problems driven by the p-Laplacian plus an indefinite potential and prove a bifurcation-type theorem describing the dependence the dependence of the set of positive solutions on the parameter λ > 0.

Abstract: We consider nonlinear Robin problems driven by the p-Laplacian plus an indefinite potential. In the reaction, we have the competing effects of a parametric concave (that is, ( p − 1 ) -sublinear) term and of a convex (that is, ( p − 1 ) -superlinear) term which need not satisfy the Ambrosetti–Rabinowitz condition. We prove a "bifurcation-type" theorem describing in a precise way the dependence the dependence of the set of positive solutions on the parameter λ > 0 . In addition, we show the existence of a smallest positive solution u λ * and determine the monotonicity and continuity properties of the map λ ↦ u λ * .

10 citations

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TL;DR: In this paper, the existence, uniqueness and multiplicity of positive solutions of a nonlinear perturbed fourth-order problem related to the curvature was proved. But the existence and uniqueness of the positive solutions were not established.

Abstract: In this paper, we prove the existence, uniqueness and multiplicity of positive solutions of a nonlinear perturbed fourth-order problem related to the curvature.

9 citations

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TL;DR: In this article , a 2D numerical model of a Y-shaped bifurcating combustor with a Helmholtz resonator attached is developed, which is applied to gain insights on the entropy generation and nonlinearity of the pulsating oscillations and the flow fluctuations across the resonator neck.

5 citations