$L_{p,q}$ spaces

Partial Differential Equations 1

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 * .

Concave-Convex Problems for the Robin p-Laplacian Plus an Indefinite Potential

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

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Some remarks on the curvature type problem on

Entropy generation and CO2 emission in presence of pulsating oscillations in a bifurcating thermoacoustic combustor with a Helmholtz resonator at off-design conditions

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.

On the generalized Hardy-Rellich inequalities

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.

The Fredholm alternative for the p-Laplacian in exterior domains

On the perturbed q-curvature problem on s4

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.

Abhishek Sarkar

Papers

On the generalized Hardy-Rellich inequalities

The Fredholm alternative for the p-Laplacian in exterior domains

On the perturbed q-curvature problem on s4

On the Generalized Hardy-Rellich Inequalities.

On the eigenvalue problem involving the weighted p-Laplacian in radially symmetric domains