arXiv subjects

Arshi Vaishnavi

Publications and source records attributed to Arshi Vaishnavi.

4 recordsLinked to original sources

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.

math.AP↗

arXiv2025

Sharp embeddings and existence results for Logarithmic $p$-Laplacian equations with critical growth

In this paper, we derive a new $p$-Logarithmic Sobolev inequality and optimal continuous and compact embeddings into Orlicz-type spaces of the function space associated with the logarithmic $p$-Laplacian. As an application of these results, we study a class of Dirichlet boundary value problems involving the logarithmic $p$-Laplacian and critical growth nonlinearities perturbed with superlinear-subcritical growth terms. By employing the method of the Nehari manifold, we prove the existence of a nontrivial weak solution. Lastly, we conduct an asymptotic analysis of a weighted nonlocal, nonlinear problem governed by the fractional $p$-Laplacian with superlinear or sublinear type non-linearity, demonstrating the convergence of least energy solutions to a non-trivial, non-negative least energy solution of a Brezis-Nirenberg type or logistic-type problem, respectively, involving the logarithmic $p$-Laplacian as the fractional parameter $s \to 0^+$. The findings in this work serve as a nonlinear analogue of the results reported in \cite{Angeles-Saldana, Arora-Giacomoni-Vaishnavi, Santamaria-Saldana}, thereby extending their scope to a broader variational framework.

math.AP↗

arXiv2025

The Brezis-Nirenberg and logistic problem for the Logarithmic Laplacian

In this work, we study the non-local analogue of Brezis-Nirenberg and logistic type elliptic equations involving the logarithmic Laplacian and critical logarithmic non-linearity with superlinear-subcritical perturbation. In the first part of this work, we derive new sharp, continuous and compact embeddings of nonlocal Sobolev spaces (of order zero) into Orlicz type spaces. As an application of these embeddings and variational analysis as carried out in \cite{Angeles-Saldana-2023, Santamaria-Saldana-2022}, we prove the existence of a least energy weak solution of the Brezis-Nirenberg and logistic type problem involving the logarithmic Laplacian. For the uniqueness of solution, we prove a new Díaz-Saa type inequality, which is of independent interest and can be applied to a larger class of problems. In the second part of the work, depending upon the growth of non-linearity and regularity of the weight function, we study the small-order asymptotic of non-local weighted elliptic equations involving the fractional Laplacian of order $2s.$ We show that least energy solutions of a weighted non-local fractional problem with superlinear or sublinear type non-linearity converge to a non-trivial, non-negative least energy solution of a Brezis-Nirenberg type or logistic-type problem, respectively, involving the logarithmic Laplacian.

math.AP↗

arXiv2025

Nonlocal elliptic equations involving logarithmic Laplacian: Existence, non-existence and uniqueness results

In this work, we study the existence, non-existence, and uniqueness results for nonlocal elliptic equations involving logarithmic Laplacian, and subcritical, critical, and supercritical logarithmic nonlinearities. The Poho\u zaev's identity and Díaz-Saa type inequality are proved, which are of independent interest and can be applied to a larger class of problems. Depending upon the growth of nonlinearities and regularity of the weight function, we study the small-order asymptotic of nonlocal weighted elliptic equations involving the fractional Laplacian of order $2s.$ We show that the least energy solutions of a weighted nonlocal problem with superlinear or sublinear growth converge to a nontrivial nonnegative least-energy solution of Brézis-Nirenberg type and logistic-type limiting problem respectively involving the logarithmic Laplacian.

math.AP↗