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Lovelesh Sharma

Publications and source records attributed to Lovelesh Sharma.

11 recordsLinked to original sources
arXiv2026

A sharp eigenvalue theorem for mixed elliptic problems under mixed boundary conditions

In this paper, we study a class of eigenvalue problems involving both local and nonlocal operators, namely the classical Laplacian and the fractional Laplacian, under mixed boundary conditions. More precisely, we consider the problem \begin{equation}\label{1} \left\{ \begin{aligned} \mathcal{L}u &= \lambda f(u), \quad u>0 &&\text{in }\Omega,\\ u&=0 &&\text{in }U^c,\\ \mathcal{N}_s(u)&=0 &&\text{in }\mathcal{N},\\ \frac{\partial u}{\partial\nu}&=0 &&\text{on }\partial\Omega\cap\overline{\mathcal{N}}, \end{aligned} \right. \tag{$P_\lambda$} \end{equation} where \( U=\Omega\cup\mathcal{N}\cup \bigl(\partial\Omega\cap\overline{\mathcal{N}}\bigr), \) \(\Omega\subseteq\mathbb{R}^n\) is a bounded open set with smooth boundary, \(\lambda>0\) is a real parameter, \(f\) is continuous function with \(f(0)=0\), and \[ \mathcal{L}=-\Delta+(-\Delta)^s, \qquad s\in(0,1). \] We establish a characterization theorem for the existence of positive weak solutions to problem (P_{\lambda}). Motivated by the classical elliptic framework developed by Molica Bisci and R\u{a}dulescu \cite{MolicaBisciRadulescu2017}, we establish a corresponding characterization result for mixed local-nonlocal operators under mixed boundary conditions.

math.AP↗
arXiv2026

H\"older regularity and Harnack inequality for the logarithmic Laplacian

In this article, we establish Schauder-type estimates for the logarithmic Laplacian. We show that for a $\kappa$-H\"older inhomogeneous term, the solution is also $\kappa$-H\"older in the interior. In fact, the interior regularity slightly exceeds $\kappa$-H\"older smoothness up to a logarithmic correction. Additionally, we prove a Harnack inequality for non-negative solutions.

math.AP↗
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.

math.AP↗
arXiv2026

The role of the mean curvature in nonlinear p-Laplacian problems with critical exponent

We deal with critical nonlinear problems involving the p-Laplacian operator on bounded domains with mixed boundary conditions. We prove the existence of least energy solutions. Our work shows a significant difference between the semi-linear case p = 2 and the quasilinear case for the existence results. Moreover, neither the results for the Laplacian can be extended to the p-Laplacian, nor the method for the p-Laplacian can apply to the Laplacian setting. Additionally, the cases (p < 2 and p > 2) present different challenges and need to be studied separately. More precisely, when p > 2, the effect of the geometry of the boundary conditions dominates that one of the potential, whereas for p < 2 the opposite behavior holds true.

math.DG↗
arXiv2026

Spectrum properties of mixed operators under the mixed boundary conditions

In this paper, we describe the spectrum properties of mixed operators, precisely the superposition of 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\: &= \lambda u,~~\text{in} ~\Omega, u&=0~~~~~\text{in} ~~{U^c}, \mathcal{N}_s(u)&=0 ~~~~~\text{in} ~~{\mathcal{N}}, \frac{\partial u}{\partial \nu}&=0 ~~~~~\text{in}~~ \partial \Omega \cap \overline{\mathcal{N}}, \end{split} \right.\tag{$P_\lambda$} \end{equation} where $U= (\Omega \cup {\mathcal{N}} \cup (\partial\Omega\cap\overline{\mathcal{N}}))$, $\Omega \subseteq \mathbb{R}^n$ is a non empty bounded open set with sufficiently smooth boundary $\partial\Omega$, say of class $C^1$, and $\mathcal{D}$, $\mathcal{N}$ are open subsets of $\mathbb{R}^n\setminus{\bar{\Omega }}$ such that $\overline{{\mathcal{D}} \cup {\mathcal{N}}}= \mathbb{R}^n\setminus{\Omega}$, $\mathcal{D} \cap {\mathcal{N}}= \emptyset $ and $\Omega\cup \mathcal{N}$ is a bounded set with sufficiently smooth boundary, $\lambda >0$ is a real parameter and $\mathcal{L}= -\Delta+(-\Delta)^{s},~ \text{for}~s \in (0, 1).$

math.AP↗
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 \Omega, u &= 0 \quad \text{in} \quad U^c, \mathcal{N}_s(u) &= 0 \quad \text{in} \quad \mathcal{N}, \frac{\partial u}{\partial \nu} &= 0 \quad \text{in} \quad \partial \Omega \cap \overline{\mathcal{N}}, \end{aligned} \right. \tag{$P_\lambda$} \end{equation*} where $U= (\Omega \cup {\mathcal{N}} \cup (\partial\Omega\cap\overline{\mathcal{N}}))$, $\Omega \subseteq \mathbb{R}^N$ is a non empty open set, $\mathcal{D}$, $\mathcal{N}$ are open subsets of $\mathbb{R}^N\setminus{\bar{\Omega }}$ such that ${\mathcal{D}} \cup {\mathcal{N}}= \mathbb{R}^N\setminus{\bar{\Omega}}$, $\mathcal{D} \cap {\mathcal{N}}= \emptyset $ and $\Omega\cup \mathcal{N}$ is a bounded set with smooth boundary, $\lambda >0$ is a real parameter and $\mathcal{L}= -\Delta+(-\Delta)^{s},~ \text{for}~s \in (0, 1).$ Here $g(u)=u^{-q}$ or $g(u)= \lambda u^{-q}+ u^p$ with $0<q<1<p\leq 2^*-1$. We study $(P_\lambda)$ 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.

math.AP↗
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\: &= \lambda u,~~u>0~ \text{in} ~\Omega, u&=0~~\text{in} ~~{U^c}, \mathcal{N}_s(u)&=0 ~~\text{in} ~~{\mathcal{N}}, \frac{\partial u}{\partial \nu}&=0 ~~\text{in}~~ \partial \Omega \cap \overline{\mathcal{N}}, \end{split} \right.\tag{$P_\lambda$} \end{equation} where $U= (\Omega \cup {\mathcal{N}} \cup (\partial\Omega\cap\overline{\mathcal{N}}))$, $\Omega \subseteq \mathbb{R}^n$ is a non empty open set, $\mathcal{D}$, $\mathcal{N}$ are open subsets of $\mathbb{R}^n\setminus{\bar{\Omega }}$ such that $\overline{{\mathcal{D}} \cup {\mathcal{N}}}= \mathbb{R}^n\setminus{\Omega}$, $\mathcal{D} \cap {\mathcal{N}}= \emptyset $ and $\Omega\cup \mathcal{N}$ is a bounded set with smooth boundary, $\lambda >0$ is a real parameter and $$\mathcal{L}= -\Delta+(-\Delta)^{s},~ \text{for}~s \in (0, 1).$$ 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 $(P_\lambda)$.

math.AP↗
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 Br\'ezis-Lieb type lemma for Choquard nonlinearity play key roles in our proofs.

math.AP↗
arXiv2023

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\: &= \lambda u^{q} + u^{p}, \quad u>0 ~~ \text{in} ~\Omega, u&=0~~\text{in} ~~{D^c}, \mathcal{N}_s(u)&=0 ~~\text{in} ~~{\Pi_2}, \frac{\partial u}{\partial \nu}&=0 ~~\text{in}~~ \partial \Omega \cap \overline{\Pi_2}. \end{split} \right.\tag{$P_\lambda$} \end{equation*} {where $D= \left(\Omega \cup {\Pi_2} \cup (\partial\Omega\cap\overline{\Pi_2})\right)$ and $D^c$ is the complement of $D$, $\Omega \subseteq \mathbb{R}^n$ is a non empty open set, $\Pi_{1}$, $\Pi_{2}$ are open subsets of $\mathbb{R}^n\setminus{\bar \Omega }$ such that $\overline{{\Pi_{1}} \cup {\Pi_{{2}}}}= \mathbb{R}^n\setminus{\Omega}$, $\Pi_{1} \cap \Pi_{{2}}= \emptyset$ and $\Omega\cup \Pi_2$ is a bounded set with smooth boundary}, $\lambda >0$ is a real parameter, $ 0 < q < 1 2$ and $\mathcal{L}= -\Delta+(-\Delta)^{s},~ \text{for}~s \in (0, 1).$ 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 $\lambda$, $q$ and $p$}. Our article also contains results related to Picone's identity, strong maximum principles and comparison principles.

math.AP↗
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} (-\Delta)^{s}u = \lambda u^{-q} + u^{2^*_s-1}, \quad u>0 \quad \text{in }\Omega, \mathcal A(u) = 0 \quad \text{on}~ \partial\Omega = \sum_{D} \cup \sum_{\mathcal{N}}, \end{cases} \tag{$P_\lambda$} \end{equation*} where $\Omega \subset \mathbb{R}^N$ is a bounded domain with smooth boundary $\partial{\Omega}$, $1/2 0$ is a real parameter, $ 0 < q < 1 $, $N>2s$, $2^*_s=2N/(N-2s)$ and $$\mathcal{A}(u)= u \mathcal{X}_{\sum_{D}} + {\partial_{\nu}u}\mathcal{X}_{ \sum_{\mathcal{N}}}, \quad{\partial_{\nu}=\frac{\partial }{\partial{\nu}}}.$$ Here $\sum_{D}$, $\sum_{\mathcal{N}}$ are smooth $(N-1)$ dimensional submanifolds of $\partial \Omega$ such that $\sum_{D} \cup \sum_{\mathcal{N}}= \partial\Omega$, $\sum_{D} \cap \sum_{\mathcal{N}}= \emptyset $ and $\sum_{D} \cap \overline{\sum_{\mathcal{N}}} = \tau'$ is a smooth $(N-2)$ dimensional submanifold of $\partial{\Omega}$. Within a suitable range of $\lambda$, we establish existence of at least two opposite energy solutions for \eqref{1} using the standard Nehari manifold technique.

math.AP↗