arXiv subjects

Shammi Malhotra

Publications and source records attributed to Shammi Malhotra.

7 recordsLinked to original sources

arXiv2026

The isocritical regime for mixed local-nonlocal $(p,q)$ Laplacian: existence of ground state, and decay estimates

We study the mixed local-nonlocal operator $\mathcal{L}_{p,q} := -Δ_p + (-Δ)_q^s$ in the isocritical regime $p^* = q_s^*$, i.e. $1 - N/p = s - N/q$, under which both operators become critical for the same nonlinearity. We consider \[ -Δ_p u + (-Δ)_q^s u = |u|^{p^*-2}u \qquad \text{in } \mathbb{R}^N, \] with $N \geq 2$, $1 < p < N$, $0 < s < 1$, $1 < sq < N$. In this regime the energy space reduces to $\mathcal{D}_0^{1,p}(\mathbb{R}^N)$, and both best Sobolev constants enter the variational structure simultaneously. We prove: $(i)$ existence of a nonnegative radial ground state via Nehari manifold methods and a double-threshold concentration-compactness analysis; $(ii)$ a logarithmic energy estimate, weak comparison principle, and strong maximum principle for all admissible exponents; $(iii)$ a weak Harnack inequality; and $(iv)$ sharp two-sided decay $U(x) \asymp |x|^{-(N-p)/(p-1)}$ for positive radial solutions, matching the fundamental solution of the $p$-Laplacian.

math.AP↗

arXiv2026

Existence and multiplicity of solutions for a critical Grushin problem with a singular nonlinearity

We investigate the existence and multiplicity of positive solutions to the problem \begin{equation} \begin{cases} \begin{aligned} - Δ_γ u &= λu^{p} + u^{-δ} &\quad \text{in } Ω, \quad u &= 0 &\quad \text{on } \partial Ω, \end{aligned} \end{cases} \end{equation} where $Δ_γ$ denotes the Grushin operator defined by \begin{equation} Δ_γ := Δ_x + (1+γ)^2 |x|^{2γ}Δ_y, \end{equation} with $γ>0$, $z=(x,y)\in \mathbb{R}^N$, $N=n+m$, $n \geq 1$, $m\geq 1$, $Ω\subset \mathbb{R}^N$ a smooth bounded domain, $λ>0$, $1 0$. The analysis depends on the exponent $p$, which may be subcritical, critical, or supercritical, that is, $p<2_γ^*-1$, $p=2_γ^*-1$, or $p>2_γ^*-1$, respectively, where $2_γ^*=\frac{2Q}{Q-2}$ is the critical Sobolev exponent associated with the Grushin operator, and $Q=m+(1+γ)n$ is the corresponding homogeneous dimension.

math.AP↗

arXiv2026

Multiplicity results for mixed local-nonlocal variable exponent problem involving singular and superlinear term

In this paper, we study a class of quasilinear elliptic equations involving both local and nonlocal operators with variable exponents. The problem exhibits singular nonlinearities along with a subcritical superlinear growth term and a parameter $λ$. We study the existence of multiple solutions with the help of variational methods by restricting the associated energy functional on appropriate subsets of the Nehari manifold. Using the topological index and the structure of the fibering maps, we analyse a key splitting property of the associated Nehari manifold. This decomposition allows us to establish the existence of two distinct solutions. Additionally, we establish the $L^\infty$-bound for the solutions.

math.AP↗

arXiv2025

Asymptotic behaviour and existence of positive solutions for mixed local nonlocal elliptic equations with Hardy potential

We investigate the existence and multiplicity of positive solutions to the following problem driven by the superposition of the Laplacian and the fractional Laplacian with Hardy potential \begin{equation*} \left\{ \begin{aligned} -Δu + (-Δ)^s u - μ\frac{u}{|x|^2} &= λ|u|^{p-2} u + |u|^{2^*-2} u \quad \text{in } Ω\subset \mathbb{R}^N, u &= 0 \quad \text{in } \mathbb{R}^N \setminus Ω, \end{aligned} \right. \end{equation*} where $ Ω\subset \mathbb{R}^N $ is a bounded domain with smooth boundary, $ 0 < s < 1 $, $ 1 < p < 2^* $, with $ 2^* = \frac{2N}{N-2} $, $ λ> 0 $, and $ μ\in (0, \barμ) $ where $\bar μ= \left( \frac{N-2}{2} \right)^2$. The aim of this paper is twofold. First, we establish uniform asymptotic estimates for solutions of the problem by means of a suitable transformation. Then, according to the value of the exponent $p$, we analyze three distinct cases and prove the existence of a positive solution. Moreover, in the sublinear regime $1 < p < 2$, we demonstrate the existence of multiple positive solutions for small perturbations of the fractional Laplacian.

math.AP↗

arXiv2025

Global Compactness Result for a Brézis-Nirenberg-Type Problem Involving Mixed Local Nonlocal Operator

This paper investigates the profile decomposition of Palais-Smale sequences associated with a Brezis-Nirenberg type problem involving a combination of mixed local nonlocal operators, given by \begin{equation*} \left\{\begin{aligned} &-Δu + (-Δ)^s u - λu = |u|^{2^*-2}u \;\;\mbox{ in } Ω, &\quad u=0\,\mbox{ in }\mathbb{R}^N\setminus Ω. \end{aligned} \right. \end{equation*} where $Ω\subseteq \mathbb{R}^{N}$ is a smooth bounded domain with $N \geq 3$, $s\in (0,1),\,λ\in\mathbb{R}$ is a real parameter and $2^* = \frac{2N}{N - 2} $ denotes the critical Sobolev exponent. As an application of the derived global compactness result, we further study the existence of positive solution of the corresponding Coron-type problem (C. R. Acad. Sci. Paris Sér I Math, 299(7):209-212, 1984) when $λ=0$.

math.AP↗

arXiv2025

On the eigenvalues and Fuč\'ık spectrum of $p$-Laplace local and nonlocal operator with mixed interpolated Hardy term

In this article, we are concerned with the eigenvalue problem driven by the mixed local and nonlocal $p$-Laplacian operator having the interpolated Hardy term \begin{equation*} \mathcal{T}(u) :=- Δ_p u + (- Δ_p)^s u - μ\frac{|u|^{p-2}u}{|x|^{p θ}}, \end{equation*} where $0<s<1<p<N$, $θ\in [s,1]$, and $μ\in (0,μ_0(θ))$. First, we establish a mixed interpolated Hardy inequality and then show the existence of eigenvalues and their properties. We also investigate the Fuč\'ık spectrum, the existence of the first nontrivial curve in the Fuč\'ık spectrum, and prove some of its properties. Moreover, we study the shape optimization of the domain with respect to the first two eigenvalues, the regularity of the eigenfunctions, the Faber-Krahn inequality, and a variational characterization of the second eigenvalue.

math.AP↗

arXiv2024

Quasilinear Schrödinger Equation involving Critical Hardy Potential and Choquard type Exponential nonlinearity

In this article, we study the following quasilinear Schrödinger equation involving Hardy potential and Choquard type exponential nonlinearity with a parameter $α$ \begin{equation*} \left\{ \begin{array}{l} - Δ_N w - Δ_N(|w|^{2α}) |w|^{2α- 2} w - λ\frac{|w|^{2αN-2}w}{\left( |x| \log\left(\frac{R}{|x|} \right) \right)^N} = \left(\int_Ω \frac{H(y,w(y))}{|x-y|^μ}dy\right) h(x,w(x))\; \mbox{in }\; Ω, w > 0 \mbox{ in } Ω\setminus \{ 0\}, \quad \quad w = 0 \mbox{ on } \partial Ω, \end{array} \right. \end{equation*} where $N\geq 2$, $α>\frac12$, $0\leq λ< \left(\frac{N-1}{N}\right)^N$, $0 < μ< N$, $h : \mathbb R^N \times \mathbb R \rightarrow \mathbb R$ is a continuous function with critical exponential growth in the sense of the Trudinger-Moser inequality and $H(x,t)= \int_{0}^{t} h(x,s) ds$ is the primitive of $h$. With the help of Mountain Pass Theorem and critical level which is obtained by the sequence of Moser functions, we establish the existence of a positive solution for a small range of $λ$. Moreover, we also investigate the existence of a positive solution for a non-homogeneous problem for every $0\leq λ<\left(\frac{N-1}{N}\right)^N.$ To the best of our knowledge, the results obtained here are new even in case of $N$-Laplace equation with Hardy potential.

math.AP↗