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On the eigenvalues and Fučík spectrum of $p$-Laplace local and nonlocal operator with mixed interpolated Hardy term

Shammi Malhotra, Sarika Goyal, K. Sreenadh

TL;DR

We address the spectrum and Fučík spectrum for the mixed local–nonlocal $p$-Laplacian with interpolated Hardy potential, establishing a variational framework on the mixed energy space $X(\Omega)$ and proving a mixed interpolated Hardy inequality. We obtain an unbounded, increasing sequence of eigenvalues $\{\lambda_k\}$ with a simple, positive, and isolated first eigenfunction $\phi_1$, and we characterize higher eigenfunctions by sign changes; the Fučík spectrum is explored through a restricted minimax, yielding the first nontrivial curve $d \mapsto (d+c(d),c(d))$ and its symmetric counterpart, with Lipschitz regularity and asymptotic convergence to $\lambda_1$ as $d\to\infty$. Shape optimization results include the Faber–Krahn inequality for the first eigenvalue and a Hong–Krahn–Szegö type result for the second eigenvalue, complemented by regularity of eigenfunctions and nodal-domain analyses. Overall, the work extends spectral theory for operators that couple local and nonlocal diffusion with singular Hardy potentials, with implications for spectral geometry and domain optimization.

Abstract

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 Fučík spectrum, the existence of the first nontrivial curve in the Fučík 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.

On the eigenvalues and Fučík spectrum of $p$-Laplace local and nonlocal operator with mixed interpolated Hardy term

TL;DR

We address the spectrum and Fučík spectrum for the mixed local–nonlocal -Laplacian with interpolated Hardy potential, establishing a variational framework on the mixed energy space and proving a mixed interpolated Hardy inequality. We obtain an unbounded, increasing sequence of eigenvalues with a simple, positive, and isolated first eigenfunction , and we characterize higher eigenfunctions by sign changes; the Fučík spectrum is explored through a restricted minimax, yielding the first nontrivial curve and its symmetric counterpart, with Lipschitz regularity and asymptotic convergence to as . Shape optimization results include the Faber–Krahn inequality for the first eigenvalue and a Hong–Krahn–Szegö type result for the second eigenvalue, complemented by regularity of eigenfunctions and nodal-domain analyses. Overall, the work extends spectral theory for operators that couple local and nonlocal diffusion with singular Hardy potentials, with implications for spectral geometry and domain optimization.

Abstract

In this article, we are concerned with the eigenvalue problem driven by the mixed local and nonlocal -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 , , and . First, we establish a mixed interpolated Hardy inequality and then show the existence of eigenvalues and their properties. We also investigate the Fučík spectrum, the existence of the first nontrivial curve in the Fučík 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.

Paper Structure

This paper contains 6 sections, 34 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Interpolated Hardy Inequality and Convergence results
  4. Eigenvalue Problem for T
  5. Fučı́k Spectrum
  6. Shape Optimization and Regularity

Key Result

Proposition 2.1

Let $p \in (1, \infty)$ and $\gamma \geq 1$, then for every $a,b,M \geq 0,$ we have where $a_M = \min \{a,M\}$ and $b_M = \min\{ b, M\}$.

Theorems & Definitions (57)

  • Proposition 2.1: Lemma $C.2$ of brasco_cheeger
  • Proposition 2.2: Formula $2.2$ of simon_inequalities
  • Proposition 2.3: Lemma $A.2$ of brasco_second_eigenvalue
  • Theorem 2.1
  • Proposition 2.4: Discrete Picone Identity amghibech_discrete_picone
  • Proposition 2.5: Continuous Picone Identity huang_continous_picone_identity
  • Lemma 2.1: Mountain Pass Lemma
  • Lemma 3.1: Mixed Interpolated Hardy Inequality
  • proof
  • Remark
  • ...and 47 more