Multiple solutions to asymptotically linear problems driven by superposition operators
Danilo Gregorin Afonso, Rossella Bartolo, Giovanni Molica Bisci
TL;DR
Problem addressed is existence and multiplicity of weak solutions to asymptotically linear problems driven by a superposition of fractional Laplacians $A_\mu$. The authors apply variational methods in the Hilbert space $X_0(\Omega)$, exploiting the spectral theory of $A_\mu$ and an abstract critical point theorem to handle lack of compactness and the indefinite nature. The main contributions are: existence of a solution when $\bar{\lambda}$ avoids the spectrum, and, under symmetry and spectral-gap conditions, the existence of at least $k-h+1$ pairs of nontrivial solutions; the results generalize classical and fractional Laplacian theory to mixed-order operators defined by signed measures on $[0,1]$. These findings have implications for nonlocal PDEs with mixed-order operators and provide a unified variational framework for multiplicity results.
Abstract
In this paper, we investigate the existence and multiplicity of weak solutions to problems involving a superposition operator of the type $$\int_{[0, 1]}(- Δ)^s u d μ(s),$$ for a signed measure $μ$ on the interval of fractional exponents $[0,1]$, when the nonlinearity is subcritical and asymptotically linear at infinity; thus, we deal with a perturbation of the eigenvalue problem for the superposition operator. We use variational tools, extending to this setting well-known results for the classical and the fractional Laplace operators.