Superlinear problems involving nonlinear superposition operators of mixed fractional order

In this work, we study a class of elliptic problems involving nonlinear superpositions of fractional operators of the form

\begin{equation*}A_{\mu,p}u := \int_{[0,1]} (-\Delta)_{p}^{s} u , d\mu(s),\end{equation*}

where

$\mu$

is a signed measure on

$[0,1]$

, coupled with nonlinearities of superlinear type. This kind of operator was introduced by S. Dipierro, K. Perera, C. Sportelli, and E. Valdinoci in [11]. Our analysis covers a variety of superlinear growth assumptions, beginning with the classical Ambrosetti–Rabinowitz condition. Within this framework, we construct a suitable variational setting and apply the Fountain Theorem to establish the existence of infinitely many weak solutions. The results obtained are novel even in the special cases of superpositions of fractional

$p$

-Laplacians, or combinations of the fractional

$p$

-Laplacian with the

$p$

-Laplacian. More generally, our approach applies to finite sums of fractional

$p$

-Laplacians with different orders, as well as to operators in which fractional Laplacians appear with ‘wrong’ signs. A distinctive contribution of the paper lies in providing a unified variational framework that systematically accommodates this broad class of operators.