An existence theory for nonlinear superposition operators of mixed fractional order

Abstract

We establish the existence of multiple solutions for a nonlinear problem of critical type. The problem considered is fractional in nature, since it is obtained by the superposition of $(s,p)$-fractional Laplacians of different orders. The results obtained are new even in the case of the sum of two different fractional $p$-Laplacians, or the sum of a fractional $p$-Laplacian and a classical $p$-Laplacian, but our framework is general enough to address also the sum of finitely, or even infinitely many, operators. In fact, we can also consider the superposition of a continuum of operators, modulated by a general signed measure on the fractional exponents. When this measure is not positive, the contributions of the individual operators to the whole superposition operator is allowed to change sign. In this situation, our structural assumption is that the positive measure on the higher fractional exponents dominates the rest of the signed measure.

References in corpus (1)

• An existence theory for superposition operators of mixed order subject to jumping nonlinearities

Cited by in corpus (5)

• A general theory for the $(s,p)$-superposition of nonlinear fractional operators
• The Neumann condition for the superposition of fractional Laplacians
• Some nonlinear problems for the superposition of fractional operators with Neumann boundary conditions
• Logistic diffusion equations governed by the superposition of operators of mixed fractional order
• Superlinear problems involving nonlinear superposition operators of mixed fractional order