We consider a superposition operator of the form ∫ [ 0 , 1 ] ( − Δ ) s u d μ ( s ) , for a signed measure µ on the interval of fractional exponents [ 0 , 1 ] , joined to a nonlinearity whose term of homogeneity equal to one is ‘jumping’, i.e. it may present different coefficients in front of the negative and positive parts. The signed measure is supposed to possess a positive contribution coming from the higher exponents that overcomes its negative contribution (if any). The problem taken into account is also of ‘critical’ type, though in this case the critical exponent needs to be carefully selected in terms of the signed measure µ. Not only the operator and the nonlinearity considered here are very general, but our results are new even in special cases of interest and include known results as particular subcases. The possibility of considering operators ‘with the wrong sign’ is also a complete novelty in this setting.

We present a new functional setting for Neumann conditions related to the superposition of (possibly infinitely many) fractional Laplace operators. We will introduce some bespoke functional framework and present minimization properties, existence and uniqueness results, asymptotic formulas, spectral analyses, rigidity results, integration by parts formulas, superpositions of fractional perimeters, as well as a study of the associated heat equation.

The aim of this paper is investigating the existence of one or more weak solutions of the coupled quasilinear elliptic system of gradient type \textup{(P)} { - div ⁡ ( A ⁢ ( x , u ) ⁢ | ∇ ⁡ u | p 1 - 2 ⁢ ∇ ⁡ u ) + 1 p 1 ⁢ A u ⁢ ( x , u ) ⁢ | ∇ ⁡ u | p 1 = G u ⁢ ( x , u , v ) in Ω , - div ⁡ ( B ⁢ ( x , v ) ⁢ | ∇ ⁡ v | p 2 - 2 ⁢ ∇ ⁡ v ) + 1 p 2 ⁢ B v ⁢ ( x , v ) ⁢ | ∇ ⁡ v | p 2 = G v ⁢ ( x , u , v ) in Ω , u = v = 0 on ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle-\operatorname{div}(A(x,u)|\nabla u|^{p_{1% }-2}\nabla u)+\frac{1}{p_{1}}A_{u}(x,u)|\nabla u|^{p_{1}}&\displaystyle=G_{u}(% x,u,v)&&\displaystyle\phantom{}\text{in~{}${\Omega}$,}\\ \displaystyle-\operatorname{div}(B(x,v)|\nabla v|^{p_{2}-2}\nabla v)+\frac{1}{% p_{2}}B_{v}(x,v)|\nabla v|^{p_{2}}&\displaystyle=G_{v}(x,u,v)&&\displaystyle% \phantom{}\text{in~{}${\Omega}$,}\\ \displaystyle u=v&\displaystyle=0&&\displaystyle\phantom{}\text{on ${\partial% \Omega}$,}\end{aligned}\right. where Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} is an open bounded domain, p 1 {p_{1}} , p 2 > 1 {p_{2}>1} and A ⁢ ( x , u ) {A(x,u)} , B ⁢ ( x , v ) {B(x,v)} are 𝒞 1 {\mathcal{C}^{1}} -Carathéodory functions on Ω × ℝ {\Omega\times\mathbb{R}} with partial derivatives A u ⁢ ( x , u ) {A_{u}(x,u)} , respectively B v ⁢ ( x , v ) {B_{v}(x,v)} , while G u ⁢ ( x , u , v ) {G_{u}(x,u,v)} , G v ⁢ ( x , u , v ) {G_{v}(x,u,v)} are given Carathéodory maps defined on Ω × ℝ × ℝ {\Omega\times\mathbb{R}\times\mathbb{R}} which are partial derivatives of a function G ⁢ ( x , u , v ) {G(x,u,v)} . We prove that, even if the coefficients make the variational approach more difficult, under suitable hypotheses functional 𝒥 {{\mathcal{J}}} , related to problem (P), admits at least one critical point in the “right” Banach spaceX. Moreover, if 𝒥 {{\mathcal{J}}} is even, then (P) has infinitely many weak bounded solutions. The proof, which exploits the interaction between two different norms, is based on a weak version of the Cerami–Palais–Smale condition, a “good” decomposition of the Banach spaceXand suitable generalizations of the Ambrosetti–Rabinowitz Mountain Pass Theorems.