Existence of infinitely many solutions for fractional p-Laplacian Schrödinger–Kirchhoff type equations with sign-changing potential

Schrödinger–Kirchhoff–Hardy <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula>–fractional equations without the Ambrosetti–Rabinowitz condition

“…Condition (6) is mainly a consequence of the fact that equation 1is set on the whole space R n , as well explained in Remark 4.1. Anyway, we observe that when a = 1 and b = µ = 0, namely without considering Kirchhoff and Hardy terms, (6) is compatible with the results in [1,6,17,23,26], since p > 1.…”

“…By combining the mountain pass theorem and the Ekeland variational principle, the authors of [20] provided the existence of two solutions. For this, they added a small perturbation and assumed the nonlinearity f satisfies the following well-known Ambrosetti-Rabinowitz condition (AR) there exists σ > pθ such that σF (x, t) = σ t 0 f (x, τ )dτ ≤ f (x, t)t for any (x, t) ∈ R n × R. After [20], an interesting challenge is trying to solve Schrödinger equations like (1) without (AR), as done for example in [1,24,26]. Indeed, in [24] a solution of (1), with µ = 0, was provided by variational and truncation arguments, when parameter b is controlled by a suitable threshold.…”

“…While, in [1] by assuming a symmetric assumption for f , the existence of infinitely many solutions of (1) was proved, when a = 1 and b = µ = 0, that is without Kirchhoff and Hardy terms. The result of [1] was then improved in [26], where the primitive of nonlinearity f could be sign-changing. Also, [6,17,23] are worthy of mention in this context, since they introduced multiplicity results for different fractional problems set on a bounded domain Ω, with nonlinear term f not satisfying (AR), but without Kirchhoff and Hardy terms.…”

“…The proof of Theorem 1.1 is mainly variational, based on the mountain pass theorem given in [10]. For this, we have to show that the Euler-Lagrange functional related to (1) satisfies the Cerami compactness condition, as done in [6,9,13,17,18,25,26]. The delicate proof of the boundedness of a Cerami sequence without (AR) forces us to consider a Kirchhoff coefficient equal to the model M (t) = a + bt θ−1 .…”

“…While, because of the presence of the Hardy term, we need parameter a > 0. In any case, Theorem 1.1 generalizes in several directions [ The last part of the paper is devoted to the proof of a multiplicity result for (1), when f satisfies a symmetric condition such as in [1,6,17,23,26]. That is, we assume that…”