Multiplicity results for the non-homogeneous fractionalp-Kirchhoff equations with concave–convex nonlinearities

Abstract

In this paper, we are interested in the multiplicity of solutions for a non-homogeneous p -Kirchhoff-type problem driven by a non-local integro-differential operator. As a particular case, we deal with the following elliptic problem of Kirchhoff type with convex–concave nonlinearities: a + b ∬ R 2 N | u ( x ) − u ( y ) | p | x − y | N + s p d x d y θ − 1 ( − Δ ) p s u = λ ω 1 ( x ) | u | q − 2 u + ω 2 ( x ) | u | r − 2 u + h ( x ) in R N , where ( − Δ ) p s is the fractional p -Laplace operator, a + b >0 with a , b ∈ R 0 + , λ>0 is a real parameter, 0 < s < 1 < p < ∞ with sp < N , 1< q < p ≤ θp < r < Np /( N − sp ), ω 1 , ω 2 , h are functions which may change sign in R N . Under some suitable conditions, we obtain the existence of two non-trivial entire solutions by applying the mountain pass theorem and Ekeland's variational principle. A distinguished feature of this paper is that a may be zero, which means that the above-mentioned problem is degenerate. To the best of our knowledge, our results are new even in the Laplacian case.

Existence of solutions for perturbed fractional p-Laplacian equations

“…[3] for the semilinear case in bounded domains. We also refer to [38] for the fractional Laplacian case in bounded domains and to [37] for the fractional p-Laplacian case in R N . (b) Under the conditions (H1) and (H3), the condition (H4) is more general than that in [5, (1.4)] for fractional Laplacian problems, see [2] for further details.…”

Existence of solutions for fractional $p$-Kirchhoff type equations with a generalized Choquard nonlinearity

“…Xiang et al in [35] investigated the existence of solutions for Kirchhoff type problems involving the fractional p−Laplacian by variational methods, where the nonlinearity is subcritical and the Kirchhoff function is non-degenerate. Combining the mountain pass theorem with Ekeland variational principle, Xiang et al in [36] established the existence of two solutions for a degenerate fractional p−Laplacian Kirchhoff equation in R N with concave-convex nonlinearity. By the same methods as in [36], Pucci et al in [28] obtained the existence of two solutions for a nonhomogenous Schrödinger-Kirchhoff type equation involving the fractional p−Laplacian in R N on a nondegenerate situation.…”

“…Combining the mountain pass theorem with Ekeland variational principle, Xiang et al in [36] established the existence of two solutions for a degenerate fractional p−Laplacian Kirchhoff equation in R N with concave-convex nonlinearity. By the same methods as in [36], Pucci et al in [28] obtained the existence of two solutions for a nonhomogenous Schrödinger-Kirchhoff type equation involving the fractional p−Laplacian in R N on a nondegenerate situation. Furthermore, nonexistence and multiplicity of solutions for a nonhomogeneous fractional p−Kirchhoff type problem involving critical exponent in R N were studied in [37].…”

Solutions of p-Kirchhoff type problems with critical nonlinearity in R N

“…In [38], the authors investigated the existence of solutions for Kirchhoff type problem involving the fractional p-Laplacian via variational methods, where the nonlinearity is subcritical and the Kirchhoff function is non-degenerate. By using the mountain pass theorem and Ekeland's variational principle, the authors in [39] studied the multiplicity of solutions to a nonhomogeneous Kirchhoff type problem driven by the fractional p-Laplacian, where the nonlinearity is convex-concave and the Kirchhoff function is degenerate. Using the same methods as in [39], Pucci et al in [28] obtained the existence of multiple solutions for the nonhomogeneous fractional p-Laplacian equations of Schrödinger-Kirchhoff type in the whole space.…”

“…By using the mountain pass theorem and Ekeland's variational principle, the authors in [39] studied the multiplicity of solutions to a nonhomogeneous Kirchhoff type problem driven by the fractional p-Laplacian, where the nonlinearity is convex-concave and the Kirchhoff function is degenerate. Using the same methods as in [39], Pucci et al in [28] obtained the existence of multiple solutions for the nonhomogeneous fractional p-Laplacian equations of Schrödinger-Kirchhoff type in the whole space. Indeed, the fractional Kirchhoff problems have been extensively studied in recent years, for instance, we also refer to [27] about non-degenerate Kirchhoff type problems and to [2,29] about degenerate Kirchhoff type problems for the recent advances in this direction.…”