Existence and multiplicity of solutions for fractional Schrödinger-p-Kirchhoff equations in ℝ N

Abstract This paper concerns the existence and multiplicity of solutions for a nonlinear Schrödinger–Kirchhoff-type equation involving the fractional p -Laplace operator in ℝ N R^{N}. Precisely, we study the Kirchhoff-type problem (a + b ⁢ ∬ ℝ 2 ⁢ N | u ⁢ (x) - u ⁢ (y) | p | x - y | N + s ⁢ p ⁢ d x ⁢ d y) ⁢ (- Δ) p s ⁢ u + V ⁢ (x) ⁢ | u | p - 2 ⁢ u = f ⁢ (x, u) in ⁢ ℝ N, (a+bₑ^₂₍|u (x) -u (y) |^p|x-y|^{N+sp}\, % dx\, dy) (-) ^s₏u+V (x) |u|^p-2u=f (x, u) % in R^N, where a, b > 0 a, b>0, (- Δ) p s (-) ^{s₏} is the fractional p -Laplacian with 0 s 1 p N s 0 V: ℝ N → ℝ {V^{N} and f: ℝ N × ℝ → ℝ f^{N} are continuous functions while V can have negative values and f fulfills suitable growth assumptions. According to the interaction between the attenuation of the potential at infinity and the behavior of the nonlinear term at the origin, using a penalization argument along with L ∞ L^{} -estimates and variational methods, we prove the existence of a positive solution. In addition, we also establish the existence of infinitely many solutions provided the nonlinear term is odd.