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 {\mathbb{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 , \Biggl{(}a+b\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\,% \mathrm{d}x\,\mathrm{d}y\Biggr{)}(-\Delta)^{s}_{p}u+V(x)|u|^{p-2}u=f(x,u)\quad% \text{in }\mathbb{R}^{N}, where a , b > 0 {a,b>0} , ( - Δ ) p s {(-\Delta)^{s}_{p}} is the fractional p-Laplacian with 0 < s < 1 < p < N s {0<s<1<p<\frac{N}{s}} , V : ℝ N → ℝ {V\colon\mathbb{R}^{N}\to\mathbb{R}} and f : ℝ N × ℝ → ℝ {f\colon\mathbb{R}^{N}\times\mathbb{R}\to\mathbb{R}} 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^{\infty}} -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.