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In this paper we consider the existence of multiple solutions for the non-homogeneous fractional p-Laplacian equations of Schrödinger–Kirchhoff type

In this paper we consider the existence of multiple solutions for the non-homogeneous fractional p-Laplacian equations of Schrödinger–Kirchhoff type M ⁢ ( ∫ ℝ n ∫ ℝ n | u ⁢ ( x ) - u ⁢ ( z ) | p | x - z | n + p ⁢ s ⁢ 𝑑 z ⁢ 𝑑 x ) ⁢ ( - Δ ) p s ⁢ u + V ⁢ ( x ) ⁢ | u | p - 2 ⁢ u = f ⁢ ( x , u ) + g ⁢ ( x ) M\Bigg{(}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(z)|^{p}}{|x-{% z}|^{n+ps}}\,dz\,dx\Bigg{)}(-\Delta)_{p}^{s}u+V(x)|u|^{p-2}u=f(x,u)+g(x) in ℝ n {\mathbb{R}^{n}} , where (-Δ ) p s )_{p}^{s} is the fractional p-Laplacian operator with 0¡s¡1¡p¡ ∞ \infty , ps¡n, f : ℝ n \mathbb{R}^{n} × \times ℝ \mathbb{R} → \to ℝ \mathbb{R} is a continuous function, V : ℝ n \mathbb{R}^{n} → \to ℝ + \mathbb{R}^{+} is a potential function and g : ℝ n \mathbb{R}^{n} → \to ℝ \mathbb{R} is a perturbation term. Assuming that the potential V is bounded from bellow, that f(x,t) satisfies the Ambrosetti–Rabinowitz condition and some other reasonable hypotheses, and that g(x) is sufficiently small in L p ′ L^{p^{\prime}} ( ℝ n \mathbb{R}^{n} ), we obtain some new criterion to guarantee that the equation above has at least two non-trivial solutions. [ABSTRACT FROM AUTHOR]

In this paper we consider the existence of multiple solutions for the non-homogeneous fractional p-Laplacian equations of Schrödinger–Kirchhoff type M ⁢ ( ∫ ℝ n ∫ ℝ n | u ⁢ ( x ) - u ⁢ ( z ) | p | x - z | n + p ⁢ s ⁢ 𝑑 z ⁢ 𝑑 x ) ⁢ ( - Δ ) p s ⁢ u + V ⁢ ( x ) ⁢ | u | p - 2 ⁢ u = f ⁢ ( x , u ) + g ⁢ ( x ) M\Bigg{(}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(z)|^{p}}{|x-{% z}|^{n+ps}}\,dz\,dx\Bigg{)}(-\Delta)_{p}^{s}u+V(x)|u|^{p-2}u=f(x,u)+g(x) in ℝ n {\mathbb{R}^{n}} , where (-Δ ) p s )_{p}^{s} is the fractional p-Laplacian operator with 0¡s¡1¡p¡ ∞ \infty , ps¡n, f : ℝ n \mathbb{R}^{n} × \times ℝ \mathbb{R} → \to ℝ \mathbb{R} is a continuous function, V : ℝ n \mathbb{R}^{n} → \to ℝ + \mathbb{R}^{+} is a potential function and g : ℝ n \mathbb{R}^{n} → \to ℝ \mathbb{R} is a perturbation term. Assuming that the potential V is bounded from bellow, that f(x,t) satisfies the Ambrosetti–Rabinowitz condition and some other reasonable hypotheses, and that g(x) is sufficiently small in L p ′ L^{p^{\prime}} ( ℝ n \mathbb{R}^{n} ), we obtain some new criterion to guarantee that the equation above has at least two non-trivial solutions. [ABSTRACT FROM AUTHOR]