Existence results for Schrödinger–Choquard–Kirchhoff equations involving the fractional p-Laplacian

Abstract The paper is concerned with existence of nonnegative solutions of a Schrödinger–Choquard–Kirchhoff-type fractional p-equation. As a consequence, the results can be applied to the special case ( a + b ⁢ ∥ u ∥ s p ⁢ ( θ - 1 ) ) ⁢ [ ( - Δ ) p s ⁢ u + V ⁢ ( x ) ⁢ | u | p - 2 ⁢ u ] = λ ⁢ f ⁢ ( x , u ) + ( ∫ ℝ N | u | p μ , s * | x - y | μ ⁢ 𝑑 y ) ⁢ | u | p μ , s * - 2 ⁢ u   in ⁢ ℝ N , (a+b\|u\|_{s}^{p(\theta-1)})[(-\Delta)^{s}_{p}u+V(x)|u|^{p-2}u]=\lambda f(x,u)% +\Bigg{(}\int_{\mathbb{R}^{N}}\frac{|u|^{p_{\mu,s}^{*}}}{|x-y|^{\mu}}\,dy% \Biggr{)}|u|^{p_{\mu,s}^{*}-2}u\quad\text{in }\mathbb{R}^{N}, where ∥ u ∥ s = ( ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y + ∫ ℝ N V ⁢ ( x ) ⁢ | u | p ⁢ 𝑑 x ) 1 p , \|u\|_{s}=\Bigg{(}\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}% \,dx\,dy+\int_{\mathbb{R}^{N}}V(x)|u|^{p}\,dx\Biggr{)}^{\frac{1}{p}}, a , b ∈ ℝ 0 + {a,b\in\mathbb{R}^{+}_{0}} , with a + b > 0 {a+b>0} , λ > 0 {\lambda>0} is a parameter, s ∈ ( 0 , 1 ) {s\in(0,1)} , N > p ⁢ s {N>ps} , θ ∈ [ 1 , N / ( N - p ⁢ s ) ) {\theta\in[1,N/(N-ps))} , ( - Δ ) p s {(-\Delta)^{s}_{p}} is the fractional p-Laplacian, V : ℝ N → ℝ + {V:\mathbb{R}^{N}\rightarrow\mathbb{R}^{+}} is a potential function, 0 < μ < N {0<\mu<N} , p μ , s * = ( p ⁢ N - p ⁢ μ / 2 ) / ( N - p ⁢ s ) {p_{\mu,s}^{*}=(pN-p\mu/2)/(N-ps)} is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality, and f : ℝ N × ℝ → ℝ {f:\mathbb{R}^{N}\times\mathbb{R}\rightarrow\mathbb{R}} is a Carathéodory function. First, via the Mountain Pass theorem, existence of nonnegative solutions is obtained when f satisfies superlinear growth conditions and λ is large enough. Then, via the Ekeland variational principle, existence of nonnegative solutions is investigated when f is sublinear at infinity and λ is small enough. More intriguingly, the paper covers a novel feature of Kirchhoff problems, which is the fact that the parameter a can be zero. Hence the results of the paper are new even for the standard stationary Kirchhoff problems.