Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity

Abstract

Abstract This paper is concerned with the following Kirchhoff-type problem with convolution nonlinearity: - ( a + b ∫ ℝ 3 | ∇ u | 2 d x ) Δ u + V ( x ) u = ( I α * F ( u ) ) f ( u ) , x ∈ ℝ 3 , u ∈ H 1 ( ℝ 3 ) , -\bigg{(}a+b\int_{\mathbb{R}^{3}}\lvert\nabla u|^{2}\,\mathrm{d}x\bigg{)}% \Delta u+V(x)u=(I_{\alpha}*F(u))f(u),\quad x\in{\mathbb{R}}^{3},\,u\in H^{1}(% \mathbb{R}^{3}), where a , b > 0 {a,b>0} , I α : ℝ 3 → ℝ {I_{\alpha}\colon\mathbb{R}^{3}\rightarrow\mathbb{R}} , with α ∈ ( 0 , 3 ) {\alpha\in(0,3)} , is the Riesz potential, V ∈ 𝒞 ⁢ ( ℝ 3 , [ 0 , ∞ ) ) {V\in\mathcal{C}(\mathbb{R}^{3},[0,\infty))} , f ∈ 𝒞 ⁢ ( ℝ , ℝ ) {f\in\mathcal{C}(\mathbb{R},\mathbb{R})} and F ⁢ ( t ) = ∫ 0 t f ⁢ ( s ) ⁢ d s {F(t)\kern-1.0pt=\kern-1.0pt\int_{0}^{t}f(s)\,\mathrm{d}s} . By using variational and some new analytical techniques, we prove that the above problem admits ground state solutions under mild assumptions on V and f. Moreover, we give a non-existence result. In particular, our results extend and improve the existing ones, and fill a gap in the case where f ⁢ ( u ) = | u | q - 2 ⁢ u {f(u)=|u|^{q-2}u} , with q ∈ ( 1 + α / 3 , 2 ] {q\in(1+\alpha/3,2]} .