Existence of positive solutions for a p-Schrödinger–Kirchhoff integro-differential equation with critical growth

We consider the p-Schrödinger–Kirchhoff-type equation for $$v\in {\textrm{W}}^{1,p}({\mathbb {R}}^{\textrm{N}}),$$ v ∈ W 1 , p ( R N ) , where $$\tilde{\sigma }(s) = \lambda f(s) + |s|^{p^*-2} s,$$ σ ~ ( s ) = λ f ( s ) + | s | p ∗ - 2 s , $$b\ge 0,$$ b ≥ 0 , $$a,\varepsilon ,\lambda >0,$$ a , ε , λ > 0 , $$\beta =p^2-Np+N$$ β = p 2 - N p + N and $$1 0.$$ M 0 = inf M > 0 . Thanks to a study of the ground state of the limit problem associated to ( $$\textrm{P}_{\varepsilon }$$ P ε ), we prove, by the method of Nehari manifold, the existence of a positive ground state of ( $$\textrm{P}_{\varepsilon }$$ P ε ). By a Ljusternik–Schnirelmann scheme it’s shown, for $$\varepsilon $$ ε small and $$\lambda $$ λ big, that ( $$\textrm{P}_{\varepsilon }$$ P ε ) has at least $${\textrm{cat}}({\mathcal {M}},{\mathcal {M}}_\delta )$$ cat ( M , M δ ) positive solutions, where $${\mathcal {M}}_\delta = \{x\in {\mathbb {R}}^{\textrm{N}} \, / \, {\textrm{dist}}(x,{\mathcal {M}}) 0.$$ δ > 0 .