Infinitely many radial solutions for the -Kirchhoff-type equation with oscillatory nonlinearities in

In this paper we consider the p ( x ) -Kirchhoff-type equation in R N of the form { a ( ∫ R N | ∇ u | p ( x ) + | u | p ( x ) p ( x ) d x ) ( − Δ p ( x ) u + | u | p ( x ) − 2 u ) = Q ( x ) f ( u ) , u ⩾ 0 , x ∈ R N , u ( x ) → 0 , as | x | → + ∞ , where 1 < p ( x ) < N for x ∈ R N , Q : R N → R + is a radial potential, f : [ 0 , + ∞ ) → R is a continuous nonlinearity which oscillates near the origin or at infinity and a is allowed to be singular at zero. By means of a direct variational method and the principle of symmetric criticality for non-smooth Szulkin-type functionals, the existence of infinitely many radially symmetric solutions of the problem is established. Meanwhile, the sequence of solutions in L ∞ -norm tends to 0 (resp., to +∞) whenever f oscillates at the origin (resp., at infinity).