Existence,multiplicity, and nonexistence of solutions for a p -Kirchhoff elliptic equation on R N $mathbb{R}^{N}$

摘要

In this paper, we study the multiplicity of solutions for the following nonhomogeneous p-Kirchhoff elliptic equation: 0.1 ( a + λ ( ∫ R N ( | ∇ u | p + | u | p ) d x ) m ) ( − Δ p u + | u | p − 2 u ) = f ( u ) + h ( x ) , x ∈ R N , $$ iggl(a+lambda iggl( int_{mathbb{R}^{N}} igl(|abla{u}|^{p}+|u|^{p} igr),dx iggr)^{m} iggr) igl(-Delta_{p}u+|u|^{p-2}u igr) =f(u)+h(x),quad xinmathbb{R}^{N}, $$ with a , λ , m 0 $a,lambda,m0$ and 1 p N $1 p N$ . By variational methods we prove that problem (0.1) admits at least two solutions under appropriate assumptions on f