Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation R ^N

Abstract This article shows the existence and multiplicity of solutions for the following p p -Kirchhoff-type equation: a + b ∫ R N ( ∣ ∇ u ∣ p + V ( x ) ∣ u ∣ p ) d x ( − △ p u + V ( x ) ∣ u ∣ p − 2 u ) = λ g ( x ) ∣ u ∣ r − 2 u − h ( x ) ∣ u ∣ q − 2 u , in R N . \left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\left({| \nabla u| }^{p}+V\left(x){| u| }^{p}){\rm{d}}x\right)\left(-{\bigtriangleup }_{p}u+V\left(x){| u| }^{p-2}u)=\lambda g\left(x){| u| }^{r-2}u-h\left(x){| u| }^{q-2}u,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}. where λ \lambda is a real parameter and 1 < p < q < ∞ 1\lt p\lt q\lt \infty , a a and b b are positive constants. Depending on the relationship of p , q p,q , and r r , we obtain the existence, multiplicity, and nonexistence of solutions to the abovementioned equation using variational methods.