Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN

Abstract

This article shows the existence and multiplicity of solutions for the following pp-Kirchhoff-type equation: a+b∫RN(∣∇u∣p+V(x)∣u∣p)dx(−△pu+V(x)∣u∣p−2u)=λg(x)∣u∣r−2u−h(x)∣u∣q−2u,inRN.\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 , aa and bb are positive constants. Depending on the relationship of p,qp,q, and rr, we obtain the existence, multiplicity, and nonexistence of solutions to the abovementioned equation using variational methods.

Keywords

• p-kirchhoff-type equation
• (ps)c condition
• symmetric mountain pass lemma
• 35a15