A study on elliptic PDE involving the p-harmonic and the p-biharmonic operators with steep potential well

Abstract

In this paper, we give an existence result pertaining to anontrivial solution to the problem${\mathrm{\Delta }}_p^{2}u-{\mathrm{\Delta }}_{p}u+\lambda V(x)|u{|}^{p-2}u=f(x,u),x\in R^N,u\in W^{2,p}(R^N)$,where $p>1$, $\lambda >0$, $V\in C(R^N,R^{+})$, $f\text{inC}(R^N\times R,R)$, $N>2p$. We alsoexplore the problem in the limiting case of $\lambda \to \mathrm{\infty }$.