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$\Delta^2_p u -\Delta_p u + \lambda V(x)|u|^{p-2}u = f(x,u)\,,\,x\in R^N,\ u \in W^{2,p}(R^N)$,where $p>1$, $\lambda>0$, $V\in C(R^N,R^+)$, $f\inC(R^N \times R,R)$, $N>2p$. We alsoexplore the problem in the limiting case of $\lambda \rightarrow\infty$.