Solutions to perturbed eigenvalue problems of the p-Laplacian in RN

Using a variational approach, we investigate the existence of solutions for non-autonomous perturbations of the p-Laplacian eigenvalue problem $$ -Delta _pu=f(x,u)quad { m in}quad {Bbb R}^N,. $$ Under the assumptions that the primitive $F(x,u)$ of $f(x,u)$ interacts only with the first eigenvalue, we look for solutions in the space $D^{1,p}({Bbb R}^N)$. Furthermore, we assume a condition that measures how different the behavior of the function $F(x,u)$ is from that of the $p$-power of $u$.