On a problem of lower limit in the study of nonresonance

Abstract

We prove the solvability of the Dirichlet problem $$\{\begin{array}{l}-{\Delta }_{p}u=f\left(u\right)+h\text{in}\Omega ,\\ u=0\text{on}\partial \Omega \end{array}$$ for every given $h$, under a condition involving only the asymptotic behaviour of the potential $F$ of $f$ with respect to the first eigenvalue of the $p$-Laplacian ${\Delta }_p$. More general operators are also considered.