This paper is devoted, with my great esteem, to Jean Mawhin. Jean Mawhin, who is for me a great teacher and a very good friend, is a fundamental reference for the research in nonlinear differential problems dealt both with topological and variational methods. Here, owing to this occasion in honor of Jean Mawhin, Dirichlet problems depending on a parameter are investigated, ensuring the existence of non-zero solutions without requiring asymptotic conditions neither at zero nor at infinity on the nonlinear term which, in addition, is not forced by subcritical or critical growth. The approach is based on a combination of variational and topological tools that in turn are developed by starting from a fundamental estimate.
