On a problem of lower limit in the study of nonresonance. (English) Zbl 0933.35067

Anane, A.; Chakrone, O.

Abstr. Appl. Anal. 2, No. 3-4, 227-237 (1997).

Summary: We prove the solvability of the Dirichlet problem \[ \begin{cases} -\Delta_pu=f(u) +h\quad &\text{in }\Omega,\\ u=0\quad &\text{on }\partial \Omega \end{cases} \] 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.

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35P30	Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35A25	Other special methods applied to PDEs

Keywords:

\(p\)-Laplacian; nonresonance; first eigenvalue