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

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
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