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Nonlinear second order elliptic partial differential equations at resonance. (English) Zbl 0686.35045

The authors consider nonlinear second order elliptic partial differential equations at resonance. More precisely, they study the solvability of selfadjoint boundary value problems of the form

$\left(1\right)\phantom{\rule{1.em}{0ex}}Lu+{\lambda }_{1}u+g\left(x,u\right)=h\phantom{\rule{1.em}{0ex}}in\phantom{\rule{1.em}{0ex}}{\Omega },\phantom{\rule{1.em}{0ex}}u/\partial {\Omega }=0,$

$\left(2\right)\phantom{\rule{1.em}{0ex}}Au+{\lambda }_{1}u+g\left(x,u\right)=h\phantom{\rule{1.em}{0ex}}in\phantom{\rule{1.em}{0ex}}{\Omega },\phantom{\rule{1.em}{0ex}}u/\partial {\Omega }=0,$
where h is a given function on ${\Omega }$ and ${\lambda }_{1}$ is the first (resp. principal) eigenvalue of a uniformly elliptic operator -L (resp. -A) on a bounded smooth domain ${\Omega }\subset {ℝ}^{N}:$
$Lu=\sum _{i,j}\partial /\partial {x}_{i}\left({a}_{ij}\left(x\right)\partial u/\partial {x}_{j}\right)-{a}_{0}\left(u\right)u,\phantom{\rule{1.em}{0ex}}Au=Lu+\sum _{i}{b}_{i}\left(x\right)\partial u/\partial {x}_{i},$
with the coefficients satisfying suitable regularity conditions. The objective is to show solvability of (1) or (2) for any h orthogonal to the first eigenfunction, in situations where the nonlinearity satisfies neither a monotonicity condition nor a Landesman- Lazer type condition. Instead, the nonlinearity is assumed to satisfy a sign condition g(x,u), $u\ge 0$, and a linear growth allowing “interaction” with the first and second eigenvalues. Moreover, some crossing of eigenvalues is allowed in certain cases.