On the solvability of a Neumann boundary value problem at resonance

http://dx.doi.org/10.4153/CMB-1997-055-7
Canad. Math. Bull. 40(1997), 464-470

  Published:1997-12-01
 Printed: Dec 1997

We study the existence of solutions of the semilinear equations (1) $\triangle u + g(x,u)=h$, ${\partial u \over \partial n} = 0$ on $\partial \Omega$ in which the non-linearity $g$ may grow superlinearly in $u$ in one of directions $u \to \infty$ and $u \to -\infty$, and (2) $-\triangle u + g(x,u)=h$, ${\partial u \over \partial n} = 0$ on $\partial \Omega$ in which the nonlinear term $g$ may grow superlinearly in $u$ as $|u| \to \infty$. The purpose of this paper is to obtain solvability theorems for (1) and (2) when the Landesman-Lazer condition does not hold. More precisely, we require that $h$ may satisfy $\int g^\delta_- (x) \, dx < \int h(x) \, dx = 0< \int g^\gamma_+ (x)\,dx$, where $\gamma, \delta$ are arbitrarily nonnegative constants, $g^\gamma_+ (x) = \lim_{u \to \infty} \inf g(x,u) |u|^\gamma$ and $g^\delta_- (x)=\lim_{u \to -\infty} \sup g(x,u)|u|^\delta$. The proofs are based upon degree theoretic arguments.