Quasilinear problems under local Landesman-Lazer condition (Q2338498)

The paper studies the problem \( -\Delta_p u=\lambda |u|^{p-2}u+\mu h_\mu (x,u)\) in \(\Omega \), \( u=0\) on \( \partial \Omega \) . Here \( \Omega \subset R^N\) is a bounded regular domain in \( R^N\), \( p\in (1,\infty )\), \( \Delta_p u=\nabla \cdot (|\nabla u|^{p-2}\nabla u)\) , \( \lambda >0\) , \( \mu \ne 0\) are real parameters and \( h_\mu :\Omega \times R\to R\) is a family of Caratheodory functions depending on \( \mu \). Under some conditions on \( h_\mu \) it is proved that there are positive numbers \( \nu_*\) and \( \mu^* \) such that for \( \mu \in (0,\mu^*) \) and \( |\lambda -\lambda_1| \in [0,\mu \nu^*)\) there exists a nonnegative weak solution of the problem. Here \(\lambda_1\) is the principal eigenvalue of the operator \( -\Delta_p\) with zero boundary condition.