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Resonance problems for the \(p\)-Laplacian. (English) Zbl 0940.35087
Using variational arguments the authors prove the existence of a weak solution for the boundary value problem \[ \begin{cases} -\Delta_p u-\lambda|u|^{p-2}u+f(x,u)=0\quad&\text{in }\Omega,\\ u=0\quad&\text{on }\partial\Omega,\end{cases} \] where \(\Delta_p u=\)div\((|Du|^{p-2}Du)\), \(p>1\), \(\Omega\) is a bounded domain of \(\mathbb R^N\), \(\lambda\in\mathbb R\) and \(f:\Omega\times\mathbb R \to \mathbb R\) is a bounded Carathéodory function such that there exist \(\lim_{t\to\pm\infty}f(x,t)= f^\pm(x)\) a.e. in \(\Omega\), with either \[ \begin{gathered}\int_{v>0}f^+ v+\int_{v<0}f^- v>0, \qquad\text{or} \tag{(LL)}\(_\lambda^+\)\\ \int_{v>0}f^+ v+\int_{v<0}f^- v<0\phantom{\qquad\text{or}} \tag{(LL)}\(_\lambda^-\)\end{gathered} \] for all \(v\in\)Ker\((-\Delta_p-\lambda)\setminus\{0\}\), and also there is \(g\in L^{p/(p-1)}(\Omega)\) such that \[ |f(x,t)|\leq g(x)\quad\text{in }\Omega\times\mathbb R. \]
The conditions (LL)\(_\lambda^\pm\) are the standard Landesman – Laser conditions for resonance problems, and of course are trivially satisfied whenever \(\lambda\) is not an eigenvalue.
Reviewer: P.Pucci (Perugia)

35J65 Nonlinear boundary value problems for linear elliptic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35A15 Variational methods applied to PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
Full Text: DOI
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