Multiplicity theorems for the Dirichlet problem involving the \(p\)-Laplacian.
Let \(\Omega\) be an open bounded domain in \(\mathbb R^N\) \((N\geq 1)\), \(p>N\), \(\lambda>0\) and \(f:\Omega\times\mathbb R\to\mathbb R\) is a Carathéodory function. The authors interested existence result for the following Dirichlet problem: \[ \Delta_pu+ \lambda f(x,u)=0 \quad\text{in }\Omega, \qquad u=0\quad\text{on }\partial\Omega, \tag{1} \] where \(\Delta_p\) is the \(p\)-Laplacian. Using variational methods the authors prove the existence of at least three weak solutions for (1) in \(W_0^{1,p}(\Omega)\).
scientific article; zbMATH DE number 1929505