On existence and multiplicity of solutions for elliptic equations involving the \(p\)-Laplacian
In this paper, under suitable assumptions on \(f(x,t)\) and \(a(x)\) the authors proved that there are at least three solutions to the following \(p\)-Laplacian eigenvalue problems with Dirichlet and Neumann boundary vaules: \[ \begin{cases} -\Delta_p (u) = \lambda\, f(x,u),&\text{in } \Omega,\\ u = 0, &\text{on } \partial \Omega, \end{cases}\tag{1} \] and \[ \begin{cases} -\Delta _p (u) + a(x)|u|^{p - 2} u = \lambda f(x,u), &\text{in } \Omega, \\ \frac{{\partial u}}{{\partial n}} = 0, &\text{on } \partial \Omega, \end{cases}\tag{2} \] for \(\lambda\) in some interval \(\Lambda\). Moreover, if \(\lambda =1\), by using the KKM principle they proved also some existence results for problems (1) and (2).
scientific article; zbMATH DE number 5562669