Afrouzi, G. A.; Heidarkhani, S. On a boundary value problem involving \(p\)-Laplacian. (English) Zbl 1159.35359 Adv. Theor. Appl. Math. 1, No. 1, 71-77 (2006). Summary: The existence of at least three weak solutions for Dirichlet problem \[ \begin{cases} -\Delta_pu+\lambda f(x,u)=0\quad &\text{in }\Omega,\\ u-0 \quad &\text{on }\partial\Omega,\end{cases} \] where \(\Delta_pu=\text{div}(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplacian operator, \(\Omega\subset\mathbb{R}^N(N\geq 1)\) is non-empty bounded open set with smooth boundary \(\partial\Omega\), \(p<N\), \(\lambda>0\) and \(f: \Omega\times\mathbb{R}\to\mathbb{R}\) is a Caratheodory function, is established. MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35J60 Nonlinear elliptic equations 35J20 Variational methods for second-order elliptic equations 35J25 Boundary value problems for second-order elliptic equations 47J30 Variational methods involving nonlinear operators Keywords:\(p\)-Laplacian; Dirichlet problem; weak solutions; critical point PDFBibTeX XMLCite \textit{G. A. Afrouzi} and \textit{S. Heidarkhani}, Adv. Theor. Appl. Math. 1, No. 1, 71--77 (2006; Zbl 1159.35359)