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On a boundary value problem involving \(p\)-Laplacian. (English) Zbl 1159.35359

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
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