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Geometry of the energy functional and the Fredholm alternative for the \(p\)-Laplacian in higher dimensions. (English) Zbl 1114.35318

Summary: We study Dirichlet boundary-value problems, for the \(p\)-Laplacian, of the form \[ -\Delta_p u- \lambda_1| u|^{p-2} u-f\quad\text{in }\Omega, \qquad u=0\quad\text{on }\partial\Omega, \] where \(\Omega\subset \mathbb R^N\) is a bounded domain with smooth boundary \(\partial\Omega\), \(N\geq 1\), \(p>1\), \(f\in C(\overline{\Omega})\) and \(\lambda_1>0\) is the first eigenvalue of \(\Delta_p\). We study the geometry of the energy functional \[ E_p(u)\;\frac{1}{p} \int_\Omega |\nabla u|^p- \frac{\lambda_1}{p} \int_\Omega| u|^p- \int_\Omega fu \] and show the difference between the case \(1< p< 2\) and the case \(p> 2\). We also give the characterization of the right hand sides \(f\) for which the Dirichlet problem above is solvable and has multiple solutions.

MSC:

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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