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.


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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