## 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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