Abstract: We study the Dirichlet boundary value problem for the $p$-Laplacian of the form $$ -\Delta _p u - \lambda _1 |u|^{p-2} u = f \mbox { in } \Omega ,\quad u = 0 \mbox { on } \partial \Omega , $$ where $\Omega \subset \R ^N$ 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 above Dirichlet problem is solvable and has multiple solutions.
Keywords: $p$-Laplacian, variational methods, PS condition, Fredholm alternative, upper and lower solutions
Classification (MSC2000): 35J60, 35P30, 35B35, 49N10
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