On Fredholm alternative for certain quasilinear boundary value problems. (English) Zbl 1074.35035

Drábek, Pavel

Math. Bohem. 127, No. 2, 197-202 (2002).

Summary: We study the Dirichlet boundary value problem for the \(p\)-Laplacian of the form \[ -\Delta _p u- \lambda _1 | u| ^{p-2} u= f \;\text{ in } \Omega ,\quad u= 0 \;\text{ on } \partial \Omega , \] 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 above Dirichlet problem is solvable and has multiple solutions.

MSC:

35J60	Nonlinear elliptic equations
35P30	Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35B35	Stability in context of PDEs
49N10	Linear-quadratic optimal control problems

Keywords:

\(p\)-Laplacian; variational methods; PS condition; Fredholm alternative; upper and lower solutions