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

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.


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
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