## 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.

### 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
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