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Zbl 0940.35087
Drábek, Pavel; Robinson, Stephen B.
Resonance problems for the $p$-Laplacian. (English)
[J] J. Funct. Anal. 169, No.1, 189-200 (1999). ISSN 0022-1236

Using variational arguments the authors prove the existence of a weak solution for the boundary value problem $$\cases -\Delta_p u-\lambda|u|^{p-2}u+f(x,u)=0\quad&\text{in }\Omega,\\ u=0\quad&\text{on }\partial\Omega,\endcases$$ where $\Delta_p u=$div$(|Du|^{p-2}Du)$, $p>1$, $\Omega$ is a bounded domain of $\Bbb R^N$, $\lambda\in\Bbb R$ and $f:\Omega\times\Bbb R \to \Bbb R$ is a bounded Carathéodory function such that there exist $\lim_{t\to\pm\infty}f(x,t)= f^\pm(x)$ a.e. in $\Omega$, with either $$\gather\int_{v>0}f^+ v+\int_{v<0}f^- v>0, \qquad\text{or} \tag(LL)$_\lambda^+$\\ \int_{v>0}f^+ v+\int_{v<0}f^- v<0\phantom{\qquad\text{or}} \tag(LL)$_\lambda^-$\endgather$$ for all $v\in$Ker$(-\Delta_p-\lambda)\setminus\{0\}$, and also there is $g\in L^{p/(p-1)}(\Omega)$ such that $$|f(x,t)|\le g(x)\quad\text{in }\Omega\times\Bbb R.$$ \par The conditions (LL)$_\lambda^\pm$ are the standard Landesman -- Laser conditions for resonance problems, and of course are trivially satisfied whenever $\lambda$ is not an eigenvalue.
[P.Pucci (Perugia)]

MSC 2000:: *35J65 (Nonlinear) BVP for (non)linear elliptic equations
35P30 Nonlinear eigenvalue problems for PD operators
35A15 Variational methods (PDE)
35A05 General existence and uniqueness theorems (PDE)

Keywords: existence; Landesman-Laser conditions; resonance problems

Cited in: Zbl 1104.35012 Zbl 1033.35078

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