an:05132868
Zbl 1220.35030
Anello, Giovanni; Cordaro, Giuseppe
Perturbation from Dirichlet problem involving oscillating nonlinearities
EN
J. Differ. Equations 234, No. 1, 80-90 (2007).
00193423
2007
j
35J20 35J25
Dirichlet problem; weak solution; strong solution; oscillating nonlinearities
The authors deal with the following perturbed Dirichlet problem
\[
-\Delta u= f(x, u)+\lambda g(x,u)\quad\text{in }\Omega,
\]
\[
u= 0\quad\text{on }\partial\Omega,
\]
where \(\Omega\) is a bounded set in \(\mathbb{R}^d\) with smooth boundary, \(f,g: \Omega\times\mathbb{R}\to \mathbb{R}\) are given functions. The key role is played by the assumptions on \(f(x,\cdot)\) that \(f\) is allowed to change sign, uniformly with respect to \(x\), in any neighborhood of zero (respectively \(+\infty\)), which in turn implies an oscillating behaviour for its potential \(F(x,t)\), \(F(x,t):= \int^t_0 f(x,s)\,ds\). Under some suitable assumptions \(f\), \(g\) and \(\lambda\), the authors prove prove existence of at least \(k\) distinct weak solutions in \(W^{1,2}_0(\Omega)\), for every \(k\in\mathbb{N}\).
Messoud A. Efendiev (Berlin)