Perturbation from Dirichlet problem involving oscillating nonlinearities. (English) Zbl 1220.35030

Anello, Giovanni; Cordaro, Giuseppe

J. Differ. Equations 234, No. 1, 80-90 (2007).

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 \(k\in\mathbb{N}\).

Reviewer: Messoud A. Efendiev (Berlin)

MSC:

35J20	Variational methods for second-order elliptic equations
35J25	Boundary value problems for second-order elliptic equations

Keywords:

Dirichlet problem; weak solution; strong solution; oscillating nonlinearities

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