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Perturbation from Dirichlet problem involving oscillating nonlinearities. (English) Zbl 1220.35030
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 $$k\in\mathbb{N}$$.

##### MSC:
 35J20 Variational methods for second-order elliptic equations 35J25 Boundary value problems for second-order elliptic equations
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##### References:
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