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Perturbations from symmetric elliptic boundary value problems. (English) Zbl 1032.35066
Summary: We study the multiplicity of solutions for the elliptic problem $-\Delta u=f(x,u)+\varepsilon g(x,y) \text{ in }\Omega\text{ and }u=0 \text{ on }\partial \Omega,$ where $$\varepsilon$$ is a parameter, $$\Omega$$ is a smooth bounded domain in $$\mathbb{R}^N$$, $$f\in C(\overline \Omega\times \mathbb{R})$$, $$f(x,t)$$ is odd with respect to $$t$$, and $$g\in C(\overline \Omega\times \mathbb{R})$$. Under suitable conditions only on $$f$$, we prove that for any $$j\in\mathbb{N}$$ there exists $$\varepsilon_j >0$$ such that if $$|\varepsilon|\leq\varepsilon_j$$ then the above problem possesses at least $$j$$ distinct solutions.

##### MSC:
 35J25 Boundary value problems for second-order elliptic equations 35B20 Perturbations in context of PDEs
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##### References:
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