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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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