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Three solutions for a perturbed Dirichlet problem. (English) Zbl 1141.35030
Summary: We prove the existence of at least three distinct solutions to the following perturbed Dirichlet problem: \[ \begin{gathered} -\Delta u= f(x,u)+\lambda g(x,u)\qquad\text{in }\Omega,\\ u= 0\qquad\text{on }\partial\Omega.\end{gathered} \] where \(\Omega\subset\mathbb{R}^N\) is an open bounded set with smooth boundary \(\partial\Omega\) and \(k\in\mathbb{R}\). Under very mild conditions on \(g\) and some assumptions on the behaviour of the potential of \(f\) at \(0\) and \(+\infty\), our result assures the existence of at least three distinct solutions to the above problem for \(\lambda\) small enough. Moreover such solutions belong to a ball of the space \(W^{1,2}_0(\Omega)\) centered in the origin and with radius not dependent on \(\lambda\).
35J65 Nonlinear boundary value problems for linear elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35J20 Variational methods for second-order elliptic equations
Full Text: DOI
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