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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 \(\lambda\).
MSC:
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
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