an:05286010
Zbl 1141.35030
Cordaro, Giuseppe; Rao, Giuseppe
Three solutions for a perturbed Dirichlet problem
EN
Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 68, No. 12, 3879-3883 (2008).
00219453
2008
j
35J65 35D05 35J20
weak solutions; critical points; weakly sequentially lower-semicontinuity; perturbed Dirichlet problem
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 \(\lambda\).