A local saddle point theorem and an application to a nonlocal PDE (Q2403766)

The following local Saddle Point Theorem is proved: Let \(E\) be a Hilbert space, \(T\in C^1(E,\mathbb R)\) has locally Lipschitz derivative and suppose that \(E=H\oplus C\), where \( C\) is finite dimensional. Assume that there exist \( r>0, R>0\) such that \[ \inf T_{|\partial B_r(0)\cap H} > \max T_{|\overline{B_R(0)}\cap C} \] and \[ \inf T_{| B_r(0)\cap H} > \max T_{| \left(\partial B_R(0)\right)\cap C}. \] Then there exists a Palais-Smale sequence \(x_n\) of \(T\) such that \(T(x_n) \to \alpha\), where \(\alpha \geq \inf T_{| B_r(0)\cap H}\) (Comp. [\textit{P. H. Rabinowitz}, Reg. Conf. Ser. Math. 65, vii, 100 p. (1986; Zbl 0609.58002)]). Then the theorem is applied to show the existence result for the elliptic Dirichlet problem: Let \(\Omega \subset \mathbb R^n\) be open bounded with smooth boundary. For any \(\lambda>0\) and \(f\in L^2(\Omega)\), there is \(\mu _0>0\) such that for all \(0<\mu<\mu _0\) there is a weak solution of the problem \[ -\Delta u=\lambda u +f+4\mu \|u\|^2_{L^2}u \qquad \text{for } x\in \Omega, \] \[ u=0 \qquad \text{for } \partial \Omega . \]