Morse theory without (PS) condition at isolated values and strong resonance problems
The authors consider the elliptic boundary value problem \[ -\Delta u = \lambda u + q(x,u) \quad\text{in }\Omega, \qquad u=0 \quad\text{on }\partial\Omega,\tag{*} \] where \(\Omega\subset{\mathbb R}^{N}\) is bounded, \(\lambda\) is an eigenvalue of \(-\Delta\), \(q(x,0)=0\) and \(q(x,t)\to 0\), \(q'_{t}(x,t)\to 0\) as \(|t|\to\infty\). Solutions of (*) correspond to critical points of the functional \(I(u) = {1\over 2}\int_{\Omega}(|\nabla u|^2-\lambda u^2) dx - \int_{\Omega}Q(x,u) dx\) in the Sobolev space \(H^1_{0}(\Omega)\); here \(Q(x,t) := \int_{0}^tq(x,s) ds\). An important role in the analysis of (*) is played by the set \(\Lambda\) of values \(c\) such that for some \(v\in \text{Ker} (-\Delta -\lambda) \setminus \{0\}\), \(-\int_{\Omega}Q(x,tv) dx\to c\) as \(t\to\infty\). If \(\Lambda\) is bounded, then problem (*) is called strongly resonant. In this paper it is shown that if \(\Lambda\) is a finite set, then the functional \(I\) satisfies the Morse inequalities, with an extra part in the Morse polynomial compensating for the loss of the Palais-Smale condition at the levels \(c\in\Lambda\). This result is subsequently employed in order to show that under different sets of hypotheses, (*) contains 2, 3 or 4 nontrivial solutions. Here the set \(\Lambda\) is a singleton if \(\dim\text{Ker} (-\Delta -\lambda) \geq 2\) and consists of 1 or 2 points if \(\dim\text{Ker} (-\Delta -\lambda) = 1\). The results of this paper considerably extend earlier work on strongly resonant problems where it was usually assumed that \(\Lambda\) is a singleton and existence of only 1 nontrivial solution was shown. The fact that strongly resonant problems can be treated by a Morse theory which is very close to the usual one is of independent interest.
scientific article; zbMATH DE number 1455253