Half-eigenvalues of periodic Sturm--Liouville problems
Here, the half-eigenvalues and the nodal properties of the eigenfunctions of the nonlinear Sturm-Liouville equation \[ \begin{aligned} &-u''+qu=au^+-bu^-+\lambda u, \;0 < x<2\pi, \tag{E1}\\ &u(0)=u(2\pi),\;u'(0)=u'(2\pi),\tag{E2} \end{aligned} \] are considered, where \(q, a, b \in L_1(0,2\pi),\) \(\lambda\) is a real parameter and \(u^{\pm}(t)=\max \{\pm u(t),0\}.\) Let \[ \begin{aligned} H&=\{u\mid u\in W_1^2(0,2\pi), \; u \text{ satisfies the condition (E2)} \},\\ S_k^1&=\{u\mid u\in H,\; \| u\| =1 \text{ and } u \text{ has exactly } k \text{ simple zeros on }[0,2\pi) \},\\ \Sigma_H&=\{\lambda\in\mathbb R: \text{(E1) has a solution on the unit ball } S^1 \text{ of }H \},\\ \Sigma_H^k&=\{\lambda\in\mathbb R: \text{(E1) has a solution in } S^1_{2k}\},\\ \tilde\Sigma_H^k&=\{(\lambda,u)\in \mathbb R \times S_{2k}^1:(\lambda,u)\text{ satisfies (E1)}\}. \end{aligned} \] The authors show that (1) For each integer \(k\geq 0\), (E1) has two distinct solutions \((\lambda_k^{min} ,\phi_k^{min}),\) \((\lambda_k^{max},\phi_k^{max})\in \mathbb R\times S^1_{2k},\) such that (a) \(\lambda_k^{min}\leq \lambda_k^{max}<\lambda_{k+1}^{min},\) (b) \(\{\lambda_k^{min},\lambda_k^{max}\}\subset \Sigma_H^k\subset [\lambda_k^{min},\lambda_k^{max}],\) and \(\Sigma_H^0=\{\lambda_0^{min},\lambda_0^{max}\}.\) (2) Denote by \((L;a,b)\) the problem (E1), (E2). As functions of \((a,b),\) the eigenvalues \(\lambda_k^{min,max}(L;a,b)\) depend continuously on \(a\) and \(b\) and possess a monotonie property, i.e., if \((a_0,b_0)\leq (a_1,b_1)\) then \[ \lambda_k^{min,max}(L;a_1,b_1)\leq \lambda_k^{min,max}(L;a_0,b_0). \] Moreover, some intrinsic properties for \(\Sigma_H\) are obtained. The results are extended to the more general nonlinear problem \[ -u''+qu={\mathbf f}(u)+h(t). \tag{E3} \]
scientific article; zbMATH DE number 2126458
scientific article