Half-eigenvalues of self-adjoint, \(2m\)th-order differential operators and semilinear problems with jumping nonlinearities
Let \(L\) be a \(2m\)th-order selfadjoint disconjugate ordinary differential operator on \([0,\pi]\) and consider the semilinear boundary value problem \[ Lu(x)=f(x,u(x))+h(x), \tag{1} \] where \(f\) is a Carathéodory function. Assuming that \(a(x)=\lim_{\xi\to\infty }f(x,\xi)/\xi\), \(b(x)=\lim_{\xi\to-\infty }f(x,\xi)/\xi\) exist a.e., the differential equation \[ Lu=au^+-bu^-+\lambda u+h,\tag{2} \] \(u^\pm(x)=\max\{\pm u(x),0\}\) is considered. A real number \(\lambda \) is called a half-eigenvalue of (2) if, with \(h=0\), there is a nontrivial solution \(u\) to (2). These half-eigenvalues can be indexed \(\lambda _{k,\nu}\), \(k\geq 1\), \(\nu\in\{+,-\}\), where \(k-1\) is the number of zeros of the corresponding solution and \(\nu\) indicates its sign near \(0\). If the first eigenvalue \(\lambda _1\) of the positive linear operator \(L\) satisfies \(Q:=\max\{a_{\max},b_{\max}\}-\min\{a_{min},b_{min}\}<\lambda _1\) or if \(a\) and \(b\) are constant, then they satisfy \(\max\{\lambda _{k,+},\lambda _{k,-}\}<\min\{\lambda _{k,+},\lambda_{k,-}\}\) and for \(\lambda \in (\max\{\lambda_{k,+},\lambda_{k,-}\},\min\{\lambda_{k+1,+},\lambda_{k+1,-}\})\) or \(\lambda \in (-\infty ,\min\{\lambda_{1,+},\lambda_{1,-}\})\) and every \(h\in L^2(0,\pi)\), (2) has a solution. On the other hand, if \(3Q<\lambda _1\), a lower bound for the number of normalized solutions, normally larger than \(1\), is given if the above interval contains \(0\). In the case \(a\), \(b\) constant, it is shown that the Fučik spectrum has a behavior as known for the case \(m=1\). Finally, it is shown that the existence and multiplicity results for (2) lead to existence and multiplicity results for (1).
scientific article; zbMATH DE number 1832861