Radial Fučik spectrum of the Laplace operator
Let \(L: D(L)\subset H\to H\) be a linear operator in a function space \(H\). The Fučik spectrum of \(L\) is defined by \(A_ 0= \{(a, b)\in \mathbb{R}^ 2\): \(Lu= au^ +- bu^ -\) for some nontrivial \(u\}\), where \(u^ += \max\{u, 0\}\), \(u^ -= \max\{- u, a\}\). Knowledge of \(A_ 0\) is important for the existence of solutions of \(Lu= g(u)+ f\), where \(g: \mathbb{R}\to \mathbb{R}\) is continuous with \(y^{-1} g(y)\to a(b)\) as \(y\to +\infty(- \infty)\). Obviously, if \(\lambda\) is a real eigenvalue of \(L\), then \((\lambda, \lambda)\in A_ 0\). In this note, the authors consider the case that \(L= \Delta\), the Laplace operator on the unit ball of \(\mathbb{R}^ N\) with Dirichlet boundary conditions, and investigate the set \(A^ R_ 0\) of those \((a, b)\in \mathbb{R}^ 2\), where \(u= u(| x|)\) is a radial function. The following is shown: If \(\xi_ 1< \xi_ 2<\cdots\) denotes the sequence of positive zeros of the Bessel function \(J_ \eta\), \(\eta= (N- 2)/N\), then \(A^ R_ 0\) consists of the union of \(\xi^ 2_ 1\times \mathbb{R}\), \(\mathbb{R}\times \xi^ 2_ 1\), and the curves \(C^ j_ i= \{(t, c^ j_ i(t): t\in (\alpha^ j_ i, \infty)\}\), \(i= 1,2,\dots, j= 1,2\), where \(c^ j_ i\) is a decreasing analytic homeomorphism from \((\alpha^ j_ i, \infty)\) onto \((\beta^ j_ i, \infty)\), where \(\beta^ j_ i= \xi^ 2_ n\) if \(i= 2n\) and \(j= 1\), \(\beta^ j_ i= \xi^ 2_{n+ 1}\) if \(i= 2n+ 1\) or \(i= 2n\) and \(j= 2\), and \(\alpha^ 1_ i= \beta^ 2_ i\), \(\alpha^ 2_ i= \beta^ 1_ i\). It is the case that \(C^{j_ 1}_ k\cap C^{j_ 2}_ i= \emptyset\) if \(i\neq k\) and \((\xi^ 2_{i+ 1}, \xi^ 2_{i+ 1})\in C^ 1_ i\cap C^ 2_ i\). Explicit numerical results have been obtained in the cases \(N= 2,3,4,5\).
scientific article; zbMATH DE number 762180