A nonlinear boundary value problem with potential oscillating around the first eigenvalue
The paper deals with the boundary value problem (1) \(u'' + u + f(t,u) = 0\), \(u(0) = u(\pi) = 0\), where \(f : [0, \pi] \times \mathbb{R} \to \mathbb{R}\) is such that \(f(t,.)\) is continuous for a.e. \(t\), \(f(x,.)\) is measurable for all \(x\) and there exists \(h \in L^ 2\) such that \(| f (t,u)|\leq h(t)\) holds for a.e. \(t\) and all \(x\). The authors prove that in the case that the potential \(F(t, u) = \int^ t_ 0 f(t,v) \text{d}v\) fulfils the conditions (2) \(\lim \sup_{a \to \pm \infty} \int^ \pi_ 0\) \(F(s, a \sin s) \text{d}s = + \infty\), \(\lim \inf_{b \to \pm \infty} \int^ \pi_ 0\) \(F(s,b \sin s) \text{d}s = - \infty\) the given problem possesses infinitely many solutions and each of them is a critical point of the functional \(\Phi (u) = \int_ 0^ \pi [{1 \over 2} (u'(t))^ 2 - {1 \over 2} (u(t))^ 2 - F(t, u(t))] \text{d}t\). Furthermore, some extensions to the case of more general boundary conditions and several examples of functions fulfilling the assumptions (2) are given.
scientific article; zbMATH DE number 756570