Non-existence of nodal and one-signed solutions for nonlinear variational equations with special symmetries
In a recent paper with \textit{P. Pucci} [Arch. Ration. Mech. Anal. 127, No. 3, 255-280 (1994; see the review above)], the authors established a lower bound of the length of the interval \((a, b)\) for which a one-signed solution \(u\) of equation \((| u'|^{m- 2} u')'+ f(t, u)= Q(t, u, u')\), \(m> 1\), \(t\in (a, b)\subset \mathbb{R}^ +\), with \(u(b)= u'(a)= 0\), exists. The purpose of this paper is to extend such a result to the much wider class of equations of the form \((G_ p(u'))'+ f(t, u)= Q(t, u, u')\). The results are applied to obtain non-existence theorems for nodal and one-signed solutions of the variational problem \((g(t) G_ p(u'))'+ g(t) f(u)= 0\), \(u(0)\neq 0\), \(u(T)= 0\), \(T> 0\), when \(T\) is sufficiently small.
scientific article; zbMATH DE number 687682