Uniqueness of positive solutions for some nonlinear fourth-order equations
The paper deals with the boundary value problem (1) \(u^{(4)}= f(u)\), \(u\in (- R, R)\), \(u(\pm R)= u'(\pm R)= 0\), \(u> 0\) in \((- R, R)\), where \(R> 0\) and \(f: [0, \infty)\to [0, \infty)\) is from \(C^1(0, \infty)\) and satisfies the condition (H): \(0< f(u)< uf'(u)\) for \(u> 0\). The author gives the relation of problem (1) with the problem (2) \(u^{(4)}= f(u)\) in \([0, R)\), \(u(R)= u'(R)= u'(0)= u'''(0)= 0\), \(u> 0\) in \([0, R)\) and with the problem (3) \(u^{(4)}(t)= f(u(t))\), \(t\geq 0\), \(u(0)= \alpha\), \(u''(0)= -\beta\), \(u'(0)= u'''(0)= 0\) for \(\alpha, \beta> 0\). He proves the theorem: Let \(f\in C^1[0, \infty)\) satisfy (H). Then for any \(\alpha> 0\) there exists a unique \((\beta(\alpha), t(\alpha))\in (0, \infty)\times (0, \infty)\) such that \(u(\alpha, \beta(\alpha), t(\alpha))= u'(\alpha, \beta(\alpha), t(\alpha))= 0\), \(u(\alpha, \beta(\alpha), t)> 0\) for \(t\in [0, t(\alpha))\) and \(u'(\alpha, \beta(\alpha), t)< 0\) for \(t\in (0, t(\alpha))\). Moreover, \(\beta\), \(t\in C^1(0, \infty)\), \(\beta'(\alpha)> 0\) for \(\alpha> 0\) and \(t'(\alpha)\neq 0\) for \(\alpha> 0\). From this theorem follows immediately the validity of the existence and uniqueness of the solution of problem (1). Some examples are given, too.
scientific article; zbMATH DE number 908623