Multiple solutions for a Dirichlet problem with jumping nonlinearities. II
[For part I see Trends in the theory and practice of nonlinear analysis, Proc. 6th Int. Conf. Arlington/Tex. 1984, North-Holland Math. Stud. 110, 97-101 (1985; Zbl 0575.34002).] Continuing their previous work the authors give a very technical proof of a theorem stating that if p: [0,1]\(\to {\mathbb{R}}\) is continuous, g:(0,\(\infty)\to (0,\infty)\) is of class \(C^ 1\), strictly increasing and such that there exist \(M\in {\mathbb{R}}\), \(r>0\) such that \(\lim_{u\to - \infty}(g(u)/u)=M,\) \(\lim_{u\to \infty}(g(u)/u^{1+r})=\infty\) then there exists an increasing sequence c(n)\(\to \infty\) such that if n is sufficiently large and \(c>c(n)\) then the problem \(-x''=g(x)-p(t)-c,\) \(t\in [0,1]\), \(x(0)=x(1)=0\) has at least two solutions with n interior zeroes; moreover, if \(g(u)-\pi^ 2u\) is bounded below then there exists \(c_*\) such that the problem has no solution for \(c<c_*\).
nonlinear second order differential equations
scientific article; zbMATH DE number 4138181