Bifurcation and nodal solutions for the half-linear problems with nonasymptotic nonlinearity at \(0\) and \(\infty\)
Summary: We study the existence of nodal solutions for the following problem: \(- x'' = \alpha x^+ + \beta x^- + r a(t) f(x)\), \(0 < t < 1\), \(x(0) = x(1) = 0\), where \(r \neq 0\) is a parameter, \(a(t) \in C([0,1],(0, \infty))\) with \(a(t) \not\equiv 0\) on any subinterval of \([0,1]\), \(x^+ = \max \{x, 0 \}, x^- = - \min \{x, 0 \}\), and \(\alpha, \beta \in C [0,1]\); \(f \in C(\mathbb{R}, \mathbb{R})\), \(s f(s) > 0\) for \(s \neq 0\), and \(f_0, f_{\infty} \notin(0, \infty)\), where \(f_0 = \lim_{| s | \to 0} f(s) / s\) and \(f_{\infty} = \lim_{| s | \to + \infty} f(s) / s\). We use bifurcation techniques to prove our main results.
scientific article; zbMATH DE number 7025588
scientific article