Imperfect bifurcations via topological methods in superlinear indefinite problems (Q260958)

The author gives a deeper insight in the bifurcation diagrams structure, showing how the secondary bifurcations break as the weight is perturbed from the symmetric situation. More precisely, the author is looking for positive solutions of the boundary value problem NEWLINE\[NEWLINE\begin{aligned} -u''& =\lambda u+a(t)u^{p}, \quad \Omega=(0,1),\\ u(0)& =u(1)=M, \end{aligned}NEWLINE\]NEWLINE where \(M \in (0,+\infty]\), \(p>1\), \(\lambda<0\) are constants and \(a(t)\) is a weight function indefinite in sign which is defined as NEWLINE\[NEWLINEa(t)=a_{\nu}(t)=\begin{cases} -c,\quad t\in [0,\alpha)\\ b, \quad t \in [\alpha,1-\alpha]\\ -\nu c, \quad (1-\alpha, 1]\end{cases}NEWLINE\]NEWLINEwith \(\alpha \in (0,0.5)\), \(b\geq 0\) and \(c\), \(\nu >0\). Thus, it is sublinear in some region of the domain, while it is superlinear in other, i.e. is superlinear infinite.NEWLINENEWLINEThe author essentially uses his previous results [\textit{J. López-Gómez} and the author, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 108, 223--248 (2014; Zbl 1302.34043); \textit{J. López-Góme} et al., Commun. Pure Appl. Anal. 13, No. 1, 1--73 (2014; Zbl 1281.34027)].