Bifurcation of positive solutions for a Neumann boundary value problem (Q2711358)

The authors study the following Neumann boundary value problem: NEWLINE\[NEWLINE-u_{xx} + q^2 u = u^2(1+\sin x), \text{ for } 0 < x < \pi, \quad u_x(0)=u_x(\pi)=0, \tag{1}NEWLINE\]NEWLINE with \(q^2 \in (0,+ \infty)\), by using analytical, approximate and numerical methods. First of all, the authors show that all nonzero solutions to (1) are positive; when \(q^2 =0\) they prove that the only possible solution is \(u\equiv 0\) and however they show that there exist nonzero solutions for \(q^2>0\) as well as the trivial solution \(u \equiv 0\). Then they study the behaviour of problem (1) for \(q^2\) small and \(q^2\) large, finding approximate solutions (also numerically) and they also show that the solutions converge in every sense to the zero function as \(q^2 \to 0\). Moreover, using the monotone convergence theorem for quadratic forms, the authors prove that the inverse of the operator on the left-hand side of (1) is strongly convergent as \(q^2 \to \infty\); this fact is sufficient to stop outer-layer behaviour occurring while does not stop inner-layer behaviour occurring. They also obtain a bifurcation diagram for (1) for \(q^2 \in (0,10)\) and they examine the linearized problem to confirm the accuracy of their bifurcation results.