Connectedness of the branch of positive solutions of some weakly nonlinear elliptic equations (Q1327688)

The author considers branches of positive solutions of \((*)\;-\Delta u= \lambda f(u)\) in \(D\), \(u| \partial D=0\). Here \(\lambda>0\) and \(D\subset \mathbb{R}^ 2\) is a smooth domain, which contains the origin and is symmetric with respect to the \(x\)- and the \(y\)-axis. Its boundary satisfies \(\partial D\cap \{(x,y)\): \(x\geq 0\), \(y\geq 0\}= \{(x, h(x))\): \(0\leq x\leq a\}\) where \(h\) is decreasing. A number of different conditions on \(f\) is discussed; one of these conditions reads as follows: \(f\in C^ 1 ([0,\infty))\), \(f(y)>0\) for all \(y\geq 0\) and \(f'(y)/ y^{q-1}\to c\in (0,\infty)\) as \(y\to\infty\) for some \(q>1\). It is shown that \(W= \{(u,\lambda)\in C(\overline{D})\times \mathbb{R}\): \(\lambda>0\), \((u,\lambda)\) is a positive solution to \((*)\}\) is a connected smooth curve. The core of the paper is the proof of the connectedness of \(W\), which is obtained by deforming the ball continuously into the domain \(D\). If the domain is a ball, every positive solution is radial and the solution curve \(W\) may be parametrized by the ``initial value'' \(u(0)>0\).