A note on a result of Mironescu and Radulescu (Q390078)

Consider the boundary value problem \(-\Delta u = \lambda f(u)\), \(u\in H^1_0(\Omega)\), where \(\Omega\subset\mathbb{R}^N\) is a bounded domain, \(f\in C^1([0,\infty),\mathbb{R})\) is convex, non-negative, \(f(0), f'(0)>0\) and \(\lambda>0\) is a parameter. It has been shown by Mironescu and Radulescu that if \(\lim_{t\to\infty}(f(t)-at) = l\geq 0\), then this problem has exactly one solution \(u_\lambda\) for \(0<\lambda<\lambda^*:=\lambda_1/a\) and no solution if \(\lambda\geq \lambda^*\) (\(\lambda_1\) is the first Dirichlet eigenvalue of \(-\Delta\) in \(\Omega\)). In this paper the author shows that \(u_\lambda\) has some additional properties. In particular, the function \(\lambda \mapsto h(\lambda) := \|u_\lambda\|^2\) is increasing on \((0,\lambda^*)\) and its range is \((0,+\infty)\). Moreover, for \(r>0\) and \(\lambda=h^{-1}(r)\), \(u_\lambda\) is the unique global maximum of \(J\) on the sphere \(\|u\|^2=r\), where \(J(u):=\int_\Omega F(u)\,dx\) and \(F\) is the primitive of \(f\). The proof uses an abstract result by \textit{B. Ricceri} [Taiwanese J. Math. 12, No. 6, 1303--1312 (2008; Zbl 1157.35076)].