Solvability of a semilinear boundary value problem with respect to the first eigenvalue. (Q1281188)

If \(\overline f(s)= \sup\{f(t): t\in [0,s]\}\) for \(s\geq 0\), \(\overline f(s)= \inf\{f(t): t\in [s,0]\}\) for \(s\leq 0\) and \(\liminf_{s\to\pm\infty} {\overline f(s)\over s}< {2\over\pi} \lambda_1\), then the problem \[ -u''= f(u)+ h(x)\text{ in }(a,b),\;u(a)= u(b)= 0 \] has at least one solution for each \(h\in L^\infty(a, b)\). Here, \(f\in C(\mathbb{R},\mathbb{R})\) and \(\lambda_1>0\) is the first eigenvalue of \(-u''= \lambda u\), \(u(a)= u(b)= 0\) in \(H^1_0(a, b)\).