In this paper, we study the existence and multiplicity of solutions to periodic boundary value problem of the extended Fisher–Kolmogorov equation P \(\begin{aligned} \left\{ \begin{array}{ll} u''''(t)+ q u''(t)-\lambda u(t)=f(t,u(t))-h(t), \quad \ t\in (0,1),\\ u(0)=u(1),\quad u'(0)=u'(1),\quad u''(0)=u''(1),\quad u'''(0)=u'''(1), \end{array} \right. \end{aligned}\) where \(q\in (-4\pi ^2,0)\) , \(\lambda \in \mathbb {R}\) is a parameter, \(h\in L^1(0,1)\) , \(f:[0,1]\times \mathbb {R}\rightarrow \mathbb {R}\) is a \(L^1\) -Carathéodory function. Also, f is nondecreasing with respect to the second variable and sublinear at \(+\infty \) . We show that if \(\lambda \) is sufficiently close to 0 from the left, where 0 is the first eigenvalue of the corresponding linear eigenvalue problem, then (P) has at least one solution. If \(\lambda \) is sufficiently close to 0 from the right, then there are at least two solutions to the problem (P), and it is proved that one of the two solutions is positive. The proof of the main result is based upon the method of lower and upper solutions and bifurcation theory.
