Classification of stable solutions for non-homogeneous higher-order elliptic PDEs

Under some assumptions on the nonlinearity f, we will study the nonexistence of nontrivial stable solutions or solutions which are stable outside a compact set of [Formula: see text] for the following semilinear higher-order problem: [Formula: see text] with [Formula: see text]. The main methods used are the integral estimates and the Pohozaev identity. Many classes of nonlinearity will be considered; even the sign-changing nonlinearity, which has an adequate subcritical growth at zero as for example [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text]. More precisely, we shall revise the nonexistence theorem of Berestycki and Lions (Arch. Ration. Mech. Anal. 82:313-345, 1983) in the class of smooth finite Morse index solutions as the well known work of Bahri and Lions (Commun. Pure Appl. Math. 45:1205-1215, 1992). Also, the case when [Formula: see text] is a nonnegative function will be studied under a large subcritical growth assumption at zero, for example [Formula: see text] or [Formula: see text], [Formula: see text] and [Formula: see text]. Extensions to solutions which are merely stable are discussed in the case of supercritical growth with [Formula: see text].