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 R n for the following semilinear higher-order problem: ( − Δ ) k u = f ( u ) in R n , with k = 1 , 2 , 3 , 4 . 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 f ( u ) = − m u + λ | u | θ − 1 u − μ | u | p − 1 u , where m ≥ 0 , λ > 0 , μ > 0 , p , θ > 1 . 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 f ( u ) u is a nonnegative function will be studied under a large subcritical growth assumption at zero, for example f ( u ) = | u | θ − 1 u ( 1 + | u | q ) or f ( u ) = | u | θ − 1 u e | u | q , θ > 1 and q > 0 . Extensions to solutions which are merely stable are discussed in the case of supercritical growth with k = 1 .