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

Abstract 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 $\mathbb {R}^{n}$ for the following semilinear higher-order problem: ( − Δ ) k u = f ( u ) in R n , $$\begin{aligned} (-\Delta)^{k} u= f(u) \quad \mbox{in }\mathbb {R}^{n}, \end{aligned}$$ with k = 1 , 2 , 3 , 4 $k=1,2,3,4$ . 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 $f(u)= -m u +\lambda|u|^{\theta-1}u-\mu |u|^{p-1}u$ , where m ≥ 0 $m\geq0$ , λ > 0 $\lambda>0$ , μ > 0 $\mu>0$ , p , θ > 1 $p, \theta>1$ . 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 $f(u)u$ 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 ) $f(u)=|u|^{\theta-1}u(1 + |u|^{q})$ or f ( u ) = | u | θ − 1 u e | u | q $f(u)= |u|^{\theta-1}u e^{|u|^{q}}$ , θ > 1 $\theta>1$ and q > 0 $q>0$ . Extensions to solutions which are merely stable are discussed in the case of supercritical growth with k = 1 $k=1$ .