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 $\mathbb {R}^{n}$ for the following semilinear higher-order problem: 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 $f(u)= -m u +\lambda|u|^{\theta-1}u-\mu |u|^{p-1}u$ , where $m\geq0$ , $\lambda>0$ , $\mu>0$ , $p, \theta>1$ . 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 $f(u)u$ is a nonnegative function will be studied under a large subcritical growth assumption at zero, for example $f(u)=|u|^{\theta-1}u(1 + |u|^{q})$ or $f(u)= |u|^{\theta-1}u e^{|u|^{q}}$ , $\theta>1$ and $q>0$ . Extensions to solutions which are merely stable are discussed in the case of supercritical growth with $k=1$ . [ABSTRACT FROM AUTHOR]