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 mathbb {R}^{n} for the following semilinear higher-order problem: (−Δ)ku=f(u)in Rn,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} (-\\Delta)^{k} u= f(u) \\quad \\mbox{in }\\mathbb {R}^{n}, \\end{aligned}$$ \\end{document} 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 mgeq0, 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.