On stable solutions of the biharmonic problem with polynomial growth (Q461299)

The authors are concerned with the nonexistence of smooth stable positive solutions for \(\Delta^2 u=u^p\) in the whole space \({\mathbb R}^N\) where \(1<p<\infty\), and the space dimension \(N\) satisfies \(N<2(1+x_0)\) where \(x_0\) is the largest root of the polynomial NEWLINE\[NEWLINE P(x)=X^4-\frac{32p(p+1)}{(p-1)^2}X^2+\frac{32p(p+1)(p+3)}{(p-1)^3}X-\frac{64p(p+1)^2}{(p-1)^4}. NEWLINE\]NEWLINE As a consequence, for \(N\leq 12\) the above equation has no positive stable solutions for all \(p>1\). A similar result is obtained for positive stable solutions of \(\Delta^2=u^p\) subject to Navier boundary conditions in a half-space. Some related results are also obtained for the problem \(\Delta^2 u=\lambda (1+u)^p\) in a smooth bounded domain \(\Omega\) subject to Navier conditions on the boundary.