The objective is to prove the existence of positive decaying radial solutions of biharmonic problems of the type \[ \Delta^ 2u=p \bigl( | x | \bigr) u^ \gamma,\;x \in \mathbb{R}^ N,\;N \geq 5,\;u\in D_ 0^{2,2}(\mathbb{R}^ N) \cap C^{4+\alpha}_{\text{loc}} (\mathbb{R}^ N), \tag{1} \] where \(\Delta^ 2\) denotes the iterated \(N\)-dimensional Laplacian, \(D_ 0 ^{2,2}(\mathbb{R}^ N)\) is a standard Sobolev space with equivalent norm \(\| \Delta u \|_ 2\), \(p \in C^ \alpha_{\text{loc}}(0,\infty)\) for some \(\alpha \in(0,1)\), \(0\not\equiv p(r) \geq 0\) in \((0,\infty)\), and \[ p(r)=O(r^ \mu)\text{ as } r \to 0,\quad p(r)=O(r^ \nu) \text{ as } r \to \infty,\;-4<\nu<\mu, \tag{2} \] where \(\mu,\nu,\gamma\) satisfy (3) \(2(N+\nu)/(N-4)< \gamma+1<2 (N+ \mu)/(N-4)\). If \(\nu<0\) and \(\mu>0\), we note that the interval (3) contains the critical Sobolev exponent \(2N/(N-4)\) in its interior, and accordingly problem (1) can be appropriately labelled “transcritical”.
Theorem. Under the stated conditions on \(p\) and \(\gamma\), problem (1) has a positive radial solution \(u(x)\) in \(\mathbb{R}^ N\) such that, for arbitrary \(\varepsilon>0\), \(u(x)=O \bigl( | x |^{4- N+\varepsilon} \bigr)\) as \(| x | \to \infty\).