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Positive entire solutions of semilinear biharmonic equations. (English) Zbl 0649.35032
The main objective is to prove the existence of infinitely many positive radially symmetric solutions of the semilinear biharmonic equation \[ \Delta^ 2u=p(| x|)u^{\gamma},\quad x\in \mathbb R^ N, \] where \(\gamma\neq 1\) is a real constant and \(p: [0,\infty)\mapsto \mathbb R\) is a continuous function. Assume that \(p\) satisfies one of the following three decay conditions:
\[ \begin{aligned} \int^{\infty}t^{2\gamma +1}| p(t)| \,dt&<\infty,\quad N\geq 3;\\ \int^{\infty}t^ 3| p(t)| \,dt&<\infty,\quad N\geq 5;\\ \int^{\infty}t^{\delta}p(t)\,dt&<\infty,\quad N\geq 5 \end{aligned} \] where \(\delta =N-1-\gamma (N-4)\), \(-1<\gamma <1\), and the additional assumption \(p(t)\geq 0\) for any \(x\geq 0\) in the last condition. Then the results include the existence of infinitely many positive entire solutions of each of the following three types:
(I) Unbounded solutions which are bounded from above and below by positive constant multiples of \(1+| x|^ 2,\)
(II) Solutions which are bounded above and below by positive constants,
(III) Solutions which decay uniformly to zero as \(| x| \to \infty.\)
The last condition yields particularly that there exist infinitely many triples \((u_ 1,u_ 2,u_ 3)\) of positive radially symmetric solutions in \(\mathbb R^ N,\) \(N\geq 5\), where \(u_ 1\) is unbounded, \(u_ 2\) is bounded from above and below by positive constants and \(u_ 3\) decays to zero for \(| x| \to \infty\). Moreover, using the additional assumption \(p(x)\geq 0\) for all \(x\geq 0\) the asymptotic behavior of the members of these triples is described in detail.
The assumptions (I), (II), or (III) are sharp in the sense that there are special situations where they are necessary for the existence of a radial solution.

MSC:
35B08 Entire solutions to PDEs
35J60 Nonlinear elliptic equations
35J30 Higher-order elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
35B35 Stability in context of PDEs
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