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A priori estimates and existence of positive solutions of a superlinear polyharmonic equation. (English) Zbl 0812.35048

The author considers subcritical semilinear polyharmonic equations in \(\Omega\subset \mathbb{R}^ n\), a prototype of which is \((-\Delta)^ K u=u^ p\). Here \(\Omega\) is a bounded smooth domain, \(p< (n+2K)/ (n-2K)\) \((n>2K)\). The solution \(u\) is subject either to Navier boundary conditions \(\Delta^ j u=0\) on \(\partial\Omega\), \(j=0,\dots, K-1\) or to Dirichlet boundary conditions: \(D^ j u=0\) on \(\partial\Omega\), \(j=0,\dots, K-1\). As the author is interested in the existence of strictly positive solutions, he needs a positivity preserving property for the Green’s operator to \((-\Delta)^ K\) under the respective boundary conditions. Under Dirichlet conditions, this property is known up to now only for balls, while under Navier conditions, \(\Omega\) may be arbitrary.
By means of \(L^ \infty\) a-priori estimates and degree theory, the existence of a positive solution \(u\in C^{2K} (\overline{\Omega})\) is shown for nonlinearities, which behave similarly as \(u\mapsto u^ p\). This result is well known for \(K=1\) and due to P. Oswald [Commentat. Math. Univ. Carol. 26, 565-577 (1985; Zbl 0612.35055)] for \(K=2\). See also R. Dalmasso [Bull. Sci. Math., II. Ser. 114, No. 2, 123-137 (1990; Zbl 0714.35030)].
Moreover the author remarks that the Dirichlet problem doesn’t have any strictly positive solution, if \(p= (n+2K)/ (n-2K)\) is critical. This follows from a Pohožaev-type identity due to P. Pucci and J. Serrin [Indiana Univ. Math. J. 35, 681-703 (1986; Zbl 0625.35027)].

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35B45 A priori estimates in context of PDEs
35J40 Boundary value problems for higher-order elliptic equations
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