Existence of positive entire solutions for higher order quasilinear elliptic equations
The purpose of this paper is to establish the existence of positive non- radial entire solutions for a class of nonlinear elliptic equations of the form \[ (- \Delta)^ Mu = f \bigl( x,u, - \Delta u, \dots, (- \Delta)^{M-1}u \bigr), \quad x \in \mathbb{R}^ N, \tag \(*\) \] where \(M \geq 2\), \(N \geq 2\), \(\Delta\) is the \(N\)-dimensional Laplace operator and \(f\) is a locally Hölder continuous function on \(\mathbb{R}^ N \times \mathbb{R}^ M\), with the aid of the higher order super-subsolution method recently discovered by one of the authors [see \textit{Y. Furusho}, Proc. R. Soc. Edinb., Sect. A 120, No. 1/2, 79-91 (1992; Zbl 0769.35019)]. A crucial step in the development of the paper is the construction of super- and subsolutions of \((*)\) as radial entire solutions of suitable chosen majorants and minorants with radial symmetry of \((*)\).
scientific article; zbMATH DE number 757987