Positive entire solutions of nonlinear polyharmonic equations in \(\mathbb R^2\). (English) Zbl 1126.35310

Summary: In this paper, the existence of positive, radially symmetric entire solutions for the equations \(\Delta^mu=f(| x| ,u,| \nabla u| )\) (\(m=2,3,\dots,\)) on \(\mathbb R^2\) where \(\nabla\) is the spatial gradient and \(\Delta\) is the Laplacian on \(\mathbb R^2\) is proved. Some properties of the solutions are obtained. The results of this paper are generalizations of these proved in [W. Walter, Math. Z 67, 32–37 (1957; Zbl 0077.30303); Arch. Math. 9, 308–312 (1958; Zbl 0087.09602); W. Walter and H. Rhee, Proc. R. Soc. Edinb., Sect. A 82, 189–192 (1979; Zbl 0402.35048); T. Kusno and C. A. Swanson, Hiroshima Math. J. 17, 13–28 (1989; Zbl 0649.35032)].


35J30 Higher-order elliptic equations
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J60 Nonlinear elliptic equations
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[1] W. Water, Ganze Losungen der Differentialglchung Δpu=f(u), Math. Z 67 (1957) 32-37
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