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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)].

##### MSC:
 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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##### References:
 [1] W. Water, Ganze Losungen der Differentialglchung Δpu=f(u), Math. Z 67 (1957) 32-37 [2] W. Water, Zur Existenz Ganzer Losungen der Differentialglchung Δpu=eu, Arch. Math. 9 (1958) 308-312 [3] Water, W.; Rhee, H., Entire solutions of δpu=f(r,u), Proc. royal soc. Edinburgh A, 82, 189-192, (1979) · Zbl 0402.35048 [4] Kusno, T.; Swanson, C.A., Positive entire solutions of semilinear biharmonic equations, Hiroshima math. J., 17, 13-28, (1989) [5] Xingye, X., The existence of positive entire solutions about singular nonlinear elliptic equations, J. math. (PRC), 15, 421-428, (1995) · Zbl 0848.35038 [6] Adams, R.A., Sobolev spaces, (1975), Academic Press Boston, MA · Zbl 0186.19101 [7] Edwards, R.E., Functional analysis, (1965), Richart and Winston New York · Zbl 0182.16101 [8] Debnath, L.; Mikusinski, P., Introduction to Hilbert spaces with applications, (1999), Academic Press Boston, MA · Zbl 0940.46001
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