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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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