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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)].

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
Full Text: DOI
[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)
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[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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