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Classification of solutions of some nonlinear elliptic equations. (English) Zbl 0768.35025

The paper concerns the equations \(\Delta u+u^ p=0\), \(x\in\mathbb{R}^ n\), \(n\geq 3\), and \(\Delta u+\exp u=0\), \(x\in\mathbb{R}^ 2\), \(\int_{\mathbb{R}^ 2} \exp u(x)dx<\infty\).

MSC:

35J60 Nonlinear elliptic equations
35C05 Solutions to PDEs in closed form
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[1] C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domain , to appear in Comm. Partial Differential Equations. · Zbl 0741.35014
[2] B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in \(\mathbf R\spn\) , Mathematical analysis and applications, Part A ed. L. Nachbin, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York, 1981, pp. 369-402. · Zbl 0469.35052
[3] H. Berestycki and L. Nirenberg, Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains , · Zbl 0705.35004
[4] M. Obata, The conjectures on conformal transformations of Riemannian manifolds , J. Differential Geometry 6 (1971/72), 247-258. · Zbl 0236.53042
[5] L. Caffarelli, B. Gidas, and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth , · Zbl 0702.35085
[6] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations , Comm. Pure Appl. Math. 34 (1981), no. 4, 525-598. · Zbl 0465.35003
[7] W. Ding, personal communication. · Zbl 1216.18015
[8] H. Brezis and F. Merle, Estimates on the solutions of \(\Delta u=v(x) \exp u(x)\) on \(R^2\) ,
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