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Symmetry of positive solutions of nonlinear elliptic equations in R^n
B. Gidas
- Vol. 7, pp 369-402
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The article was published on 1981-01-01 and is currently open access. It has received 982 citations till now. The article focuses on the topics: Symmetry (physics) & Nonlinear system.read more
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Nonlinear dispersive equations : local and global analysis
TL;DR: In this paper, the Korteweg de Vries equation was used for ground state construction in the context of semilinear dispersive equations and wave maps from harmonic analysis.
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Uniqueness of positive solutions of Δu−u+up=0 in Rn
TL;DR: In this article, the uniqueness of the positive, radially symmetric solution to the differential equation Δu−u+up=0 (with p>1) in a bounded or unbounded annular region in Rn for all n ≥ 1, with the Neumann boundary condition on the inner ball and the Dirichlet boundary condition decaying to zero in the case of an unbounded region, was established.
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The concentration-compactness principle in the calculus of variations. The locally compact case, part 2
TL;DR: In this paper, the authors investigated further applications of the concentration-compactness principle to the solution of various minimization problems in unbounded domains, in particular the problem of minimizing nonlinear field equations.
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On the method of moving planes and the sliding method
Henri Berestycki,Louis Nirenberg +1 more
TL;DR: In this paper, a simplified version of the sliding method is used to prove mono- tonicity or symmetry in the zl direction for solutions of nonlinear elliptic equations F(x, u, Du, D~u) = 0 in a bounded domain (2 in R n).
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Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics
TL;DR: In this paper, the authors considered a semilinear elliptic problem and proved the existence of a positive groundstate solution of the Choquard or nonlinear Schr\"odinger--Newton equation for an optimal range of parameters.