Interior bubbling solutions for the critical Lin-Ni-Takagi problem in dimension 3
- 16 Downloads
We consider the problem of finding positive solutions of the problem Δu − λu + u5 = 0 in a bounded, smooth domain Ω in ℝ3, under zero Neumann boundary conditions. Here λ is a positive number. We analyze the role of Green’s function of −Δ + λ in the presence of solutions exhibiting single bubbling behavior at one point of the domain when λ is regarded as a parameter. As a special case of our results, we find and characterize a positive value λ* such that if λ − λ* > 0 is sufficiently small, then this problem is solvable by a solution uλ which blows-up by bubbling at a certain interior point of Ω as λ ↓ λ*.
Unable to display preview. Download preview PDF.
- M. G. Adimurthi, The Neumann problem for elliptic equations with critical nonlinearity, in Nonlinear Analysis, Publications of the Scuola Normale Superiore of Pisa, Pisa, 1991, pp. 9–25.Google Scholar
- O. Druet, F. Robert and J. Wei, The Lin-Ni’s problem for mean convex domains, Mem. Amer. Math. Soc. 218 (2012), no. 1027.Google Scholar
- C.-S. Lin and W.-M. Ni, On the diffusion coefficient of a semilinear Neumann problem, in Calculus of Variations and PartialDifferential Equations (Trento, 1986), Lecture Notes inMath., Vol. 1340, Springer, Berlin, 1988, pp. 160–174Google Scholar
- W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semi-linear Neumann problem, Duke Math. J. 70 (1993), 247–281.Google Scholar