Abstract
The structure of the set of positive solutions of the semilinear elliptic boundary value problem
depends on a certain non-degeneracy condition, which was proved by K.J. Brown [2] and T. Ouyang and J. Shi [12], with a shorter proof given later by P. Korman [8]. In this note we present a more general result, communicated to us by L. Nirenberg [13]. We also discuss the extensions in cases when the domain D is in R
2, and it is either symmetric or convex.
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References
J. Bebernes abd D. Eberly, Mathematical Problems from Combustion Theory, Spring er-Verlag (1989).
K.J. Brown, Curves of solutions through points of neutral stability, Result. Math. 37, 315–321 (2000).
L. Damascelli, M. Grossi and F. Pacella, Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle, Ann. Inst. H. Poincare Anal. Non Linéaire 16, no. 5, 631–652 (1999).
B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys. 68, 209–243 (1979).
M. Holzmann and H. Kielhofer, Uniqueness of global positive solution branches of nonlinear elliptic problems, Math. Ann. 300, no. 2, 221–241 (1994).
P. Korman, Solution curves for semilinear equations on a ball, Proc. Amer. Math. Soc. 125, 1997–2005 (1997).
P. Korman, A global solution curve for a class of semilinear equations, Proceedings of the Third Mississippi State Conference on Differential Equations and Computational Simulations 119–127, Electron. J. Differ. Equ. Conf, 1 (1998).
P. Korman, A remark on the non-degeneracy condition, Result. Math. 41, 334–336 (2002).
P. Korman, Y. Li and T. Ouyang, Exact multiplicity results for boundary-value problems with nonlinearities generalising cubic, Proc. Royal Soc. Edinburgh, Ser. A 126A, 599–616 (1996).
C.S. Lin, Uniqueness of least energy solutions to a semilinear elliptic equation in R2, Manuscripta Mathematica 84, 13–19 (1994).
C.S. Lin and W.-M. Ni, A counterexample to the nodal domain conjecture and related semilinear equation, Proc. Amer. Math. Soc. 102, 271–277 (1988).
T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problems. J. Differential Equations 146, 121–156 (1998).
L. Nirenberg, Letter of 03/13/2003.
J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43, 304–318 (1971).
M. Tang, Uniqueness of positive radial solutions for An - u + up - 0 on an annulus. J. Differential Equations 189, no. 1, 148–160 (2003).
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Korman, P. Further remarks on the non-degeneracy condition. Results. Math. 45, 293–298 (2004). https://doi.org/10.1007/BF03323383
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DOI: https://doi.org/10.1007/BF03323383