Solution curves for semilinear equations on a ball

Korman, Philip

doi:10.1090/S0002-9939-97-04119-1

Solution curves for semilinear equations on a ball
HTML articles powered by AMS MathViewer

by Philip Korman PDF

Proc. Amer. Math. Soc. 125 (1997), 1997-2005 Request permission

Abstract:

We show that the set of positive solutions of semilinear Dirichlet problem on a ball of radius $R$ in $R^n$ \[ \Delta u+\lambda f(u)=0 \; \; \text {for} \; \; |x|<R, \; \; u=0 \; \; \text {on} \; \; |x|=R \] consists of smooth curves. Our results can be applied to compute the direction of bifurcation. We also give an easy proof of a uniqueness theorem due to Smoller and Wasserman (1984).

References

Michael G. Crandall and Paul H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal. 52 (1973), 161–180. MR 341212, DOI 10.1007/BF00282325
E. N. Dancer, On the structure of solutions of an equation in catalysis theory when a parameter is large, J. Differential Equations 37 (1980), no. 3, 404–437. MR 590000, DOI 10.1016/0022-0396(80)90107-2
L.C. Evans, Partial Differential Equations, Berkeley Lecture Notes in Mathematics. Vol. 3 A&B (1994).
R. Gardner and L. A. Peletier, The set of positive solutions of semilinear equations in large balls, Proc. Roy. Soc. Edinburgh Sect. A 104 (1986), no. 1-2, 53–72. MR 877892, DOI 10.1017/S0308210500019065
B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. MR 544879
Marco Holzmann and Hansjörg Kielhöfer, Uniqueness of global positive solution branches of nonlinear elliptic problems, Math. Ann. 300 (1994), no. 2, 221–241. MR 1299061, DOI 10.1007/BF01450485
P. Korman, Steady states and long time behavior of some convective reaction-diffusion equations, To appear in Funkcialaj Ekvacioj.
P. Korman, Y. Li and T. Ouyang, Exact multiplicity results for boundary-value problems with nonlinearities generalizing cubic, Proc. Royal Soc. Edinburgh Ser. A 126A, 599-616 (1996).
P. Korman, Y. Li and T. Ouyang, An exact multiplicity result for a class of semilinear equations, To appear in Comm. in PDE.
Chang Shou Lin and Wei-Ming Ni, A counterexample to the nodal domain conjecture and a related semilinear equation, Proc. Amer. Math. Soc. 102 (1988), no. 2, 271–277. MR 920985, DOI 10.1090/S0002-9939-1988-0920985-9
L. A. Peletier and James Serrin, Uniqueness of positive solutions of semilinear equations in $\textbf {R}^{n}$, Arch. Rational Mech. Anal. 81 (1983), no. 2, 181–197. MR 682268, DOI 10.1007/BF00250651
Joel A. Smoller and Arthur G. Wasserman, Existence, uniqueness, and nondegeneracy of positive solutions of semilinear elliptic equations, Comm. Math. Phys. 95 (1984), no. 2, 129–159. MR 760329