Positive solutions of nonlinear elliptic equations in the Euclidean plane

Ufuktepe, U.; Zhao, Z.

doi:10.1090/S0002-9939-98-04985-5

Positive solutions of nonlinear elliptic equations in the Euclidean plane
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Proc. Amer. Math. Soc. 126 (1998), 3681-3692 Request permission

Abstract:

In the present paper, we study the existence of solutions to the problem \[ \begin {cases} \Delta u+f(x,u)=0 & \text {in $D$}\\ u>0&\text {in $D$}\\ u=0 & \text {on $\partial D$} \end {cases} \] where $D$ is an unbounded domain in $\mathbb {R}^2$ with a compact nonempty boundary $\partial D$ consisting of finitely many Jordan curves. The goal is to prove an existence theorem for the above problem in a general setting by using Brownian path integration and potential theory.

References

Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic function theory, Graduate Texts in Mathematics, vol. 137, Springer-Verlag, New York, 1992. MR 1184139, DOI 10.1007/b97238
Kai Lai Chung and Zhong Xin Zhao, From Brownian motion to Schrödinger’s equation, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 312, Springer-Verlag, Berlin, 1995. MR 1329992, DOI 10.1007/978-3-642-57856-4
John B. Conway, A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. MR 1070713
M. Cranston, E. Fabes, and Z. Zhao, Conditional gauge and potential theory for the Schrödinger operator, Trans. Amer. Math. Soc. 307 (1988), no. 1, 171–194. MR 936811, DOI 10.1090/S0002-9947-1988-0936811-2
F. Gesztesy and Z. Zhao, On positive solutions of critical Schrödinger operators in two dimensions, J. Funct. Anal. 127 (1995), no. 1, 235–256. MR 1308624, DOI 10.1006/jfan.1995.1010
Ira W. Herbst and Zhong Xin Zhao, Green’s functions for the Schrödinger equation with short-range potentials, Duke Math. J. 59 (1989), no. 2, 475–519. MR 1016900, DOI 10.1215/S0012-7094-89-05922-X
Carlos E. Kenig and Wei-Ming Ni, An exterior Dirichlet problem with applications to some nonlinear equations arising in geometry, Amer. J. Math. 106 (1984), no. 3, 689–702. MR 745147, DOI 10.2307/2374291
W.-M. Ni, “On the elliptic equation $\Delta {u}+K(x) u^{(n+2)/(n-2)}=0$, its generalizations and applications in geometry ”, Indiana Univ. Math. J., 31(1995), 493-529.
Sidney C. Port and Charles J. Stone, Brownian motion and classical potential theory, Probability and Mathematical Statistics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0492329
Zhong Xin Zhao, The local Feynman-Kac semigroup, J. Systems Sci. Math. Sci. 2 (1982), no. 4, 314–326 (English, with Chinese summary). MR 820137
Zhong Xin Zhao, Green function for Schrödinger operator and conditioned Feynman-Kac gauge, J. Math. Anal. Appl. 116 (1986), no. 2, 309–334. MR 842803, DOI 10.1016/S0022-247X(86)80001-4
Z. Zhao, Green functions and conditioned gauge theorem for a $2$-dimensional domain, Seminar on Stochastic Processes, 1987 (Princeton, NJ, 1987) Progr. Probab. Statist., vol. 15, Birkhäuser Boston, Boston, MA, 1988, pp. 283–294. MR 1046423
Z. Zhao, On the existence of positive solutions of nonlinear elliptic equations—a probabilistic potential theory approach, Duke Math. J. 69 (1993), no. 2, 247–258. MR 1203227, DOI 10.1215/S0012-7094-93-06913-X
Z. Zhao, Positive solutions of nonlinear second order ordinary differential equations, Proc. Amer. Math. Soc. 121 (1994), no. 2, 465–469. MR 1185276, DOI 10.1090/S0002-9939-1994-1185276-5