## Duke Mathematical Journal

### On the existence of positive solutions of nonlinear elliptic equations—a probabilistic potential theory approach

Z. Zhao

#### Article information

Source
Duke Math. J. Volume 69, Number 2 (1993), 247-258.

Dates
First available in Project Euclid: 20 February 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1077293570

Digital Object Identifier
doi:10.1215/S0012-7094-93-06913-X

Mathematical Reviews number (MathSciNet)
MR1203227

Zentralblatt MATH identifier
0793.35032

#### Citation

Zhao, Z. On the existence of positive solutions of nonlinear elliptic equations—a probabilistic potential theory approach. Duke Math. J. 69 (1993), no. 2, 247--258. doi:10.1215/S0012-7094-93-06913-X. http://projecteuclid.org/euclid.dmj/1077293570.

#### References

• [1] R. Benguria, S. Lorca, and C. Yarur, Nonexistence results for solutions of semilinear elliptic equations, preprint.
• [2] K. L. Chung, Lectures from Markov Processes to Brownian Motion, Grundlehren Math. Wiss., vol. 249, Springer-Verlag, Berlin, 1982.
• [3] 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.
• [4] J. L. Doob, Classical Potential Theory and its Probabilistic Counterpart, Grundlehren Math. Wiss., vol. 262, Springer-Verlag, New York, 1984.
• [5] E. M. Harrell and B. Simon, Schrödinger operator methods in the study of a certain nonlinear P.D.E, Proc. Amer. Math. Soc. 88 (1983), no. 2, 376–377.
• [6] I. W. Herbst and Z. Zhao, Green's functions for the Schrödinger equation with short-range potentials, Duke Math. J. 59 (1989), no. 2, 475–519.
• [7] C. Kenig and W.-M. Ni, An exterior Dirichlet problem with applications to some nonlinear equations arising in geometry, Amer. J. Math. 106 (1984), no. 3, 689–702.
• [8] F.-H. Lin, On the elliptic equation $D\sb i[a\sb ij(x)D\sb jU]-k(x)U+K(x)U\sp p=0$, Proc. Amer. Math. Soc. 95 (1985), no. 2, 219–226.
• [9] W.-M. Ni, On the elliptic equation $\Delta u+K(x)u\sp(n+2)/(n-2)=0$, its generalizations, and applications in geometry, Indiana Univ. Math. J. 31 (1982), no. 4, 493–529.
• [10] S. Port and C. Stone, Brownian Motion and Classical Potential Theory, Probab. Math. Statist., Academic Press, New York, 1978.
• [11] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 447–526.
• [12] E. Zeidler, Nonlinear Functional Analysis and its Applications, Volume 1, Springer-Verlag, New York, 1986.
• [13] Z. Zhao, Subcriticality, positivity and gaugeability of the Schrödinger operator, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 513–517.