On the existence of positive solutions for semilinear elliptic equations with Neumann boundary conditions

SummaryWe give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in unbounded Lipschitz domainsD ⊂ ℝd(d≥3), having compact boundary, with nonlinear Neumann boundary conditions on the boundary ofD. For this we use an implicit probabilistic representation, Schauder's fixed point theorem, and a recently proved Sobolev inequality forW1,2(D). Special cases include equations arising from the study of pattern formation in various models in mathematical biology and from problems in geometry concerning the conformal deformation of metrics.

[1]  H. Amann.  On the Existence of Positive Solutions of Nonlinear Elliptic Boundary value Problems , 1971 .

[2]  F. W. Warner,et al.  Curvature Functions for Compact 2-Manifolds , 1974 .

[3]  M. Fukushima.  Dirichlet forms and Markov processes , 1980 .

[4]  K. Chung,et al.  Feynman-Kac Functional and the Schrödinger Equation , 1981 .

[5]  M. Fukushima.  On absolute continuity of multidimensional symmetrizable diffusions , 1982 .

[6]  K. Chung.  Lectures from Markov processes to Brownian motion , 1982 .

[7]  C. Kenig,et al.  An exterior Dirichlet problem with applications to some nonlinear equations arising in geometry , 1984 .

[8]  R. Durrett.  Brownian motion and martingales in analysis , 1984 .

[9]  Pei Hsu,et al.  Probabilistic approach to the neumann problem , 1985 .

[10]  M. Freidlin,et al.  Functional Integration and Partial Differential Equations. (AM-109), Volume 109 , 1985 .

[11]  J. Glover,et al.  Solving semilinear partial differential equations with probabilistic potential theory , 1985 .

[12]  M. Freidlin.  Functional Integration And Partial Differential Equations , 1985 .

[13]  W. Ni,et al.  On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type , 1986 .

[14]  P. Hsu.  On the Poisson Kernel for the Neumann Problem of Schrödinger Operators , 1987 .

[15]  R. Getoor,et al.  Revuz measures and time changes , 1988 .

[16]  L. Rogers.  GENERAL THEORY OF MARKOV PROCESSES , 1989 .

[17]  P. Fitzsimmons.  Time changes of symmetric Markov processes and a Feynman-Kac formula , 1989 .

[18]  Zhi-Ming Ma,et al.  Probabilistic Methods in Schrödinger Equations , 1990 .

[19]  V. Papanicolaou.  The probabilistic solution of the third boundary value problem for second order elliptic equations , 1990 .

[20]  Xu-jia Wang.  Neumann problems of semilinear elliptic equations involving critical Sobolev exponents , 1991 .

[21]  E. B. Dynkin,et al.  A probabilistic approach to one class of nonlinear differential equations , 1991 .

[22]  JosEi F. Escobar,et al.  Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary , 1992 .

[23]  José F. Escobar,et al.  The Yamabe problem on manifolds with boundary , 1992 .

[24]  F. Pacella,et al.  Interaction between the Geometry of the Boundary and Positive Solutions of a Semilinear Neumann Problem with Critical Nonlinearity , 1993 .

[25]  Z. Zhao.  On the existence of positive solutions of nonlinear elliptic equations—a probabilistic potential theory approach , 1993 .

[26]  Zhen-Qing Chen,et al.  A Sobolev Inequality and Neumann Heat Kernel Estimate for Unbounded Domains , 1994 .

[27]  Z. Zhao,et al.  On the existence of positive solutions of semilinear elliptic equations with Dirichlet boundary conditions , 1994 .

[28]  Kai Lai Chung,et al.  From Brownian Motion To Schrödinger's Equation , 1995 .