On a singular nonlinear elliptic boundary-value problem

Lazer, A. C.; McKenna, P. J.

doi:10.1090/S0002-9939-1991-1037213-9

On a singular nonlinear elliptic boundary-value problem
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by A. C. Lazer and P. J. McKenna

Proc. Amer. Math. Soc. 111 (1991), 721-730

DOI: https://doi.org/10.1090/S0002-9939-1991-1037213-9

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Abstract:

We consider the singular boundary-value problem $\Delta u + p(x){u^{ - \gamma }} = 0$ in $\Omega ,u|\partial \Omega = 0$, where $\gamma > 0$. Under the assumption $p(x) > 0$ and certain smoothness assumptions, we show that there exists a solution which is smooth on $\Omega$ and continuous on $\bar \Omega$.

References

M. G. Crandall, P. H. Rabinowitz, and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations 2 (1977), no. 2, 193–222. MR 427826, DOI 10.1080/03605307708820029
J. A. Gatica, Vladimir Oliker, and Paul Waltman, Singular nonlinear boundary value problems for second-order ordinary differential equations, J. Differential Equations 79 (1989), no. 1, 62–78. MR 997609, DOI 10.1016/0022-0396(89)90113-7
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 0473443
D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1971/72), 979–1000. MR 299921, DOI 10.1512/iumj.1972.21.21079
David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 567696

A nonlinear singular boundary value problem in the theory of pseudoplastic fluids

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C. A. Stuart, Existence theorems for a class of non-linear integral equations, Math. Z. 137 (1974), 49–66. MR 348416, DOI 10.1007/BF01213934
Steven D. Taliaferro, A nonlinear singular boundary value problem, Nonlinear Anal. 3 (1979), no. 6, 897–904. MR 548961, DOI 10.1016/0362-546X(79)90057-9