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Superlinear Equations Involving Nonlinearities Limited by Asymptotically Homogeneous Functions
Journal of Inequalities and Applications volume 2007, Article number: 058363 (2007)
We obtain a solution of the quasilinear equation in,, on. Here the nonlinearity is superlinear at zero, and it is located near infinity between two functions that belong to a class of functions where the Ambrosetti-Rabinowitz condition is satisfied. More precisely, we consider the class of functions that are asymptotically homogeneous of index.
Díaz JI: Nonlinear Partial Differential Equations and Free Boundaries. Vol. I. Elliptic Equations, Research Notes in Mathematics. Volume 106. Pitman, Boston, Mass, USA; 1985:vii+323.
Ambrosetti A, Rabinowitz PH: Dual variational methods in critical point theory and applications. Journal of Functional Analysis 1973,14(4):349–381. 10.1016/0022-1236(73)90051-7
de Figueiredo DG, Gossez J-P, Ubilla P: Local superlinearity and sublinearity for indefinite semilinear elliptic problems. Journal of Functional Analysis 2003,199(2):452–467. 10.1016/S0022-1236(02)00060-5
Gidas B, Spruck J: Global and local behavior of positive solutions of nonlinear elliptic equations. Communications on Pure and Applied Mathematics 1981,34(4):525–598. 10.1002/cpa.3160340406
Azizieh C, Clément P: A priori estimates and continuation methods for positive solutions of-Laplace equations. Journal of Differential Equations 2002,179(1):213–245. 10.1006/jdeq.2001.4029
de Figueiredo DG, Yang J: On a semilinear elliptic problem without (PS) condition. Journal of Differential Equations 2003,187(2):412–428. 10.1016/S0022-0396(02)00055-4
Ruiz D: A priori estimates and existence of positive solutions for strongly nonlinear problems. Journal of Differential Equations 2004,199(1):96–114. 10.1016/j.jde.2003.10.021
García-Huidobro M, Manásevich R, Ubilla P: Existence of positive solutions for some Dirichlet problems with an asymptotically homogeneous operator. Electronic Journal of Differential Equations 1995,1995(10):1–22.
Resnick SI: Extreme Values, Regular Variation, and Point Processes, Applied Probability. A Series of the Applied Probability Trust. Volume 4. Springer, New York, NY, USA; 1987:xii+320.
Seneta E: Regularly Varying Functions, Lecture Notes in Mathematics. Volume 508. Springer, Berlin, Germany; 1976:v+112.
Trudinger NS: On Harnack type inequalities and their application to quasilinear elliptic equations. Communications on Pure and Applied Mathematics 1967,20(4):721–747. 10.1002/cpa.3160200406
Serrin J, Zou H: Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta Mathematica 2002,189(1):79–142. 10.1007/BF02392645
Lieberman GM: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Analysis 1988,12(11):1203–1219. 10.1016/0362-546X(88)90053-3
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Lorca, S., Souto, M.A. & Ubilla, P. Superlinear Equations Involving Nonlinearities Limited by Asymptotically Homogeneous Functions. J Inequal Appl 2007, 058363 (2007). https://doi.org/10.1155/2007/58363
- Homogeneous Function
- Quasilinear Equation
- Nonlinearity Limited
- Asymptotically Homogeneous
- Equation Involve