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EXISTENCE OF WEAK NON-NEGATIVE SOLUTIONS FOR A CLASS OF NONUNIFORMLY BOUNDARY VALUE PROBLEM

Hoang Quoc Toan|Trinh Thi Minh Hang
대한수학회 (The Korean Mathematical Society)|2012.07|대한수학회보 (Bulletin of the KMS)|제49권 제4호|pp. 737-748

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참고문헌

키워드

mountain pass theoremthe weakly continuously differentiable functional
주제분류수학

초록

The goal of this paper is to study the existence of non-trivial non-negative weak solution for the nonlinear elliptic equation: $$-div(h(x){\nabla}u)=f(x,u)\;in\;{\Omega}$$ with Dirichlet boundary condition in a bounded domain ${\Omega}{\subset}\mathbb{R}^N$, $N{\geq}3$, where $h(x){\in}L^1_{loc}({\Omega})$, $f(x,s)$ has asymptotically linear behavior. The solutions will be obtained in a subspace of the space $H^1_0({\Omega})$ and the proofs rely essentially on a variation of the mountain pass theorem in [12].

Abstract

The goal of this paper is to study the existence of non-trivial non-negative weak solution for the nonlinear elliptic equation: $$-div(h(x){\nabla}u)=f(x,u)\;in\;{\Omega}$$ with Dirichlet boundary condition in a bounded domain ${\Omega}{\subset}\mathbb{R}^N$, $N{\geq}3$, where $h(x){\in}L^1_{loc}({\Omega})$, $f(x,s)$ has asymptotically linear behavior. The solutions will be obtained in a subspace of the space $H^1_0({\Omega})$ and the proofs rely essentially on a variation of the mountain pass theorem in [12].

참고문헌 (24)

  1. M. Alif, "On a p-Laplace Neumann problem with asymptotically asymmetric perturbations", Nonlinear Anal, 51(2), pp.369~389 (2002)
  2. H. Amann, "Positive solutions of convex nonlinear eigenvalue problems", Indiana Univ. Math. J, 25(3), pp.259~270 (1976)
  3. A. Ambrosetti, "Positive solutions of asymptotically linear elliptic eigenvalue problems", J. Math. Anal. Appl, 73(2), pp.411~422 (1980)
  4. G. Anello, "Existence of infinitely many weak solutions for a Neumann problem", Nonlinear Anal, 57(2), pp.199~209 (2004)
  5. H. Berestycki, "Variational methods for indefinite superlinear homogeneous elliptic problems", NoDEA, 2(4), pp.553~572 (1995)
  6. H. Berestycki, "Superlinear indefinite elliptic problems and nonlinear Liouville theorems", Topol. Methods Nonlinear Anal, 4(1), pp.59~78 (1994)
  7. K. J. Brown, "On the bifurcation of radially symmetric steady-state solutions arising in population genetics", SIAM J. Math. Anal, 22(2), pp.400~413 (1991)
  8. K. C. Chang, "Solutions of asymptotically linear operator equations via Morse theory", Comm. Pure Appl. Math, 34(5), pp.693~712 (1981)
  9. N. T. Chung, "Existence result for nonuniformly degenerate semilinear elliptic systems in RN", Glassgow. Math. J, 51, pp.561~570 (2009)
  10. D. G. Costa, "Variational elliptic problems which are nonquadratic at infinity", Nonlinear Anal, 23(11), pp.1401~1412 (1994)
  11. T.-L. Dinu, "Subcritical perturbations of resonant linear problem with sign-changing po- tential", Electron. J. Differential Equations, 2005(117), pp.15~ (2005)
  12. D. M. Duc, "Nonlinear singular elliptic equation", J. London Math. Soc. (2), 40(3), pp.420~440 (1989)
  13. W. H. Fleming, "A selection-Migration model in population genetics", J. Math. Biol, 2(3), pp.219~233 (1975)
  14. D. Gilbarg, "Elliptic Partial Differential Equations of Second Order" (1998), Springer Verlag
  15. P. Hess, "Multiple solutions of asymptotically linear elliptic boundary value problems, Proc. Equadiff IV, Prague 1977", Lecture notes in Math, 703 (1979), Springer Verlag
  16. P. Hess, "On some linear and nonlinear eigenvalue problems with an indef- inite weight function", Comm. Partial Differential Equations, 5(10), pp.999~1030 (1980)
  17. M. Lucia, "A Dirichlet problem with asymptotically linear and changing sign nonlinearity", Rev. Mat. Complut, 16(2), pp.465~481 (2003)
  18. B. Ricceri, "Infinity many solutions of the Neumann problem for elliptic equations in- volving the p-Laplacian", Bull. London Math. Soc, 33, pp.331~340 (2001)
  19. C. L. Tang, "Solvability of Neumann problem for elliptic equations at resonnance", Nonlinear Anal, 44, pp.325~335 (2001)
  20. C. L. Tang, "Some existence theorems for the sublinear Neumann boundary value problem", Nonlinear Anal, 48(7), pp.1003~1011 (2002)
  21. H. Q. Toan, "Multiplicity of weak solutions for a class of nonuniformly nonlinear elliptic equation of p-Laplacian type", Nonlinear Anal, 70, pp.1536~1546 (2009)
  22. H. Q. Toan, "Existence of weak solutions for a class of nonuniformly nonlinear elliptic equations in unbounded domains", Nonlinear Anal, 70(11), pp.3987~3996 (2009)
  23. H. S. Zhou, "Positive solutions for a semilinear elliptic equation which is almost linear at infinity", Z. Angew. Math. Phys, 49(6), pp.896~906 (1998)
  24. H. S. Zhou, "An application of a mountain pass theorem", Acta Math. Sinica, 18(1), pp.27~36 (2002)

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