DOI QR코드

DOI QR Code

ON EXISTENCE OF WEAK SOLUTIONS OF NEUMANN PROBLEM FOR QUASILINEAR ELLIPTIC EQUATIONS INVOLVING p-LAPLACIAN IN AN UNBOUNDED DOMAIN

  • 투고 : 2010.06.11
  • 발행 : 2011.11.30

초록

In this paper we study the existence of non-trivial weak solutions of the Neumann problem for quasilinear elliptic equations in the form $$-div(h(x){\mid}{\nabla}u{\mid}^{p-2}{\nabla}u)+b(x){\mid}u{\mid}^{p-2}u=f(x,\;u),\;p{\geq}2$$ in an unbounded domain ${\Omega}{\subset}\mathbb{R}^N$, $N{\geq}3$, with sufficiently smooth bounded boundary ${\partial}{\Omega}$, where $h(x){\in}L_{loc}^1(\overline{\Omega})$, $\overline{\Omega}={\Omega}{\cup}{\partial}{\Omega}$, $h(x){\geq}1$ for all $x{\in}{\Omega}$. The proof of main results rely essentially on the arguments of variational method.

과제정보

연구 과제 주관 기관 : NAFOSTED

참고문헌

  1. M. Alif and P. Omari, On a p-Neumann problem with asymptotically asymmetric perturbations, Nonlinear Anal. 51 (2002), no. 3, 369-389. https://doi.org/10.1016/S0362-546X(01)00835-5
  2. G. Anello, Existence of infinitely many weak solutions for a Neumann problem, Nonlinear Anal. 57 (2004), no. 2, 199-209. https://doi.org/10.1016/j.na.2004.02.009
  3. P. A. Binding, P. Drabek, and Y. X. Huang, Existence of multiple solutions of critical quasilinear elliptic Neumann problems, Nonlinear Anal. 42 (2000), no. 4, 613-629. https://doi.org/10.1016/S0362-546X(99)00118-2
  4. G. Bonanno and P. Candito, Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian, Arch. Math. (Basel) 80 (2003), no. 4, 424-429. https://doi.org/10.1007/s00013-003-0479-8
  5. N. T. Chung and H. Q. Toan, Existence results for uniformly degenerate semilinear elliptic systems in $R^N$, Glassgow Mathematical Journal 51 (2009), 561-570. https://doi.org/10.1017/S0017089509005175
  6. D. M. Duc, Nonlinear singular elliptic equations, J. London Math. Soc. (2) 40 (1989), no. 3, 420-440. https://doi.org/10.1112/jlms/s2-40.3.420
  7. D. M. Duc and N. T. Vu, Nonuniformly elliptic equations of p-Laplacian type, Nonlinear Anal. 61 (2005), no. 8, 1483-1495. https://doi.org/10.1016/j.na.2005.02.049
  8. E. Giusti, Direct Methods in the Calculus of Variation World Scientific, New Jersey, 2003.
  9. M. Mihailescu, Existence and multiplicity of weak solution for a class of degenerate nonlinear elliptic equations, Boundary Value Problems 2006 (2006), Art ID 41295, 17pp.
  10. B. Ricceri, Infinitely many solutions of the Neumann problem for elliptic equations involving the p-Laplacian, Bull. London Math. Soc. 33 (2001), no. 3, 331-340. https://doi.org/10.1017/S0024609301008001
  11. M. Struwe, Variational Methods, Second Edition, Springer-Verlag, 2000.
  12. C. L. Tang, Solvability of Neumann problem for elliptic equations at resonance, Nonlinear Anal. 44 (2001), no. 3, 323-335. https://doi.org/10.1016/S0362-546X(99)00266-7
  13. C. L. Tang, Some existence theorems for the sublinear Neumann boundary value problem, Nonlinear Anal. 48 (2002), no. 7, 1003-1011. https://doi.org/10.1016/S0362-546X(00)00230-3
  14. H. Q. Toan and N. T. Chung, Existence of weak solutions for a class of nonuniformly nonlinear elliptic equations in unbounded domains, Nonlinear Anal. 70 (2009), no. 11, 3987-3996. https://doi.org/10.1016/j.na.2008.08.007
  15. H. Q. Toan and Q. A. Ngo, Multiplicity of weak solutions for a class of nonuniformly elliptic equations of p-Laplacian type, Nonlinear Anal. 70 (2009), no. 4, 1536-1546. https://doi.org/10.1016/j.na.2008.02.033
  16. X. Wu and K.-K. Tan, On existence and multiplicity of solutions of Neumann boundary value problems for quasi-linear elliptic equations, Nonlinear Anal. 65 (2006), no. 7, 1334-1347. https://doi.org/10.1016/j.na.2005.10.010

피인용 문헌

  1. ON A NEUMANN PROBLEM AT RESONANCE FOR NONUNIFORMLY SEMILINEAR ELLIPTIC SYSTEMS IN AN UNBOUNDED DOMAIN WITH NONLINEAR BOUNDARY CONDITION vol.51, pp.6, 2014, https://doi.org/10.4134/BKMS.2014.51.6.1669