Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

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

  • Received : 2010.06.11
  • Published : 2011.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

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.

Keywords

References

  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

Cited by

  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