Document Type : Research Paper

**Authors**

^{1}
Faculty of Information Technology, Le Quy Don Technical University, 236 Hoang Quoc Viet, Bac Tu Liem, Hanoi, Vietnam.

^{2}
Department of Mathematics, Hanoi University of Science 334 Nguyen Trai, Thanh Xuan, Ha Noi, Vietnam.

**Abstract**

This article shows the existence of weak solutions of a resonance problem for nonuniformly p-Laplacian system in a bounded domain in $\mathbb{R}^N$. Our arguments are based on the minimum principle and rely on a generalization of the Landesman-Lazer type condition.

**Keywords**

**Main Subjects**

G.A. Afrouzi, M. Mirzapour and Q. Zhang, Simplicity and stability of the first eigenvalue of a (*p*; *q*) Laplacian system, *Electron. J. Differential Equations ***2012 **(2012), no. 08, 6 pages.

A. Anane and J.P. Gossez, Strongly nonlinear elliptic problems near resonance: a variational approach, *Comm. Partial Differential Equation ***15 **(1990), no. 8, 1141--1159.

D. Arcoya and L. Orsina, Landesman-Lazer condition and quasilinear elliptic equations, *Nonlinear Anal. ***28 **(1997), no. 10, 1623--1632.

L. Boccando, P. Drabek and M. Kucera, Landesman-Lazer conditions for strongly non-linear boundary value problems, *Comment. Math. Univ. Carolin. ***30 **(1989), no. 3, 411--427.

N.T. Chung and H.Q. Toan, Existence result for nonuniformly degenerate semilinear elliptic systems in R*N*, *Glasgow Math. J. ***51 **(2009), no. 3, 561--570.

D.M. Duc, Nonlinear singular elliptic equation, *J. Lond. Math. Soc. ***40 **(1989) 420-440.

T.T.M. Hang and H.Q.Toan, On existence of weak solutions of Neumann problem for quasilinear elliptic equations involving p-Laplacian in an unbounded domain, *Bull. Korean Math. Soc. ***48 **(2011), no. 6, 1169--1182.

B.Q. Hung and H.Q. Toan, On existence of weak solutions for a p-Laplacian system at resonance, *Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. RACSAM ***110 **(2016), no. 1, 33--47.

D. Gilbarg and N.Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.

D.A. Kandilakis and M. Magiropoulos, A *p*-Laplacian system with resonance and non-linear boundary conditions on an unbounded domain, *Comment. Math. Univ. Carolin. ***48 **(2007), no. 1, 59--68.

M. Lucia, P. Magrone and Huan-Songzhou, A Dirichlet problem with asymptotically linear and changing sign nonlinearity, *Rev. Mat. Complut. ***16 **(2003), no. 2, 465--481.

Q.A. Ng^o and H.Q. Toan, Existence of solutions for a resonant problem under Landesman-Lazer condition, *Electronic J. Differential Equations ***2008 **(2008), no. 98, 10 pages.

Q.A. Ng^o and H.Q. Toan, Some Remarks on a class of Nonuniformly Elliptic Equations of p-Laplacian type, *Acta Appl. Math. ***106 **(2009), no. 2, 229--239.

Z.Q. Ou, C.L. Tang, Resonance problems for the p-Laplacian systems, *J. Math. Anal. Appl. ***345 **(2008), no. 1, 511--521.

N.M. Stavrakakis and N.B. Zographopoulos, Existence results for quasilinear elliptic systems in R*N*, *Electron. J. Differential Equations ***1999 **(1999), no. 39, 15 pages.

M. Struwe, Variational Methods, Springer-Verlag, 2nd edition, Berlin, Heidelberg, 2008.

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.

H.Q. Toan and B.Q. Hung, On a generalization of the Landesman-Lazer condition and Neumann problem for nonuniformly semilinear elliptic equations in an unbounded domain with nonlinear boundary condition, *Bull. Math. Soc. Sci. Math. Roumanie*, **57(105) **(2014), no. 3, 301--317.

P. Tomiczek, A generalization of the Landesman-Lazer condition, *Electron. J. Differential Equations ***2001 **(2001), no. 4, 11 pages.

N.B. Zographopoulos, *p*-Laplacian systems on resonance. *Appl. Anal. ***83 **(2004), no. 5, 509--519.

N.B. Zographopoulos, On a class of degenerate potential elliptic system, *Nonlinear Differential Equations Appl. ***11 **(2004), no. 2, 191--199.

September and October 2017

Pages 1511-1528

**Receive Date:**28 February 2016**Revise Date:**13 July 2016**Accept Date:**14 July 2016