On a p-Laplacian system and a generalization of the Landesman-Lazer type condition

Document Type : Research Paper


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.


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.


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 RN, 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 RN, 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. Roumanie57(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.