# A Remark on Positive Solution for a Class of p,q-Laplacian Nonlinear System with Sign-changing Weight and Combined Nonlinear Effects

Volume 3, Issue 2, pp 126--134
• 2144 Views

### Authors

S. H. Rasouli - Department of Mathematics, Faculty of Basic Sciences, Babol Noshirvani University of Technology, Babol, Iran Z. Halimi - Department of Mathematics, Islamic Azad University Ghaemshahr branch, Iran Z. Mashhadban - Department of Mathematics, Islamic Azad University Ghaemshahr branch, Iran

### Abstract

In this article, we study the existence of positive solution for a class of (p; q)- Laplacian system $\begin{cases} -\Delta_{p}u=\lambda a(x)f(u)h(v),\,\,\,\,\, x\in \Omega,\\ -\Delta_{p}v=\lambda b(x)g(u)k(v),\,\,\,\,\, x\in \Omega,\\ u=v=0,\,\,\,\,\, x\in \partial \Omega. \end{cases}$ where $\Delta_p$ denotes the p-Laplacian operator defined by $\Delta_pz=div(|\nabla z|^{p-2} \nabla z), p>1,\Omega>0$ is a parameter and $\Omega$ is a bounded domain in $R^N(N > 1)$ with smooth boundary $\partial \Omega$. Here $a(x)$ and $b(x)$ are $C^1$ sign-changing functions that maybe negative near the boundary and $f, g, k, h$ are $C^1$ nondecreasing functions such that $f; g; h; k : [0,\infty)\rightarrow [0,\infty) ; f(s), k(s), h(s), g(s) > 0 ; s > 0$ and $\lim_{x\rightarrow \infty}\frac{h(A(g(x))^{\frac{1}{q-1}})(f(x))^{p-1}}{x^{p-1}}=0$ for every $A > 0$. We discuss the existence of positive solution when $h, k, f, g, a(x)$ and $b(x)$ satisfy certain additional conditions. We use the method of sub-super solutions to establish our results.

### Share and Cite

##### ISRP Style

S. H. Rasouli, Z. Halimi, Z. Mashhadban, A Remark on Positive Solution for a Class of p,q-Laplacian Nonlinear System with Sign-changing Weight and Combined Nonlinear Effects, Journal of Mathematics and Computer Science, 3 (2011), no. 2, 126--134

##### AMA Style

Rasouli S. H., Halimi Z., Mashhadban Z., A Remark on Positive Solution for a Class of p,q-Laplacian Nonlinear System with Sign-changing Weight and Combined Nonlinear Effects. J Math Comput SCI-JM. (2011); 3(2):126--134

##### Chicago/Turabian Style

Rasouli, S. H., Halimi, Z., Mashhadban, Z.. "A Remark on Positive Solution for a Class of p,q-Laplacian Nonlinear System with Sign-changing Weight and Combined Nonlinear Effects." Journal of Mathematics and Computer Science, 3, no. 2 (2011): 126--134

### Keywords

• (p،q)- Laplacian system
• Sign-changing weight.

•  35J48
•  35B09

### References

• [1] C. Atkinson, K. El-Ali, Some boundary value problems for the Bingham model, J. Non-Newtonian Fluid Mech., 41 (1992), 339--363

• [2] J. F. Escobar, Uniqueness theorems on conformal deformations of metrics, Sobolev inequalities, and an eigenvalue estimate, Comm. Pure Appl. Math., 43 (1990), 857--883

• [3] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126--150

• [4] G. S. Ladde, V. Lakshmikantham, A. S. Vatsale, Existence of coupled quase-solutions of systems of nonlinear elliptic boundary value problems, Nonlinear Anal., 8 (1984), 501--515

• [5] N. Dancer, Competing species systems with diffusion and large interaction, Rendiconti del Seminario Matematico e Fisico di Milano (Milan Journal of Mathematics), 65 (1995), 23--33

• [6] J. Ali, R. Shivaji, An existence result for a semipositone problem with a sign-changing weight, Abstr. Appl. Anal., 2006 (2006), 5 pages

• [7] M. Chhetri, S. oruganti, R. Shivaji, Existence results for a class of p-Laplacian problems with sign-changing weiht, Diff. Int. Equs., 18 (2005), 991--996

• [8] R. Dalmasso, Existence and uniqueness of positive solutions of semilinear elliptic systems, Nonlinear Analysis: Theory, Methods & Applications, 39 (2000), 559--568

• [9] J. Ali, R. Shivaji, M. Ramaswamy, Multiple positive solutions for a class of elliptic systems with combined nonlinear effects, Differential and Integral Equations, 19 (2006), 669--680

• [10] D. D. Hai, R. Shivaji, An existence result on positive solutions for a class of semilinear elliptic systems, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 134 (2004), 137--141

• [11] D. D. Hai, R. Shivaji, An existence result on positive solutions for a class of p-Laplacian systems, Nonlinear Analysis: Theory, Methods & Applications, 56 (2004), 1007--1010

• [12] A. Canada, P. Drabek, J. L. Gamez, Existence of positive solutions for some problems with nonlinear diffusion, Trans. Amer. Math. Soc., 349 (1997), 4231--4249

• [13] P. Drabek, J. Hernandez, Existence and uniqueness of positive solutions for some quasilinear elliptic problem, Nonlinear Anal., 44 (2001), 189--204

• [14] A. Ambrosetti, J. G. Azorero, I. Peral, Existence and multiplicity results for some nonlinear elliptic equations: a survey, Rend. Mat. Appl., 20 (2000), 167--198

• [15] C. O. Alves, D. G. De Figueiredo, Nonvariational elliptic systems, Discr. Contin. Dyn. Systems-A, 8 (2002), 289--302

• [16] G. A. Afrouzi, S. H. Rasouli, A remark on the existence of multiple solutions to a multiparameter nonlinear elliptic system, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 445--455

• [17] G. A. Afrouzi, S. H. Rasouli, A remark on the linearized stability of positive solutions for systems involving the p-Laplacian, Positivity, 11 (2007), 351--356

• [18] A. Djellit, S. Tas, On some nonlinear elliptic systems, Nonlinear Analysis: Theory, Methods & Applications, 59 (2004), 695--706

• [19] D. D. Hai, Uniqueness of positive solutions for a class of semilinear elliptic systems, Nonlinear Analysis: Theory, Methods & Applications, 52 (2003), 596--603