ON THE EXISTENCE OF POSITIVE SOLUTION FOR A CLASS OF NONLINEAR ELLIPTIC SYSTEM WITH MULTIPLE PARAMETERS AND SINGULAR WEIGHTS

Title & Authors
ON THE EXISTENCE OF POSITIVE SOLUTION FOR A CLASS OF NONLINEAR ELLIPTIC SYSTEM WITH MULTIPLE PARAMETERS AND SINGULAR WEIGHTS
Rasouli, S.H.;

Abstract
This study concerns the existence of positive solution for the following nonlinear system $\small{\{-div(|x|^{-ap}|{\nabla}u|^{p-2}{\nabla}u)=|x|^{-(a+1)p+c_1}({\alpha}_1f(v)+{\beta}_1h(u)),x{\in}{\Omega},\\-div(|x|^{-bq}|{\nabla}v|q^{-2}{\nabla}v)=|x|^{-(b+1)q+c_2}({\alpha}_2g(u)+{\beta}_2k(v)),x{\in}{\Omega},\\u=v=0,x{\in}{\partial}{\Omega}}$, where $\small{{\Omega}}$ is a bounded smooth domain of $\small{\mathbb{R}^N}$ with $\small{0{\in}{\Omega}}$, 1 < $\small{p,q}$ < N, $\small{0{{\leq}}a}$<$\small{\frac{N-p}{p}}$, $\small{0{{\leq}}b}$<$\small{\frac{N-q}{q}}$ and $\small{c_1}$, $\small{c_2}$, $\small{{\alpha}_1}$, $\small{{\alpha}_2}$, $\small{{\beta}_1}$, $\small{{\beta}_2}$ are positive parameters. Here $\small{f,g,h,k}$ : $\small{[0,{\infty}){\rightarrow}[0,{\infty})}$ are nondecresing continuous functions and $\small{\lim_{s{\rightarrow}{\infty}}\frac{f(Ag(s)^{\frac{1}{q-1}})}{s^{p-1}}=0}$ for every A > 0. We discuss the existence of positive solution when $\small{f,g,h}$ and $\small{k}$ satisfy certain additional conditions. We use the method of sub-super solutions to establish our results.
Keywords
singular weights;nonlinear elliptic system;multiple parameters;
Language
English
Cited by
1.
Positive solutions of singular elliptic systems with multiple parameters and Caffarelli–Kohn–Nirenberg exponents, Periodica Mathematica Hungarica, 2015, 70, 2, 145
References
1.
G. A. Afrouzi and S. H. Rasouli, A remark on the existence of multiple solutions to a multiparameter nonlinear elliptic system, Nonlinear Anal. 71 (2009), no. 1-2, 445-455.

2.
G. A. Afrouzi and S. H. Rasouli, A remark on the linearized stability of positive solutions for systems involving the p-Laplacian, Positivity. 11 (2007), no. 2, 351-356.

3.
J. Ali and R. Shivaji, Positive solutions for a class of p-laplacian systems with multiple parameters, J. Math. Anal. Appl. 335 (2007), 1013-1019.

4.
J. Ali and R. Shivaji, An existence result for a semipositone problem with a sign-changing weight, Abstr. Appl. Anal. 2006 (2006), Art. ID 70692, 5 pp.

5.
J. Ali, R. Shivaji, and M. Ramaswamy, Multiple positive solutions for classes of elliptic systems with combined nonlinear effects, Differential Integral Equations 19 (2006), no. 6, 669-680.

6.
C. O. Alves and D. G. de Figueiredo, Nonvariational elliptic systems, Discrete Contin. Dyn. Syst. 8 (2002), no. 2, 289-302.

7.
A. Ambrosetti, J. G. Azorero, and I. Peral, Existence and multiplicity results for some nonlinear elliptic equations: a survey, Rend. Mat. Appl. (7) 20 (2000), 167-198.

8.
C. Atkinson and K. El Kalli, Some boundary value problems for the Bingham model, J. Non-Newtonian Fluid Mech. 41 (1992), 339-363.

9.
H. Bueno, G. Ercole, W. Ferreira, and A. Zumpano, Existence and multiplicity of positive solutions for the p-Laplacian with nonlocal coefficient, J. Math. Anal. Appl. 343 (2008), no. 1, 151-158.

10.
L. Caffarelli, R. Kohn, and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math. 53 (1984), no. 3, 259-275.

11.
A. Canada, P. Drabek, and J. L. Gamez, Existence of positive solutions for some prob- lems with nonlinear diffusion, Trans. Amer. Math. Soc. 349 (1997), no. 10, 4231-4249.

12.
M. Chhetri, S. Oruganti, and R. Shivaji, Existence results for a class of p-Laplacian problems with sign-changing weight, Differential Integral Equations 18 (2005), no. 9, 991-996.

13.
F. Cstea, D. Motreanu, and V. Radulescu, Weak solutions of quasilinear problems with nonlinear boundary condition, Nonlinear Anal. 43 (2001), no. 5, Ser. A: TheoryMethods, 623-636.

14.
R. Dalmasso, Existence and uniqueness of positive solutions of semilinear elliptic systems, Nonlinear Anal. 39 (2000), no. 5, Ser. A: Theory Methods, 559-568.

15.
E. N. Dancer, Competing species systems with diffusion and large interaction, Rend. Sem. Mat. Fis. Milano 65 (1995), 23-33.

16.
P. Drabek and J. Hernandez, Existence and uniqueness of positive solutions for some quasilinear elliptic problem, Nonlinear Anal. 44 (2001), no. 2, Ser. A: Theory Methods, 189-204.

17.
J. F. Escobar, Uniqueness theorems on conformal deformations of metrics, Sobolev inequalities, and an eigenvalue estimate, Comm. Pure Appl. Math. 43 (1990), no. 7, 857-883.

18.
F. Fang and S. Liu, Nontrivial solutions of superlinear p-Laplacian equations, J. Math. Anal. Appl. 351 (2009), no. 1, 138-146.

19.
D. D. Hai and R. Shivaji, An existence result on positive solutions for a class of semi-linear elliptic systems, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), no. 1, 137-141.

20.
D. D. Hai and R. Shivaji, An existence result on positive solutions for a class of p-Laplacian systems, Nonlinear Anal. 56 (2004), 1007-1010.

21.
G. S. Ladde, V. Lakshmikantham, and A. S. Vatsale, Existence of coupled quasisolutions of systems of nonlinear elliptic boundary value problems, Nonlinear Anal. 8 (1984), no. 5, 501-515.

22.
O. H. Miyagaki and R. S. Rodrigues, On positive solutions for a class of singular quasi- linear elliptic systems, J. Math. Anal. Appl. 334 (2007), no. 2, 818-833.

23.
P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126-150.

24.
B. Xuan, The eigenvalue problem for a singular quasilinear elliptic equation, Electron. J. Differential Equations 2004 (2004), no. 16, 11 pp.

25.
B. Xuan, The solvability of quasilinear Brezis-Nirenberg-type problems with singular weights, Nonlinear Anal. 62 (2005), no. 4, 703-725.