ON THE EXISTENCE OF POSITIVE SOLUTION FOR A CLASS OF NONLINEAR ELLIPTIC SYSTEM WITH MULTIPLE PARAMETERS AND SINGULAR WEIGHTS
2
조회
0
다운로드
2
인용
0
참고문헌
키워드
초록
This study concerns the existence of positive solution for the following nonlinear system $$ \left\{\begin{array}{ll} -div(|x|^{-ap}\,|\nabla u|^{p-2}\,\nabla u) = |x|^{-(a+1)p+c_{1}}\,(\alpha_{1} f(v)+\beta_{1} h(u)), x\in \Omega,\\ -div(|x|^{-bq}\,|\nabla v|^{q-2}\,\nabla v) = |x|^{-(b+1)q+c_{2}}\,(\alpha_{2} g(u)+\beta_{2} k(v)), x\in \Omega,\\ u = v = 0, x\in\partial \Omega, \end{array}\right. $$ where $\Omega$ is a bounded smooth domain of $\mathbb{R}^N$ with $0\in \Omega,$ $1<p,q<N,$ $0\leq a<\frac{N-p}{p}$, $0\leq b<\frac{N-q}{q}$ and $c_{1},c_{2},\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}$ are positive parameters. Here $f,g,h,k:[0,\infty)\to [0,\infty)$ are nondecresing continuous functions and \begin{eqnarray*} \lim_{s\to\infty}\frac{f(A g(s)^{\frac{1}{q-1}})}{s^{p-1}} = 0 \end{eqnarray*} for every $A>0.$ We discuss the existence of positive solution when $f$, $g$,$h$ and $k$ satisfy certain additional conditions. We use the method of sub-super solutions to establish our results.
Abstract
This study concerns the existence of positive solution for the following nonlinear system $$ \left\{\begin{array}{ll} -div(|x|^{-ap}\,|\nabla u|^{p-2}\,\nabla u) = |x|^{-(a+1)p+c_{1}}\,(\alpha_{1} f(v)+\beta_{1} h(u)), x\in \Omega,\\ -div(|x|^{-bq}\,|\nabla v|^{q-2}\,\nabla v) = |x|^{-(b+1)q+c_{2}}\,(\alpha_{2} g(u)+\beta_{2} k(v)), x\in \Omega,\\ u = v = 0, x\in\partial \Omega, \end{array}\right. $$ where $\Omega$ is a bounded smooth domain of $\mathbb{R}^N$ with $0\in \Omega,$ $1<p,q<N,$ $0\leq a<\frac{N-p}{p}$, $0\leq b<\frac{N-q}{q}$ and $c_{1},c_{2},\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}$ are positive parameters. Here $f,g,h,k:[0,\infty)\to [0,\infty)$ are nondecresing continuous functions and \begin{eqnarray*} \lim_{s\to\infty}\frac{f(A g(s)^{\frac{1}{q-1}})}{s^{p-1}} = 0 \end{eqnarray*} for every $A>0.$ We discuss the existence of positive solution when $f$, $g$,$h$ and $k$ satisfy certain additional conditions. We use the method of sub-super solutions to establish our results.
참고문헌 (25)
- G. A. Afrouzi, "A remark on the existence of multiple solutions to a multiparameter nonlinear elliptic system", Nonlinear Anal, 71(1-2), pp.445~455 (2009)
- G. A. Afrouzi, "A remark on the linearized stability of positive solutions for systems involving the p-Laplacian", Positivity, 11(2), pp.351~356 (2007)
- J. Ali, "Positive solutions for a class of p-laplacian systems with multiple parameters", J. Math. Anal. Appl, 335, pp.1013~1019 (2007)
- J. Ali, "An existence result for a semipositone problem with a sign-changing weight", Abstr. Appl. Anal, 2006, pp.5~ (2006)
- J. Ali, "Multiple positive solutions for classes of elliptic systems with combined nonlinear effects", Differential Integral Equations, 19(6), pp.669~680 (2006)
- C. O. Alves, "Nonvariational elliptic systems", Discrete Contin. Dyn. Syst, 8(2), pp.289~302 (2002)
- A. Ambrosetti, "Existence and multiplicity results for some nonlinear elliptic equations: a survey", Rend. Mat. Appl, 20, pp.167~198 (2000)
- C. Atkinson, "Some boundary value problems for the Bingham model", J. Non-Newtonian Fluid Mech, 41, pp.339~363 (1992)
- H. Bueno, "Existence and multiplicity of positive solutions for the p-Laplacian with nonlocal coefficient", J. Math. Anal. Appl, 343(1), pp.151~158 (2008)
- L. Caffarelli, "First order interpolation inequalities with weights", Compositio Math, 53(3), pp.259~275 (1984)
- A. Canada, "Existence of positive solutions for some problems with nonlinear diffusion", Trans. Amer. Math. Soc, 349(10), pp.4231~4249 (1997)
- M. Chhetri, "Existence results for a class of p-Laplacian problems with sign-changing weight", Differential Integral Equations, 18(9), pp.991~996 (2005)
- F. Cstea, "no. 5, Ser. A : TheoryMethods", Nonlinear Anal, 43(5), pp.623~636 (2001)
- R. Dalmasso, "Existence and uniqueness of positive solutions of semilinear elliptic systems", Nonlinear Anal, 39(5), pp.559~568 (2000)
- E. N. Dancer, "Competing species systems with diffusion and large interaction", Rend. Sem. Mat. Fis. Milano, 65, pp.23~33 (1995)
- P. Drabek, "Existence and uniqueness of positive solutions for some quasilinear elliptic problem", Nonlinear Anal, 44(2), pp.189~204 (2001)
- J. F. Escobar, "Uniqueness theorems on conformal deformations of metrics, Sobolev inequalities, and an eigenvalue estimate", Comm. Pure Appl. Math, 43(7), pp.857~883 (1990)
- F. Fang, "Nontrivial solutions of superlinear p-Laplacian equations", J. Math. Anal. Appl, 351(1), pp.138~146 (2009)
- D. D. Hai, "An existence result on positive solutions for a class of semi- linear elliptic systems", Proc. Roy. Soc. Edinburgh Sect. A, 134(1), pp.137~141 (2004)
- D. D. Hai, "An existence result on positive solutions for a class of p-Laplacian systems", Nonlinear Anal, 56, pp.1007~1010 (2004)
- G. S. Ladde, "Existence of coupled quasisolutions of systems of nonlinear elliptic boundary value problems", Nonlinear Anal, 8(5), pp.501~515 (1984)
- O. H. Miyagaki, "On positive solutions for a class of singular quasi- linear elliptic systems", J. Math. Anal. Appl, 334(2), pp.818~833 (2007)
- P. Tolksdorf, "Regularity for a more general class of quasilinear elliptic equations", J. Differential Equations, 51(1), pp.126~150 (1984)
- B. Xuan, "The eigenvalue problem for a singular quasilinear elliptic equation, Electron", J. Differential Equations, 2004(16), pp.11~ (2004)
- B. Xuan, "The solvability of quasilinear Brezis-Nirenberg-type problems with singular weights", Nonlinear Anal, 62(4), pp.703~725 (2005)