Properties of unique positive solution for a class of nonlocal semilinear elliptic equation
Keywords
Nonlocal elliptic equation, positive solution, existence and uniqueness, monotone operator, mixed monotone operatorAbstract
We study a class of nonlocal elliptic equations $$ -M\bigg(\int_{\Omega}|u|^{\gamma}dx\bigg)\Delta u=\lambda f(x,u) $$ with the Dirichlet boundary conditions in bounded domain. Under suitable assumptions on $M$ and the nonlinear term $f$, the existence and new properties of a unique positive solutions are obtained via a monotone operator method and a mixed monotone operator method.References
H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620–709.
A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.
C.O. Alves, F.J.S.A Corrêa and T.F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005), 85–93.
C.O. Alves, F.J.S.A Corrêa, T.F. Ma and D.P. Covei, Existence of solution for a class of nonlocal elliptic problem via sub-supersolution method, Nonlinear Anal. Real World Appl. 23 (2015), 1–8.
C.Y.Chen, Y.C. Kuo and T.F. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations 250 (2011), 1876–1908.
F.J.S.A. Corrêa, On positive solutions of nonlocal and nonvariational elliptic problems, Nonlinear Anal. 59 (2004), 1147–1155.
F.J.S.A. Corrêa and D.C.D.M. Filho, On a class of nonlocal elliptic problems via Galerkin method, J. Math. Anal. Appl. 310 (2005), 177–187.
F.J.S.A. Corrêa, D.C.D.M. Filho, S.D.B. Menezes and J. Ferreira, On a class of problems involving a nonlocal operator, Appl. Math. Comput. 147 (2004), 475–489.
D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, 1988.
Y.Y. Lan and C.L. Tang, Existence of solutions to a class of semilinear elliptic equations involving general subcritical growth, Proc. Roy. Soc. Edinburgh 144 (2014), 809–818.
Y.H. Li, F.Y. Li and J.P. Shi, Existence of a posit solutions to Kirchhoff type problems without compactness conditons, J. Differential Equations 253 (2012), 2285–2294.
T.F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal. 63 (2005), 1967–1977.
O.H. Miyagaki and M.A.S. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations 245 (2008), 3628–3638.
B.Q. Yan and D.C. Wang, The multiplicity of positive solutions for a class of nonlocal elliptic problem, J. Math. Anal. Appl. 442 (2016), 72–102.
C.B. Zhai and F. Wang, Properties of positive solutions for the operator equation Ax = λx and applications to fractional differential equations with integral boundary conditions, Adv. Differential Equations 2015 (2015), 1–10.
C.B. Zhai, F. Wang and L.L. Zhang, New fixed point theorems for mixed monotone operators and local existence-uniqueness of positive solutions for nonlinear boundary value problems, J. Math. Anal. Appl. 382 (2011), 594–614.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
