Nonhomogeneous elliptic Kirchhoff equations of $p$-Laplacian type

A.  Benaissa; A.  Matallah

A. Benaissa Laboratory of Analysis and Control of PDEs, Djillali Liabes Univ., Sidi Bel Abbes, Algeria
A. Matallah Ecole Pr´eparatoire en Sciences Economiques, Commerciales et Sciences de Gestion, Tlemcen, Algeria

Keywords: Kirchhoff equations, laplacian type

Abstract

UDC 517.9

We use variational methods to study the existence and multiplicity of solutions for an nonhomogeneous $p$-Kirchhoff equation involving the critical Sobolev exponent.

References

Alves, C. O.; Corrêa, F. J. S. A.; Ma, T. F. Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49 (2005), no. 1, 85--93. doi: 10.1016/j.camwa.2005.01.008

Ambrosetti, Antonio; Rabinowitz, Paul H. Dual variational methods in critical point theory and applications. J. Functional Analysis 14 (1973), 349--381. doi: 10.1016/0022-1236(73)90051-7

Brézis, Haïm; Lieb, Elliott. A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 88 (1983), no. 3, 486--490. doi: 10.2307/2044999

Chen, Caisheng; Huang, Jincheng; Liu, Lihua. Multiple solutions to the nonhomogeneous $p$-Kirchhoff elliptic equation with concave-convex nonlinearities. Appl. Math. Lett. 26 (2013), no. 7, 754--759. doi: 10.1016/j.aml.2013.02.011

Chen, Shang-Jie; Li, Lin. Multiple solutions for the nonhomogeneous Kirchhoff equation on $bold{R}^N$. Nonlinear Anal. Real World Appl. 14 (2013), no. 3, 1477--1486. doi: 10.1016/j.nonrwa.2012.10.010

Corrêa, Francisco Júlio S. A.; Figueiredo, Giovany M. On an elliptic equation of $p$-Kirchhoff type via variational methods. Bull. Austral. Math. Soc. 74 (2006), no. 2, 263--277. doi: 10.1017/S000497270003570X

Ekeland, I. On the variational principle. J. Math. Anal. Appl. 47 (1974), 324--353. doi: 10.1016/0022-247X(74)90025-0

Wang, Li. On a quasilinear Schrödinger-Kirchhoff-type equation with radial potentials. Nonlinear Anal. 83 (2013), 58--68. doi: 10.1016/j.na.2012.12.012

Sun, Juan; Liu, Shibo. Nontrivial solutions of Kirchhoff type problems. Appl. Math. Lett. 25 (2012), no. 3, 500--504. doi: 10.1016/j.aml.2011.09.045

Talenti, Giorgio. Best constant in Sobolev inequality. Ann. Mat. Pura Appl. (4) 110 (1976), 353--372. doi: 10.1007/BF02418013

Wu, Xian. Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in ${bf R}^N$. Nonlinear Anal. Real World Appl. 12 (2011), no. 2, 1278--1287. doi: 10.1016/j.nonrwa.2010.09.023