|
|
|
| Weighted and Resonance Quasilinear Elliptic Problems with Jumping Nonlinearities |
| Gao JIA1, Chun-yan DAI2, Jie CHEN3 |
1 College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China;
2 Shanghai Tianshan Second Middle School, Shanghai 200336, China;
3 Xiang Yifu Rong Second Middle School, Changsha 410000, China |
|
|
|
|
Abstract In this paper, the existence of a nontrivial solution for a class of quasilinear elliptic equations with a disturbance term in a weighted Sobolev space is proved. The proofs rely on Galerkin method, Brouwer's theorem and a new weighted compact Sobolev-type embedding theorem established by V.L. Shapiro.
|
|
Received: 22 March 2012
|
|
|
| Fund:Supported by the National Natural Science Foundation of China (No. 11171220)and Shanghai Leading Academic Discipline Project (XTKX2012). |
| About author:: 35H30;35J50;65L60 |
|
|
|
| [1] |
Berestycki, H., Figueiredo, De. Double resonance in semilinear elliptic problems. Communications in Partial Differential Equations, 6:91-120(1981)
|
| [2] |
Chang, K.C. Infinite dimensional Morse theorem and multiple solution problems. Birkhäuser, Boston, 1992
|
| [3] |
Dancer, E.N. On the Dirichlet problem for weakly non-linear elliptic partial differential equations. Proceedings of the Royal Society of Edinburgh, 76:283-300(1977)
|
| [4] |
Kufner, A. Some Applications of Weighted Sobolev Spaces. Teubner, California, 1987
|
| [5] |
Kesavan, S. Topics in Functional Analysis and Applications. John Wiley, New York, 1989
|
| [6] |
Landersman, E.M., Robinson, S.B., Rumbos, A. Multiple solutions of semilinear elliptic problems at resonance. Nonlinear Analysis, 24:1049-1059(1995)
|
| [7] |
Rabinowitz, P., Paul, H. Minmax method in critical point theory with applications to differential equations. American Mathematical Society, Vol. 65, Providence, R.I., 1986
|
| [8] |
Robinson, S.B., Rumbos, A., Shapiro, V.L. One-Sided resonance problems for quasilinear elliptic operators. Journal of Mathematical Analysis and Applications, 256:636-649(2001)
|
| [9] |
Rumbos, A.J. Shapiro, V.L. Jumping nonlinearities and weighted Sobolev spaces. Journal of Differential Equations. 214:326-357(2005)
|
| [10] |
Shapiro, V.L. Quasilinear ellipticity below the first eigenvalue. Proceedings of the American Mathematical Society, 129:1955-1962(2001)
|
| [11] |
Shapiro, V.L. Quasilinear ellipticity and the 1st eigenvalue. Communications in Partial Differential Equations, 16:1819-1855(1991)
|
| [12] |
Shapiro, V.L. Singular quasilinearity and higher eigenvalues. Memoirs of the American Mathematical Society, Vol.153(726), Providence, R.I., 2001
|
| [13] |
Turesson, B.O. Nonlinear Potential Theory and Weighted Sobolev Spaces. Springer Verlag, 2000
|
|
|
|
2018, Vol. 34 
