Existence of a Nontrivial Solution for a Class of Superquadratic Elliptic Problems

Affiliation(s)

Department of Biotechnology, Beijing City University, Beijing, China.

School of Mathematics and Statistics, Central South University, Changsha, Hunan, China.

Employee’s College of Dongcheng in Beijing, Beijing, China.

School of Mathematical Sciences, Qufu Normal University, Shandong, China.

Department of Biotechnology, Beijing City University, Beijing, China.

School of Mathematics and Statistics, Central South University, Changsha, Hunan, China.

Employee’s College of Dongcheng in Beijing, Beijing, China.

School of Mathematical Sciences, Qufu Normal University, Shandong, China.

ABSTRACT

We consider the existence of a nontrivial solution for the Dirichlet boundary value problem*-△u+a(x)u=g(x,u),*in Ω *u*=0, on Ω We prove an abstract result on the existence of a critical point for the functional *f* on a Hilbert space via the local linking theorem. Different from the works in the literature, the new theorem is constructed under the*(C)** condition instead of *(PS)** condition.

We consider the existence of a nontrivial solution for the Dirichlet boundary value problem

Cite this paper

X. Mo, P. Jing, Y. Zhao and A. Mao, "Existence of a Nontrivial Solution for a Class of Superquadratic Elliptic Problems,"*Advances in Pure Mathematics*, Vol. 2 No. 5, 2012, pp. 314-317. doi: 10.4236/apm.2012.25043.

X. Mo, P. Jing, Y. Zhao and A. Mao, "Existence of a Nontrivial Solution for a Class of Superquadratic Elliptic Problems,"

References

[1] S. J. Li and M. Willem, “Applications of Local Linking to Critical Point Theory,” Journal of Mathematical Analysis and Applications, Vol. 189, No. 1, 1995, pp. 6-32. doi:10.1006/jmaa.1995.1002

[2] X.-L. Fan and Y.-Z. Zhao, “Linking and Multiplicity Results for the p-Laplacian on Unbounded Cylinders,” Journal of Mathematical Analysis and Applications, Vol. 260, No. 2, 2001, pp. 479-489. doi:10.1006/jmaa.2000.7468

[3] Q. S. Jiu, J. B. Su, “Existence and Multiplicity Results for Dirichlet Problems with p-Laplacian,” Journal of Mathematical Analysis and Applications, Vol. 281, No. 2, 2003, pp. 587-601. doi:10.1016/S0022-247X(03)00165-3

[4] P. H. Rabinowitz, “Periodic Solutions of Hamiltonian Systems,” Communications on Pure and Applied Mathematics, Vol. 31, No. 2, 1978, pp. 157-184. doi:10.1002/cpa.3160310203

[5] Q. Jiang and C. L. Tang, “Existence of a Nontrivial Solution for a Class of Superquadratic Elliptic Problems,” Nonlinear Analysis, Vol. 69, No. 2, 2008, pp. 523-529. doi:10.1016/j.na.2007.05.038

[6] S. X. Luan and A. M. Mao, “Periodic Solutions for a Class of Non-Autonomous Hamiltonian Systems,” Nonlinear Analysis, Vol. 61, No. 8, 2005, pp. 1413-1426. doi:10.1016/j.na.2005.01.108

[1] S. J. Li and M. Willem, “Applications of Local Linking to Critical Point Theory,” Journal of Mathematical Analysis and Applications, Vol. 189, No. 1, 1995, pp. 6-32. doi:10.1006/jmaa.1995.1002

[2] X.-L. Fan and Y.-Z. Zhao, “Linking and Multiplicity Results for the p-Laplacian on Unbounded Cylinders,” Journal of Mathematical Analysis and Applications, Vol. 260, No. 2, 2001, pp. 479-489. doi:10.1006/jmaa.2000.7468

[3] Q. S. Jiu, J. B. Su, “Existence and Multiplicity Results for Dirichlet Problems with p-Laplacian,” Journal of Mathematical Analysis and Applications, Vol. 281, No. 2, 2003, pp. 587-601. doi:10.1016/S0022-247X(03)00165-3

[4] P. H. Rabinowitz, “Periodic Solutions of Hamiltonian Systems,” Communications on Pure and Applied Mathematics, Vol. 31, No. 2, 1978, pp. 157-184. doi:10.1002/cpa.3160310203

[5] Q. Jiang and C. L. Tang, “Existence of a Nontrivial Solution for a Class of Superquadratic Elliptic Problems,” Nonlinear Analysis, Vol. 69, No. 2, 2008, pp. 523-529. doi:10.1016/j.na.2007.05.038

[6] S. X. Luan and A. M. Mao, “Periodic Solutions for a Class of Non-Autonomous Hamiltonian Systems,” Nonlinear Analysis, Vol. 61, No. 8, 2005, pp. 1413-1426. doi:10.1016/j.na.2005.01.108