ON SUPERLINEAR p(x)-LAPLACIAN-LIKE PROBLEM WITHOUT AMBROSETTI AND RABINOWITZ CONDITION

Title & Authors
ON SUPERLINEAR p(x)-LAPLACIAN-LIKE PROBLEM WITHOUT AMBROSETTI AND RABINOWITZ CONDITION
Bin, Ge;

Abstract
This paper deals with the superlinear elliptic problem without Ambrosetti and Rabinowitz type growth condition of the form: \{-div$$(1+\frac{|{\nabla}u|^{p(x)}}{\sqrt{1+|{\nabla}u|^{2p(x)}}}})|{\nabla}u|^{p(x)-2}{\nabla}u$$
Keywords
superlinear problem;p(x)-Laplacian;variational method;variable exponent Sobolev space;
Language
English
Cited by
1.
Existence and Multiplicity of Solutions for a Class of Elliptic Equations Without Ambrosetti–Rabinowitz Type Conditions, Journal of Dynamics and Differential Equations, 2017
References
1.
E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal. 164 (2002), no. 3, 213-259.

2.
Y. Chen, S. Levine, and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), no. 4, 1383-1406.

3.
L. Diening, Riesz potential and Sobolev embedding on generalized Lebesque and Sobolev space $L^{p({\cdot})}$ and $W^{k,p({\cdot})}$, Math. Nachr. 268 (2004), 31-43.

4.
D. E. Edmunds and J. Rakosnic, Sobolev embbeding with variable exponent II, Math. Nachr. 246/247 (2002), 53-67.

5.
X. L. Fan and Q. H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003), no. 8, 1843-1852.

6.
X. L. Fan and D. Zhao, On the generalized Orlicz-Sobolev spaces $W^{k,p(x)}({\Omega})$, J. Gansu Educ. College 12 (1998), no. 1, 1-6.

7.
X. L. Fan and D. Zhao, On the space $L^{p(x)}({\Omega})$ and $W^{k,p(x)}({\Omega})$, J. Math. Anal. Appl. 263 (2001), no. 2, 424-446.

8.
X. L. Fan, Y. Z. Zhao, and D. Zhao, Compact imbedding theorems with symmetry of Strauss-Lions type for the space $W^{1,p(x)}({\Omega})$, J. Math. Anal. Appl. 255 (2001), no. 1, 333-348.

9.
C. Ji, On the superlinear problem involving the p(x)-Laplacian, Electron. J. Qual. Theory Differ. 40 (2011), 1-9.

10.
O. Kovacik and J. Rakosuik, On spaces $L^{p(x)}({\Omega})$ and $W^{k,p(x)}({\Omega})$, Czechoslovak Math. J. 41 (1991), no. 4, 592-618.

11.
M. M. Rodrigues, Multiplicity of solutions on a nonlinear eigenvalue problem for p(x)-Laplacian-like operators, Mediterr. J. Math. 9 (2012), no. 1, 211-222.

12.
M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000.

13.
M. Struwe and G. Tarantello, On multivortex solutions in Chern-Simons gauge theory, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1 (1998), no. 1, 109-121.

14.
G. Wang and J. Wei, Steady state solutions of a reaction-diffusion system modeling chemotaxis, Math. Nachr. 233/234 (2002), 221-236.