ON SUPERLINEAR p(x)-LAPLACIAN-LIKE PROBLEM WITHOUT AMBROSETTI AND RABINOWITZ CONDITION
1
조회
0
다운로드
1
인용
0
참고문헌
키워드
초록
This paper deals with the superlinear elliptic problem with- out Ambrosetti and Rabinowitz type growth condition of the form:｛ −div(1 + |∇u|p(x) √1+|∇u|2p(x) )|∇u|p(x)−2∇u)= f(x, u), a.e. in , u = 0, on αΩ, where ⊂ RN is a bounded domain with smooth boundary αΩ , > 0 is a parameter. The purpose of this paper is to obtain the existence results of nontrivial solutions for every parameter . Firstly, by using the moun- tain pass theorem a nontrivial solution is constructed for almost every parameter > 0. Then we consider the continuation of the solutions. Our results are a generalization of that of Manuela Rodrigues.
Abstract
This paper deals with the superlinear elliptic problem with- out Ambrosetti and Rabinowitz type growth condition of the form:｛ −div(1 + |∇u|p(x) √1+|∇u|2p(x) )|∇u|p(x)−2∇u)= f(x, u), a.e. in , u = 0, on αΩ, where ⊂ RN is a bounded domain with smooth boundary αΩ , > 0 is a parameter. The purpose of this paper is to obtain the existence results of nontrivial solutions for every parameter . Firstly, by using the moun- tain pass theorem a nontrivial solution is constructed for almost every parameter > 0. Then we consider the continuation of the solutions. Our results are a generalization of that of Manuela Rodrigues.
참고문헌 (14)
- E. Acerbi, "Regularity results for stationary electro-rheological fluids", Arch. Ration. Mech. Anal, 164(3), pp.213~259 (2002)
- Y. Chen, "Variable exponent, linear growth functionals in image restoration", SIAM J. Appl. Math, 66(4), pp.1383~1406 (2006)
- L. Diening, "Riesz potential and Sobolev embedding on generalized Lebesque and Sobolev space Lp(·) and Wk,p(·)", Math. Nachr, 268, pp.31~43 (2004)
- D. E. Edmunds, "Sobolev embbeding with variable exponent II", Math. Nachr, 246, pp.247 (2002), 53~67
- X. L. Fan, "Existence of solutions for p(x)-Laplacian Dirichlet problem", Nonlinear Anal, 52(8), pp.1843~1852 (2003)
- X. L. Fan, "On the generalized Orlicz-Sobolev spaces Wk,p(x)()", J. Gansu Educ. College, 12(1), pp.1~6 (1998)
- X. L. Fan, "On the space Lp(x)() and Wk,p(x)()", J. Math. Anal. Appl, 263(2), pp.424~446 (2001)
- X. L. Fan, "Compact imbedding theorems with symmetry of Strauss-Lions type for the space W1,p(x)()", J. Math. Anal. Appl, 255(1), pp.333~348 (2001)
- C. Ji, "On the superlinear problem involving the p(x)-Laplacian", Electron. J. Qual. The-ory Differ, 40, pp.1~9 (2011)
- O. Kovacik, "On spaces Lp(x)() and Wk,p(x)()", Czechoslovak Math. J, 41(4), pp.592~618 (1991)
- M. M. Rodrigues, "Multiplicity of solutions on a nonlinear eigenvalue problem for p(x)-Laplacian-like operators", Mediterr. J. Math, 9(1), pp.211~222 (2012)
- M. Ru˘zi˘cka, "Electrorheological Fluids: Modeling and Mathematical Theory" (2000), Springer-Verlag
- M. Struwe, "On multivortex solutions in Chern-Simons gauge theory", Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 1(1), pp.109~121 (1998)
- G. Wang, "Steady state solutions of a reaction-diffusion system modeling chemotaxis", Math. Nachr, 233, pp.234 (2002), 221~236