A local mountain pass theorem and applications to a double perturbed p(x)-Laplacian equations

In this paper, we prove a local mountain pass theorem without (P.S) condition. Using this theorem and Ricceri's variational principle, we consider a double perturbed Neumann problem with nonlinear boundary condition of the [email protected]"p"("x")u+a(x)|u|^p^(^x^)^-^2u=f(x,u)[email protected]"1(x,u)[email protected],|@?u|^p^(^x^)^-^[email protected][email protected][email protected]=g(x,u)[email protected]"2(x,u)[email protected][email protected] least seven solutions are obtained under different assumptions.

[1]  Julián Fernández Bonder,et al.  Existence Results for the p-Laplacian with Nonlinear Boundary Conditions☆☆☆ , 2001 .

[2]  Michel Chipot,et al.  On the solutions to some elliptic equations with nonlinear Neumann boundary conditions , 1996, Advances in Differential Equations.

[3]  I. Shafrir,et al.  Existence of Positive Solutions of a Semilinear Elliptic Equation in Rn + with a Nonlinear Boundary Condition , 1998 .

[4]  M. Ruzicka,et al.  Electrorheological Fluids: Modeling and Mathematical Theory , 2000 .

[5]  J. Rossi,et al.  Weak solutions for the p-Laplacian with a nonlinear boundary condition at resonance , 2003 .

[6]  Julián Fernández Bonder,et al.  Existence for an elliptic system with nonlinear boundary conditions via fixed-point methods , 2001, Advances in Differential Equations.

[7]  Xianling Fan,et al.  Remarks on Ricceri’s variational principle and applications to the p(x)-Laplacian equations , 2007 .

[8]  Sublevel sets and global minima of coercive functionals and local minima of their perturbations , 2004, math/0402444.

[9]  S. Terracini.  Symmetry properties of positive solutions to some elliptic equations with nonlinear boundary conditions , 1995, Differential and Integral Equations.

[10]  Xianling Fan,et al.  Existence and multiplicity of solutions for p(x)p(x)-Laplacian equations in RNRN☆ , 2004 .

[11]  M. Willem.  Minimax Theorems , 1997 .

[12]  V. Zhikov,et al.  AVERAGING OF FUNCTIONALS OF THE CALCULUS OF VARIATIONS AND ELASTICITY THEORY , 1987 .

[13]  Yunmei Chen,et al.  Variable Exponent, Linear Growth Functionals in Image Restoration , 2006, SIAM J. Appl. Math..

[14]  Xianling Fan,et al.  Sobolev Embedding Theorems for Spaces Wk, p(x)(Ω) , 2001 .

[15]  Chun-Lei Tang,et al.  Resonance problems for the p-Laplacian with a nonlinear boundary condition , 2006 .

[16]  Qihu Zhang,et al.  Existence of solutions for p(x) -Laplacian dirichlet problem , 2003 .

[17]  P. Rabinowitz,et al.  Dual variational methods in critical point theory and applications , 1973 .

[18]  Jiří Rákosník,et al.  On spaces $L^{p(x)}$ and $W^{k, p(x)}$ , 1991 .

[19]  Patrizia Pucci,et al.  A mountain pass theorem , 1985 .

[20]  Bei Hu.  Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition , 1994, Differential and Integral Equations.

[21]  Xianling Fan,et al.  On the Spaces Lp(x)(Ω) and Wm, p(x)(Ω) , 2001 .

[22]  J. Serrin,et al.  Extensions of the mountain pass theorem , 1984 .

[23]  Klaus Pfluger.  Existence and multiplicity of solutions to a p-Laplacian equation with nonlinear boundary condition , 1998 .

[24]  Existence Results for Hamiltonian Elliptic Systems with Nonlinear Boundary Conditions , 1999 .

[25]  Vicentiu D. Rădulescu,et al.  Existence and non-existence results for a quasilinear problem with nonlinear boundary condition , 2000 .

[26]  Xianling Fan,et al.  Existence and multiplicity of solutions for p(x)-Laplacian equations in RN , 2004 .

[27]  Shao-Gao Deng,et al.  Eigenvalues of the p(x)-Laplacian Steklov problem☆ , 2008 .

[28]  Xianling Fan,et al.  Solutions for p(x)-Laplacian Dirichlet problems with singular coefficients , 2005 .