# American Institute of Mathematical Sciences

2015, 2015(special): 276-286. doi: 10.3934/proc.2015.0276

## Existence of nontrivial solutions for equations of $p(x)$-Laplace type without Ambrosetti and Rabinowitz condition

 1 Department of Mathematical Sciences, Seoul National University, Seoul 151-742, South Korea 2 Department of Mathematics Education, Sangmyung University, Seoul 110--743

Received  September 2014 Revised  August 2015 Published  November 2015

We study the following elliptic equations with variable exponents \begin{equation*} \begin{cases} -\text{div}(\varphi(x,\nabla u))+{|u|}^{p(x)-2}u= f(x,u) \quad &\text{in } \Omega \\ \varphi(x,\nabla u) \frac{\partial u}{\partial n}= g(x,u) & \text{on }\partial\Omega. \end{cases} \tag{P} \end{equation*} Under suitable conditions on $\phi$, $f$, and $g$, by employing the mountain pass theorem, the problem (P) has at least one nontrivial weak solution without assuming the Ambrosetti and Rabinowitz type condition.
Citation: Eun Bee Choi, Yun-Ho Kim. Existence of nontrivial solutions for equations of $p(x)$-Laplace type without Ambrosetti and Rabinowitz condition. Conference Publications, 2015, 2015 (special) : 276-286. doi: 10.3934/proc.2015.0276
##### References:
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##### References:
 [1] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 14 (1973), 349. Google Scholar [2] M. M. Boureanu and F. Preda, Infinitely many solutions for elliptic problems with variable exponent and nonlinear boundary conditions,, Nonlinear Differ. Equ. Appl., 19 (2012), 235. Google Scholar [3] N. T. Chung, Multiple solutions for quasilinear elliptic problems with nonlinear boundary conditions,, Electron. J. Diff. Eqns., 2008 (2008), 1. Google Scholar [4] L. Diening, P. Harjulehto and P. Hästö, M. R.užička, Lebesgue and Sobolev Spaces with Variable Exponents,, in: Lecture Notes in Mathematics, (2017). Google Scholar [5] D. E. Edmunds and J. Rákosník, Sobolev embedding with variable exponent,, Studia Math., 143 (2000), 267. Google Scholar [6] X. Fan, Boundary trace embedding theorems for variable exponent Sobolev spaces,, J. Math. Anal. Appl., 339 (2008), 1395. Google Scholar [7] X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$,, J. Math. Anal. Appl., 263 (2001), 424. Google Scholar [8] X. Fan and Q. H. Zhang, Existence of solutions for $p(x)$-Laplacian Dirichlet problem,, Nonlinear Anal., 52 (2003), 1843. Google Scholar [9] B. Ge, On the superlinear problems involving the $p(x)$-Laplacian and a non-local term without AR-condition,, Nonlinear Anal., 102 (2014), 133. Google Scholar [10] C. Ji, On the superlinear problem involving the $p(x)$-Laplacian,, Electron. J. Qual. Theory Differ., 40 (2011), 1. Google Scholar [11] I. H. Kim and Y. H. Kim, Mountain pass type solutions and positivity of the infimum eigenvalue for quasilinear elliptic equations with variable exponents,, Manuscripta Math., 147 (2015), 169. Google Scholar [12] V. K. Le, On a sub-supersolution method for variational inequalities with Leray-Lions operators in variable exponent spaces,, Nonlinear Anal., 71 (2009), 3305. Google Scholar [13] S. Liu, On superlinear problems without the Ambrosetti and Rabinowitz condition,, Nonlinear Anal., 73 (2010), 788. Google Scholar [14] F. Y. Lu and G. Q. Deng, Infinitely many weak solutions of the $p$-Laplacian equation with nonlinear boundary conditions,, The Scientific World Journal, 2014 (2014), 1. Google Scholar [15] M. Mihăilescu and V. Rădulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 2625. Google Scholar [16] O. H. Miyagaki and M. A. S. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition,, J. Differential Equations, 245 (2008), 3628. Google Scholar [17] P. Pucci and Q. Zhang, Existence of entire solutions for a class of variable exponent elliptic equations,, J. Differential Equations, 257 (2014), 1529. Google Scholar [18] M. R.užička, Electrorheological Fluids: Modeling and Mathematical Theory,, in: Lecture Notes in Mathematics, (1748). Google Scholar [19] I. Sim and Y. H. Kim, Existence of solutions and positivity of the infimum eigenvalue for degenerate elliptic equations with variable exponents,, Discrete Contin. Dyn. Syst. Supplement, 2013 (2013), 695. Google Scholar [20] Z. Tan and F. Fang, On superlinear $p(x)$-Laplacian problems without Ambrosetti and Rabinowitz condition,, Nonlinear Anal., 75 (2012), 3902. Google Scholar [21] P. Winkert, Multiplicity results for a class of elliptic problems with nonlinear boundary condition,, Commun. Pure Appl. Anal., 12 (2013), 785. Google Scholar [22] J. Yao, Solutions for Neumann boundary value problems involving $p(x)$-Laplace operators,, Nonlinear Anal., 68 (2008), 1271. Google Scholar [23] J. H. Zhao and P. H. Zhao, Existence of infinitely many weak solutions for the $p$-Laplacian with nonlinear boundary conditions,, Nonlinear Anal., 69 (2008), 1343. Google Scholar
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