# 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:
 [1] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. [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), 235-251. [3] N. T. Chung, Multiple solutions for quasilinear elliptic problems with nonlinear boundary conditions, Electron. J. Diff. Eqns., 2008 (2008), 1-6. [4] L. Diening, P. Harjulehto and P. Hästö, M. R.užička, Lebesgue and Sobolev Spaces with Variable Exponents, in: Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, 2011. [5] D. E. Edmunds and J. Rákosník, Sobolev embedding with variable exponent, Studia Math., 143 (2000), 267-293. [6] X. Fan, Boundary trace embedding theorems for variable exponent Sobolev spaces, J. Math. Anal. Appl., 339 (2008), 1395-1412. [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), 424-446. [8] X. Fan and Q. H. Zhang, Existence of solutions for $p(x)$-Laplacian Dirichlet problem, Nonlinear Anal., 52 (2003), 1843-1852. [9] B. Ge, On the superlinear problems involving the $p(x)$-Laplacian and a non-local term without AR-condition, Nonlinear Anal., 102 (2014), 133-143. [10] C. Ji, On the superlinear problem involving the $p(x)$-Laplacian, Electron. J. Qual. Theory Differ., 40 (2011), 1-9. [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), 169-191. [12] V. K. Le, On a sub-supersolution method for variational inequalities with Leray-Lions operators in variable exponent spaces, Nonlinear Anal., 71 (2009), 3305-3321. [13] S. Liu, On superlinear problems without the Ambrosetti and Rabinowitz condition, Nonlinear Anal., 73 (2010), 788-795. [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), 1-5. [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), 2625-2641 [16] O. H. Miyagaki and M. A. S. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations, 245 (2008), 3628-3638. [17] P. Pucci and Q. Zhang, Existence of entire solutions for a class of variable exponent elliptic equations, J. Differential Equations, 257 (2014), 1529-1566. [18] M. R.užička, Electrorheological Fluids: Modeling and Mathematical Theory, in: Lecture Notes in Mathematics, vol. 1748, Springer-Verlag, Berlin, 2000. [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), 695-707. [20] Z. Tan and F. Fang, On superlinear $p(x)$-Laplacian problems without Ambrosetti and Rabinowitz condition, Nonlinear Anal., 75 (2012), 3902-3915. [21] P. Winkert, Multiplicity results for a class of elliptic problems with nonlinear boundary condition, Commun. Pure Appl. Anal., 12 (2013), 785-802. [22] J. Yao, Solutions for Neumann boundary value problems involving $p(x)$-Laplace operators, Nonlinear Anal., 68 (2008), 1271-1283. [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), 1343-1355.

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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), 349-381. [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), 235-251. [3] N. T. Chung, Multiple solutions for quasilinear elliptic problems with nonlinear boundary conditions, Electron. J. Diff. Eqns., 2008 (2008), 1-6. [4] L. Diening, P. Harjulehto and P. Hästö, M. R.užička, Lebesgue and Sobolev Spaces with Variable Exponents, in: Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, 2011. [5] D. E. Edmunds and J. Rákosník, Sobolev embedding with variable exponent, Studia Math., 143 (2000), 267-293. [6] X. Fan, Boundary trace embedding theorems for variable exponent Sobolev spaces, J. Math. Anal. Appl., 339 (2008), 1395-1412. [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), 424-446. [8] X. Fan and Q. H. Zhang, Existence of solutions for $p(x)$-Laplacian Dirichlet problem, Nonlinear Anal., 52 (2003), 1843-1852. [9] B. Ge, On the superlinear problems involving the $p(x)$-Laplacian and a non-local term without AR-condition, Nonlinear Anal., 102 (2014), 133-143. [10] C. Ji, On the superlinear problem involving the $p(x)$-Laplacian, Electron. J. Qual. Theory Differ., 40 (2011), 1-9. [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), 169-191. [12] V. K. Le, On a sub-supersolution method for variational inequalities with Leray-Lions operators in variable exponent spaces, Nonlinear Anal., 71 (2009), 3305-3321. [13] S. Liu, On superlinear problems without the Ambrosetti and Rabinowitz condition, Nonlinear Anal., 73 (2010), 788-795. [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), 1-5. [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), 2625-2641 [16] O. H. Miyagaki and M. A. S. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations, 245 (2008), 3628-3638. [17] P. Pucci and Q. Zhang, Existence of entire solutions for a class of variable exponent elliptic equations, J. Differential Equations, 257 (2014), 1529-1566. [18] M. R.užička, Electrorheological Fluids: Modeling and Mathematical Theory, in: Lecture Notes in Mathematics, vol. 1748, Springer-Verlag, Berlin, 2000. [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), 695-707. [20] Z. Tan and F. Fang, On superlinear $p(x)$-Laplacian problems without Ambrosetti and Rabinowitz condition, Nonlinear Anal., 75 (2012), 3902-3915. [21] P. Winkert, Multiplicity results for a class of elliptic problems with nonlinear boundary condition, Commun. Pure Appl. Anal., 12 (2013), 785-802. [22] J. Yao, Solutions for Neumann boundary value problems involving $p(x)$-Laplace operators, Nonlinear Anal., 68 (2008), 1271-1283. [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), 1343-1355.
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