Existence ofnontrivial solutions for equations of $p(x)$-Laplace type withoutAmbrosetti and Rabinowitz condition

Eun Bee  Choi; Yun-Ho Kim

doi:10.3934/proc.2015.0276

Article Contents

2015, Volume 2015: 276-286. Doi: 10.3934/proc.2015.0276 open access

This issue Previous Article Interaction of oscillatory packets of water waves Next Article On the properties of solutions set for measure driven differential inclusions

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

Eun Bee Choi^1, and
Yun-Ho Kim^2,

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

Received: August 31, 2014

Revised: July 31, 2015

Published: October 2015

Abstract Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: 35B38, 35D30, 35J20, 35J60, 35J66.

Citation:

Related Papers

Cited by

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.