Nonhomogeneous Dirichlet problems without the Ambrosetti-Rabinowitz condition

Gang Li, Vicenţiu D. Rădulescu, Dušan D. Repovš, Qihu Zhang

DOI: http://dx.doi.org/10.12775/TMNA.2017.037

Abstract


We consider the existence of solutions of the following $p(x)$-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition: \begin{equation*} \begin{cases} -\mbox{div}\hspace{.07em}(|\nabla u|^{p(x)-2}\nabla u)=f(x,u) &\text{ in }\Omega , \\ u=0 &\text{ on }\partial \Omega .% \end{cases} \end{equation*} We give a new growth condition and we point out its importance for checking the Cerami compactness condition. We prove the existence of solutions of the above problem via the critical point theory, and also provide some multiplicity properties. The present paper extend previous results of Q. Zhang and C. Zhao (Existence of strong solutions of a $p(x)$-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition, \textit{Computers and Mathematics with Applications}, 2015) and we establish the existence of solutions under weaker hypotheses on the nonlinear term.

Keywords


Nonhomogeneous differential operator; Ambrosetti-Rabinowitz condition; Cerami compactness condition; Sobolev space with variable exponent

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References


E. Acerbi and G. Mingione, Regularity results for a class of functionals with nonstandard growth, Arch. Ration. Mech. Anal. 156 (2001), 121–140.

E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal. 164 (2002), 213–259.

C.O. Alves and S.B. Liu, On superlinear p(x)-Laplacian equations in RN , Nonlinear Anal. 73 (2010), 2566–2579.

S. Antontsev and S. Shmarev, Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties of solutions, Nonlinear Anal. 65 (2006), 728–761.

A. Ayoujil, On the superlinear Steklov problem involving the p(x)-Laplacian, Electron. J. Qual. Theory Differ. Equ. (2014), no. 38, 1–13.

J.F. Bonder, N. Saintier and A. Silva, On the Sobolev embedding theorem for variable exponent spaces in the critical range, J. Differential Equations 253 (2012), 1604–1620.

J.F. Bonder, N. Saintier and A. Silva, Existence of solution to a critical equation with variable exponent, Ann. Acad. Sci. Fenn. Math. 37 (2012), 579–594.

J. Chabrowski and Y.Q. Fu, Existence of solutions for p(x)-Laplacian problems on a bounded domain, J. Math. Anal. Appl. 306 (2005), 604–618.

K.C. Chang, Critical Point Theory and Applications, Shanghai Scientific and Technology Press, Shanghai, 1986.

Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), 1383–1406.

A. Coscia and G. Mingione, Hölder continuity of the gradient of p(x)-harmonic mappings, C.R. Acad. Sci. Paris, Sér. I 328 (1999), 363–368.

L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017, Springer, Berlin, 2011.

X.L. Fan, Global C 1,α regularity for variable exponent elliptic equations in divergence form, J. Differential Equations 235 (2007), 397–417.

X.L. Fan, On the sub-supersolution method for p(x)-Laplacian equations, J. Math. Anal. Appl. 330 (2007), 665–682.

X.L. Fan and Q.H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003), 1843–1852.

X.L. Fan, Q.H. Zhang and D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005), 306–317.

X.L. Fan and D. Zhao, On the spaces Lp(x) (Ω) and W m,p(x) (Ω), J. Math. Anal. Appl. 263 (2001), 424–446.

Y.Q. Fu, The principle of concentration-compactness in Lp(x) spaces and its application, Nonlinear Anal. 71 (2009), 1876–1892.

L. Gasiński and N. Papageorgiou, Anisotropic nonlinear Neumann problems, Calc. Var. Partial Differential Equations 42 (2011), 323–354.

L. Gasiński and N. Papageorgiou, A pair of positive solutions for the Dirichlet p(z)-Laplacian with concave and convex nonlinearities, J. Global Optim. 56 (2013), 1347–1360.

A. El Hamidi, Existence results to elliptic systems with nonstandard growth conditions, J. Math. Anal. Appl. 300 (2004), 30–42.

P. Harjulehto, P. Hästö, Ú. V. Lê, M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal. 72 (2010), 4551–4574.

P. Harjulehto, T. Kuusi, T. Lukkari, N. Marola and M. Parviainen, Harnack’s inequality for quasi-minimizers with non-standard growth conditions, J. Math. Anal. Appl. 344 (2008), 504–520.

T. Kopaliani, Interpolation theorems for variable exponent Lebesgue spaces, J. Funct.l Anal. 257 (2009), 3541–3551.

O. Kováčik and J. Rákosnı́k, On spaces Lp(x) (Ω) and W k,p(x) (Ω), Czechoslovak Math. J. 41 (1991), 592–618.

G.B. Li and C.Y. Yang, The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of p-Laplacian type without the Ambrosetti–Rabinowitz condition, Nonlinear Anal. 72 (2010), 4602–4613.

S.B. Liu, On superlinear problems without the Ambrosetti and Rabinowitz condition, Nonlinear Anal. 73 (2010), 788–795.

T. Lukkari, Singular solutions of elliptic equations with nonstandard growth, Math. Nachr. 282 (12) (2009), 1770–1787.

M. Mihailescu, V.D. Rădulescu and D.D. Repovš, On a non-homogeneous eigenvalue problem involving a potential: an Orlicz–Sobolev space setting, J. Math. Pures Appl. 93 (2010), 132–148.

M. Mihailescu and D.D. Repovš, Multiple solutions for a nonlinear and nonhomogeneous problem in Orlicz–Sobolev spaces, Appl. Math. Comput. 217 (2011), 6624–6632.

O.H. Miyagaki and M.A.S. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations 245 (2008), 3628–3638.

G. Molica Bisci and D.D. Repovš, Multiple solutions for elliptic equations involving a general operator in divergence form, Ann. Acad. Sci. Fenn. Math. 39 (2014), 259–273.

P. Pucci and Q.H. Zhang, Existence of entire solutions for a class of variable exponent elliptic equations, J. Differential Equations 257 (2014), 1529–1566.

V.D. Rădulescu, Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Anal. 121 (2015), 336–369.

V.D. Rădulescu and D.D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, CRC Press, Taylor & Francis, Boca Raton, 2015.

V.D. Rădulescu and I. Stăncut, Combined concave-convex effects in anisotropic elliptic equations with variable exponent, Nonlinear Differential Equations and Applications (NoDEA) 22 (2015), 391–410.

M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, vol. 1748, Springer, Berlin, 2000.

S.G. Samko, Denseness of C0∞ (RN ) in the generalized Sobolev spaces W m,p(x) (RN ), Dokl. Ross. Akad. Nauk 369 (1999), no. 4, 451–454.

M. Willem and W. Zou, On a Schrödinger equation with periodic potential and spectrum point zero, Indiana Univ. Math. J. 52 (2003), 109–132.

X.C. Xu and Y.K. An, Existence and multiplicity of solutions for elliptic systems with nonstandard growth condition in RN , Nonlinear Anal. 68 (2008), 956–968.

N. Yoshida, Picone identities for half-linear elliptic operators with p(x)-Laplacians and applications to Sturmian comparison theory, Nonlinear Anal. 74 (2011), 5631–5642.

A.B. Zang, p(x)-Laplacian equations satisfying Cerami condition, J. Math. Anal. Appl. 337 (2008), 547–555.

Q.H. Zhang, A strong maximum principle for differential equations with nonstandard p(x)-growth conditions, J. Math. Anal. Appl. 312 (2005), 24–32.

Q.H. Zhang, Existence and asymptotic behavior of positive solutions for variable exponent elliptic systems, Nonlinear Anal. 70 (2009), 305–316.

Q.H. Zhang and C.S. Zhao, Existence of strong solutions of a p(x)-Laplacian Dirichlet problem without the Ambrosetti–Rabinowitz condition, Comput. Math. Appl. 69 (2015), 1–12.

X.X. Zhang and X.P. Liu, The local boundedness and Harnack inequality of p(x)-Laplace equation, J. Math. Anal. Appl. 332 (2007), 209–218.

J.F. Zhao, Structure Theory of Banach Spaces, Wuhan University Press, Wuhan, 1991 (in Chinese).


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