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Critical points theorems via the generalized Ekeland variational principle and its application to equations of \(p(x)\)-Laplace type in \(\mathbb{R}^{N}\). (English) Zbl 1409.58004

Summary: In this paper, we investigate abstract critical point theorems for continuously Gâteaux differentiable functionals satisfying the Cerami condition via the generalized Ekeland variational principle developed by C.-K. Zhong [J. Math. Anal. Appl. 205, No. 1, 239–250 (1997; Zbl 0870.49015)]. As applications of our results, under certain assumptions, we show the existence of at least one or two weak solutions for nonlinear elliptic equations with variable exponents \[ -\operatorname{div} (\phi(x, \nabla u)) + V(x) |u|^{p(x)-2} u = \lambda f(x,u) \text{ in } \mathbb{R}^{N}, \] where the function \(\phi(x,v)\) is of type \(|v|^{p(x)-2}v\) with a continuous function \(p \colon \mathbb{R}^{N} \rightarrow (1,\infty)\), \(V \colon \mathbb{R}^{N} \rightarrow (0,\infty)\) is a continuous potential function, \(\lambda\) is a real parameter, and \(f \colon \mathbb{R}^{N} \times \mathbb{R} \rightarrow \mathbb{R}\) is a Carathéodory function. Especially, we localize precisely the intervals of \(\lambda\) for which the above equation admits at least one or two nontrivial weak solutions by applying our critical points results.

MSC:

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35D30 Weak solutions to PDEs
35J15 Second-order elliptic equations
35J60 Nonlinear elliptic equations
58E30 Variational principles in infinite-dimensional spaces
49J50 Fréchet and Gateaux differentiability in optimization

Citations:

Zbl 0870.49015
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References:

[1] C. O. Alves and S. Liu, On superlinear \(p(x)\)-Laplacian equations in \(\mathbb{R}^N \), Nonlinear Anal. 73 (2010), no. 8, 2566-2579. · Zbl 1194.35142
[2] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349-381. · Zbl 0273.49063
[3] S. Aouaoui, On some degenerate quasilinear equations involving variable exponents, Nonlinear Anal. 75 (2012), no. 4, 1843-1858. · Zbl 1237.35011
[4] D. Arcoya and L. Boccardo, Critical points for multiple integrals of the calculus of variations, Arch. Rational Mech. Anal. 134 (1996), no. 3, 249-274. · Zbl 0884.58023
[5] D. Arcoya and J. Carmona, A nondifferentiable extension of a theorem of Pucci and Serrin and applications, J. Differential Equations 235 (2007), no. 2, 683-700. · Zbl 1134.35052
[6] G. Barletta and A. Chinn\`\i, Existence of solutions for a Neumann problem involving the \(p(x)\)-Laplacian, Electron. J. Differential Equations 2013 (2013), no. 158, 12 pp. · Zbl 1288.35211
[7] G. Barletta, A. Chinn\`\i and D. O’Regan, Existence results for a Neumann problem involving the \(p(x)\)-Laplacian with discontinuous nonlinearities, Nonlinear Anal. Real World Appl. 27 (2016), 312-325. · Zbl 1332.35128
[8] G. Bonanno, A characterization of the mountain pass geometry for functionals bounded from below, Differential Integral Equations 25 (2012), no. 11-12, 1135-1142. · Zbl 1274.49015
[9] ——–, A critical point theorem via the Ekeland variational principle, Nonlinear Anal. 75 (2012), no. 5, 2992-3007. · Zbl 1239.58011
[10] ——–, Relations between the mountain pass theorem and local minima, Adv. Nonlinear Anal. 1 (2012), no. 3, 205-220. · Zbl 1277.35170
[11] G. Bonanno and A. Chinn\`\i, Discontinuous elliptic problems involving the \(p(x)\)-Laplacian, Math. Nachr. 284 (2011), no. 5-6, 639-652. · Zbl 1214.35076
[12] ——–, Existence and multiplicity of weak solutions for elliptic Dirichlet problems with variable exponent, J. Math. Anal. Appl. 418 (2014), no. 2, 812-827. · Zbl 1312.35111
[13] G. Bonanno, G. D’Agu\`\i and P. Winkert, Sturm-Liouville equations involving discontinuous nonlinearities, Minimax Theory Appl. 1 (2016), no. 1, 125-143. · Zbl 1357.34048
[14] G. Bonanno and S. A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Appl. Anal. 89 (2010), no. 1, 1-10. · Zbl 1194.58008
[15] M.-M. Boureanu and D. N. Udrea, Existence and multiplicity results for elliptic problems with \(p(·)\)-growth conditions, Nonlinear Anal. Real World Appl. 14 (2013), no. 4, 1829-1844. · Zbl 1271.35045
[16] H. Brezis and L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math. 44 (1991), no. 8-9, 939-963. · Zbl 0751.58006
[17] G. Cerami, An existence criterion for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend. A 112 (1978), no. 2, 332-336, (1979) (Italian). · Zbl 0436.58006
[18] E. B. Choi, J.-M. Kim and Y.-H. Kim, Infinitely many solutions for equations of \(p(x)\)-Laplace type with the nonlinear Neumann boundary condition, Proc. Roy. Soc. Edinburgh Sect. A 148 (2018), no. 1, 1-31. · Zbl 1391.35163
[19] F. Colasuonno, P. Pucci and C. Varga, Multiple solutions for an eigenvalue problem involving \(p\)-Laplacian type operators, Nonlinear Anal. 75 (2012), no. 12, 4496-4512. · Zbl 1251.35059
[20] D. G. Costa and O. H. Miyagaki, Nontrivial solutions for perturbations of the \(p\)-Laplacian on unbounded domains, J. Math. Anal. Appl. 193 (1995), no. 3, 737-755. · Zbl 0856.35040
[21] L. Diening, P. Harjulehto, P. Hästö and M. R, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics 2017, Springer, Berlin, 2011. · Zbl 1222.46002
[22] D. E. Edmunds and J. Rákosník, Sobolev embeddings with variable exponent, Studia Math. 143 (2000), no. 3, 267-293. · Zbl 0974.46040
[23] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-353. · Zbl 0286.49015
[24] X. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces \(W^{k,p(x)}(Ω)\), J. Math. Anal. Appl. 262 (2001), no. 2, 749-760. · Zbl 0995.46023
[25] X. Fan and D. Zhao, On the spaces \(L^{p(x)}(Ω)\) and \(W^{m,p(x)}(Ω)\), J. Math. Anal. Appl. 263 (2001), no. 2, 424-446. · Zbl 1028.46041
[26] N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory, Cambridge Tracts in Mathematics 107, Cambridge University Press, Cambridge, 1993. · Zbl 0790.58002
[27] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on \(\mathbb{R}^N \), Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), no. 4, 787-809. · Zbl 0935.35044
[28] 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), no. 1-2, 169-191. · Zbl 1322.35009
[29] J.-M. Kim, Y.-H. Kim and J. Lee, Existence and multiplicity of solutions for equations of \(p(x)\)-Laplace type in \(\mathbb{R}^N\) without AR-condition, Differential Integral Equations 31 (2018), no. 5-6, 435-464. · Zbl 1449.35164
[30] N. C. Kourogenis and N. S. Papageorgiou, A weak nonsmooth Palais-Smale condition and coercivity, Rend. Circ. Mat. Palermo (2) 49 (2000), no. 3, 521-526. · Zbl 1225.49021
[31] V. K. Le, On a sub-supersolution method for variational inequalities with Leray-Lions operators in variable exponent spaces, Nonlinear Anal. 71 (2009), no. 7-8, 3305-3321. · Zbl 1179.35148
[32] S. D. Lee, K. Park and Y.-H. Kim, Existence and multiplicity of solutions for equations involving nonhomogeneous operators of \(p(x)\)-Laplace type in \(\mathbb{R}^N \), Bound. Value Probl. 2014 (2014), no. 261, 17 pp. · Zbl 1316.35139
[33] G. Li and C. 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), no. 12, 4602-4613. · Zbl 1190.35104
[34] S. A. Marano and D. Motreanu, Infinitely many critical points of non-differentiable functions and applications to a Neumann-type problem involving the \(p\)-Laplacian, J. Differential Equations 182 (2002), no. 1, 108-120. · Zbl 1013.49001
[35] ——–, On a three critical points theorem for non-differentiable functions and applications to nonlinear boundary value problems, Nonlinear Anal. 48 (2002), no. 1, Ser. A: Theory Methods, 37-52. · Zbl 1014.49004
[36] O. H. Miyagaki and M. A. S. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations 245 (2008), no. 12, 3628-3638. · Zbl 1158.35400
[37] B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), no. 1-2, 401-410. · Zbl 0946.49001
[38] ——–, On a three critical points theorem, Arch. Math. (Basel) 75 (2000), no. 3, 220-226. · Zbl 0979.35040
[39] M. R, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics 1748, Springer-Verlag, Berlin, 2000.
[40] M. Schechter, Linking Methods in Critical Point Theory, Birkhäuser Boston, Boston, MA, 1999. · Zbl 0915.35001
[41] Z. Tan and F. Fang, On superlinear \(p(x)\)-Laplacian problems without Ambrosetti and Rabinowitz condition, Nonlinear Anal. 75 (2012), no. 9, 3902-3915. · Zbl 1241.35047
[42] C.-K. Zhong, On Ekeland’s variational principle and a minimax theorem, J. Math. Anal. Appl. 205 (1997), no. 1, 239-250. · Zbl 0870.49015
[43] ——–, A generalization of Ekeland’s variational principle and application to the study of the relation between the weak P.S. condition and coercivity, Nonlinear Anal. 29 (1997), no. 12, 1421-1431. · Zbl 0912.49021
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