Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

MULTIPLICITY OF SOLUTIONS FOR QUASILINEAR SCHRÖDINGER TYPE EQUATIONS WITH THE CONCAVE-CONVEX NONLINEARITIES

  • Kim, In Hyoun (Department of Mathematics Incheon National University) ;
  • Kim, Yun-Ho (Department of Mathematics Education Sangmyung University) ;
  • Li, Chenshuo (Questrom School of Business Boston University) ;
  • Park, Kisoeb (Department of IT Convergence Software Seoul Theological University)
  • Received : 2021.02.05
  • Accepted : 2021.08.04
  • Published : 2021.11.01
Download PDF
⟨ Previous Next ⟩

Abstract

We deal with the following elliptic equations: $\{-div({\varphi}^{\prime}(\left|{\nabla}z\right|^2){\nabla}z)+V(x)\left|z\right|^{{\alpha}-2}z={\lambda}{\rho}(x)\left|z\right|^{r-2}z+h(x,z),\\z(x){\rightarrow}0,\;as\;\left|x\right|{\rightarrow}{\infty},$ in ℝN , where N ≥ 2, 1 < p < q < N, 1 < α ≤ p*q'/p', α < q, 1 < r < min{p, α}, φ(t) behaves like tq/2 for small t and tp/2 for large t, and p' and q' the conjugate exponents of p and q, respectively. Here, V : ℝN → (0, ∞) is a potential function and h : ℝN × ℝ → ℝ is a Carathéodory function. The present paper is devoted to the existence of at least two distinct nontrivial solutions to quasilinear elliptic problems of Schrödinger type, which provides a concave-convex nature to the problem. The primary tools are the well-known mountain pass theorem and a variant of Ekeland's variational principle.

Keywords

Acknowledgement

The first author was supported by the Incheon National University Research Grant in 2017. The authors gratefully thank to the Referee for the constructive comments and recommendations which definitely help to improve the readability and quality of the paper.

References

  1. K. Ait-Mahiout and C. O. Alves, Existence and multiplicity of solutions for a class of quasilinear problems in Orlicz-Sobolev spaces, Complex Var. Elliptic Equ. 62 (2017), no. 6, 767-785. https://doi.org/10.1080/17476933.2016.1243669
  2. C. O. Alves and A. R. da Silva, Multiplicity and concentration of positive solutions for a class of quasilinear problems through Orlicz-Sobolev space, J. Math. Phys. 57 (2016), no. 11, 111502, 22 pp. https://doi.org/10.1063/1.4966534
  3. A. Ambrosetti, H. Brezis, and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), no. 2, 519-543. https://doi.org/10.1006/jfan.1994.1078
  4. A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349-381. https://doi.org/10.1016/0022-1236(73)90051-7
  5. A. Azzollini, Minimum action solutions for a quasilinear equation, J. Lond. Math. Soc. (2) 92 (2015), no. 3, 583-595. https://doi.org/10.1112/jlms/jdv050
  6. A. Azzollini, P. d'Avenia, and A. Pomponio, Quasilinear elliptic equations in ℝN via variational methods and Orlicz-Sobolev embeddings, Calc. Var. Partial Differential Equations 49 (2014), no. 1-2, 197-213. https://doi.org/10.1007/s00526-012-0578-0
  7. M. Badiale, L. Pisani, and S. Rolando, Sum of weighted Lebesgue spaces and nonlinear elliptic equations, NoDEA Nonlinear Differential Equations Appl. 18 (2011), no. 4, 369-405. https://doi.org/10.1007/s00030-011-0100-y
  8. J.-H. Bae and Y.-H. Kim, Critical points theorems via the generalized Ekeland variational principle and its application to equations of p(x)-Laplace type in ℝN, Taiwanese J. Math. 23 (2019), no. 1, 193-229. https://doi.org/10.11650/tjm/181004
  9. G. Bonanno, G. Molica Bisci, and V. Radulescu, Existence of three solutions for a non-homogeneous Neumann problem through Orlicz-Sobolev spaces, Nonlinear Anal. 74 (2011), no. 14, 4785-4795. https://doi.org/10.1016/j.na.2011.04.049
  10. C. Brandle, E. Colorado, A. de Pablo, and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), no. 1, 39-71. https://doi.org/10.1017/S0308210511000175
  11. M. L. M. Carvalho, E. D. da Silva, and C. Goulart, Quasilinear elliptic problems with concave-convex nonlinearities, Commun. Contemp. Math. 19 (2017), no. 6, 1650050, 25 pp. https://doi.org/10.1142/S0219199716500504
  12. W. Chen and S. Deng, The Nehari manifold for nonlocal elliptic operators involving concave-convex nonlinearities, Z. Angew. Math. Phys. 66 (2015), no. 4, 1387-1400. https://doi.org/10.1007/s00033-014-0486-6
  13. N. Chorfi and V. D. Radulescu, Standing wave solutions of a quasilinear degenerate Schrodinger equation with unbounded potential, Electron. J. Qual. Theory Differ. Equ. 2016 (2016), Paper No. 37, 12 pp. https://doi.org/10.14232/ejqtde.2016.1.37
  14. N. T. Chung, Existence of solutions for a class of Kirchhoff type problems in Orlicz-Sobolev spaces, Ann. Polon. Math. 113 (2015), no. 3, 283-294. https://doi.org/10.4064/ap113-3-5
  15. P. Clement, B. de Pagter, G. Sweers, and F. de Thelin, Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces, Mediterr. J. Math. 1 (2004), no. 3, 241-267. https://doi.org/10.1007/s00009-004-0014-6
  16. P. Clement, M. Garcia-Huidobro, R. Manasevich, and K. Schmitt, Mountain pass type solutions for quasilinear elliptic equations, Calc. Var. Partial Differential Equations 11 (2000), no. 1, 33-62. https://doi.org/10.1007/s005260050002
  17. E. D. da Silva, M. L. M. Carvalho, J. V. Goncalves, and C. Goulart, Critical quasilinear elliptic problems using concave-convex nonlinearities, Ann. Mat. Pura Appl. (4) 198 (2019), no. 3, 693-726. https://doi.org/10.1007/s10231-018-0794-0
  18. I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-353. https://doi.org/10.1016/0022-247X(74)90025-0
  19. F. Fang and Z. Tan, Existence and multiplicity of solutions for a class of quasilinear elliptic equations: an Orlicz-Sobolev space setting, J. Math. Anal. Appl. 389 (2012), no. 1, 420-428. https://doi.org/10.1016/j.jmaa.2011.11.078
  20. F. Fang and Z. Tan, Existence of three solutions for quasilinear elliptic equations: an Orlicz-Sobolev space setting, Acta Math. Appl. Sin. Engl. Ser. 33 (2017), no. 2, 287-296. https://doi.org/10.1007/s10255-017-0659-0
  21. G. Figueiredo and J. A. Santos, Existence of least energy nodal solution with two nodal domains for a generalized Kirchhoff problem in an Orlicz-Sobolev space, Math. Nachr. 290 (2017), no. 4, 583-603. https://doi.org/10.1002/mana.201500286
  22. G. A. Seregin and J. Frehse, Regularity of solutions to variational problems of the deformation theory of plasticity with logarithmic hardening, in Proceedings of the St. Petersburg Mathematical Society, Vol. V, 127-152, Amer. Math. Soc. Transl. Ser. 2, 193, Amer. Math. Soc., Providence, RI, 1999. https://doi.org/10.1090/trans2/193/06
  23. M. Fuchs and V. Osmolovski, Variational integrals on Orlicz-Sobolev spaces, Z. Anal. Anwendungen 17 (1998), no. 2, 393-415. https://doi.org/10.4171/ZAA/829
  24. M. Fuchs and G. Seregin, Variational methods for fluids of Prandtl-Eyring type and plastic materials with logarithmic hardening, Math. Methods Appl. Sci. 22 (1999), no. 4, 317-351. https://doi.org/10.1002/(SICI)1099-1476(19990310)22:4<317::AID-MMA43>3.0.CO;2-A
  25. N. Fukagai and K. Narukawa, Nonlinear eigenvalue problem for a model equation of an elastic surface, Hiroshima Math. J. 25 (1995), no. 1, 19-41. http://projecteuclid.org/euclid.hmj/1206127823 https://doi.org/10.32917/hmj/1206127823
  26. N. Fukagai and K. Narukawa, On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems, Ann. Mat. Pura Appl. (4) 186 (2007), no. 3, 539-564. https://doi.org/10.1007/s10231-006-0018-x
  27. J.-P. Gossez, A strongly nonlinear elliptic problem in Orlicz-Sobolev spaces, in Nonlinear functional analysis and its applications, Part 1 (Berkeley, Calif., 1983), 455-462, Proc. Sympos. Pure Math., 45, Part 1, Amer. Math. Soc., Providence, RI, 1986.
  28. C. He and G. Li, The existence of a nontrivial solution to the p & q-Laplacian problem with nonlinearity asymptotic to up-1 at infinity in ℝN, Nonlinear Anal. 68 (2008), 1100-1119. https://doi.org/10.1016/j.na.2006.12.008
  29. K. Ho and I. Sim, Existence and multiplicity of solutions for degenerate p(x)-Laplace equations involving concave-convex type nonlinearities with two parameters, Taiwanese J. Math. 19 (2015), no. 5, 1469-1493. https://doi.org/10.11650/tjm.19.2015.5187
  30. L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on RN, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), no. 4, 787-809. https://doi.org/10.1017/S0308210500013147
  31. Y.-H. Kim, Existence and multiplicity of solutions to a class of fractional p-Laplacian equations of Schrodinger type with concave-convex nonlinearities in ℝN, Mathematics 8 (2020), 1792. https://doi.org/10.3390/math8101792
  32. J.-M. Kim, Y.-H. Kim, and J. Lee, Radially Symmetric Solutions for Quasilinear Elliptic Equations Involving Nonhomogeneous Operators in an Orlicz-Sobolev Space Setting, Acta Math. Sci. Ser. B (Engl. Ed.) 40 (2020), no. 6, 1679-1699. https://doi.org/10.1007/s10473-020-0605-8
  33. I. H. Kim, Y.-H. Kim, and K. Park, Existence and multiplicity of solutions for Schrodinger-Kirchhoff type problems involving the fractional p(·)-Laplacian in ℝN, Bound. Value Probl. 2020, Paper No. 121, 24 pp. https://doi.org/10.1186/s13661-020-01419-z
  34. J. I. Lee and Y.-H. Kim, Multiplicity of radially symmetric small energy solutions for quasilinear elliptic equations involving nonhomogeneous operators, Mathematics 8 (2020), 128. https://doi.org/10.3390/math8010128
  35. J. Lee, J.-M. Kim, and Y.-H. Kim, Existence and multiplicity of solutions for Kirchhoff-Schrodinger type equations involving p(x)-Laplacian on the entire space ℝN, Nonlinear Anal. Real World Appl. 45 (2019), 620-649. https://doi.org/10.1016/j.nonrwa.2018.07.016
  36. 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 ℝN, Bound. Value Probl. 2014 (2014), 261, 17 pp. https://doi.org/10.1186/s13661-014-0261-9
  37. M. Mihailescu and V. Radulescu, Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: an Orlicz-Sobolev space setting, J. Math. Anal. Appl. 330 (2007), no. 1, 416-432. https://doi.org/10.1016/j.jmaa.2006.07.082
  38. B. T. K. Oanh and D. N. Phuong, On multiplicity solutions for a non-local fractional p-Laplace equation, Complex Var. Elliptic Equ. 65 (2020), no. 5, 801-822. https://doi.org/10.1080/17476933.2019.1631287
  39. P. Pucci, M. Xiang, and B. Zhang, Multiple solutions for nonhomogeneous Schrodinger-Kirchhoff type equations involving the fractional p-Laplacian in ℝN, Calc. Var. Partial Differential Equations 54 (2015), no. 3, 2785-2806. https://doi.org/10.1007/s00526-015-0883-5
  40. M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, 146, Marcel Dekker, Inc., New York, 1991.
  41. M. Shao and A. Mao, Schrodinger-Poisson system with concave-convex nonlinearities, J. Math. Phys. 60 (2019), no. 6, 061504, 11 pp. https://doi.org/10.1063/1.5087490
  42. M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhauser Boston, Inc., Boston, MA, 1996. https://doi.org/10.1007/978-1-4612-4146-1
  43. T. Wu, Multiple positive solutions for a class of concave-convex elliptic problems in ℝN involving sign-changing weight, J. Funct. Anal. 258 (2010), no. 1, 99-131. https://doi.org/10.1016/j.jfa.2009.08.005
  44. M. Xiang, B. Zhang, and M. Ferrara, Multiplicity results for the non-homogeneous fractional p-Kirchhoff equations with concave-convex nonlinearities, Proc. A. 471 (2015), no. 2177, 20150034, 14 pp. https://doi.org/10.1098/rspa.2015.0034
  45. Q. Zhang and V. D. Radulescu, Double phase anisotropic variational problems and combined effects of reaction and absorption terms, J. Math. Pures Appl. (9) 118 (2018), 159-203. https://doi.org/10.1016/j.matpur.2018.06.015