\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Double phase obstacle problems with multivalued convection and mixed boundary value conditions

  • *Corresponding author: Vicenţiu D. Rădulescu

    *Corresponding author: Vicenţiu D. Rădulescu 
Abstract Full Text(HTML) Related Papers Cited by
  • In this paper, we consider a mixed boundary value problem with a double phase partial differential operator, an obstacle effect and a multivalued reaction convection term. Under very general assumptions, an existence theorem for the mixed boundary value problem under consideration is proved by using a surjectivity theorem for multivalued pseudomonotone operators together with the approximation method of Moreau-Yosida. Then, we introduce a family of the approximating problems without constraints corresponding to the mixed boundary value problem. Denoting by $ \mathcal S $ the solution set of the mixed boundary value problem and by $ \mathcal S_n $ the solution sets of the approximating problems, we establish the following convergence relation

    $ \begin{align*} \emptyset\neq w-\limsup\limits_{n\to\infty}{\mathcal S}_n = s-\limsup\limits_{n\to\infty}{\mathcal S}_n\subset \mathcal S, \end{align*} $

    where $ w $-$ \limsup_{n\to\infty}\mathcal S_n $ and $ s $-$ \limsup_{n\to\infty}\mathcal S_n $ stand for the weak and the strong Kuratowski upper limit of $ \mathcal S_n $, respectively.

    Mathematics Subject Classification: Primary: 35J20, 35J25, 35J60.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] C. O. AlvesP. Garain and V. D. Rădulescu, High perturbations of quasilinear problems with double criticality, Math. Z., 299 (2021), 1875-1895.  doi: 10.1007/s00209-021-02757-z.
    [2] V. Ambrosio and V. D. Rădulescu, Fractional double-phase patterns: Concentration and multiplicity of solutions, J. Math. Pures Appl. (9), 142 (2020), 101-145.  doi: 10.1016/j.matpur.2020.08.011.
    [3] J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2, Birkhäuser Boston, Inc., Boston, MA, 1990.
    [4] A. BahrouniV. D. Rădulescu and D. D. Repovš, Double phase transonic flow problems with variable growth: Nonlinear patterns and stationary waves, Nonlinearity, 32 (2019), 2481-2495.  doi: 10.1088/1361-6544/ab0b03.
    [5] A. BahrouniV. D. Rădulescu and P. Winkert, A critical point theorem for perturbed functionals and low perturbations of differential and nonlocal systems, Adv. Nonlinear Stud., 20 (2020), 663-674.  doi: 10.1515/ans-2020-2095.
    [6] A. Bahrouni, V. D. Rădulescu and P. Winkert, Double phase problems with variable growth and convection for the Baouendi-Grushin operator, Z. Angew. Math. Phys., 71 (2020), 15 pp. doi: 10.1007/s00033-020-01412-7.
    [7] P. BaroniM. Colombo and G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal., 121 (2015), 206-222.  doi: 10.1016/j.na.2014.11.001.
    [8] P. Baroni and M. Colombo amd G. Mingione, Non-autonomous functionals, borderline cases and related function classes, St. Petersburg Math. J., 27 (2016), 347-379.  doi: 10.1090/spmj/1392.
    [9] P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations, 57 (2018), 48 pp. doi: 10.1007/s00526-018-1332-z.
    [10] J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121, Cambridge University Press, Cambridge, 1996.
    [11] M. CenceljV. D. Rădulescu and D. D. Repovš, Double phase problems with variable growth, Nonlinear Anal., 177 (2018), 270-287.  doi: 10.1016/j.na.2018.03.016.
    [12] F. Colasuonno and M. Squassina, Eigenvalues for double phase variational integrals, Ann. Mat. Pura Appl. (4), 195 (2016), 1917-1959.  doi: 10.1007/s10231-015-0542-7.
    [13] M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal., 218 (2015), 219-273.  doi: 10.1007/s00205-015-0859-9.
    [14] M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal., 215 (2015), 443-496.  doi: 10.1007/s00205-014-0785-2.
    [15] Á. Crespo-BlancoL. GasińskiP. Harjulehto and P. Winkert, A new class of double phase variable exponent problems: Existence and uniqueness, J. Differential Equations, 323 (2022), 182-228.  doi: 10.1016/j.jde.2022.03.029.
    [16] G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Grundlehren der Mathematischen Wissenschaften, 219, Springer-Verlag, Berlin-New York, 1976.
    [17] C. Farkas and P. Winkert, An existence result for singular Finsler double phase problems, J. Differential Equations, 286 (2021), 455-473.  doi: 10.1016/j.jde.2021.03.036.
    [18] M. Galewski, Basic Monotonicity Methods with Some Applications, Compact Textbooks in Mathematics, Birkhäuser/Springer, Cham, 2021. doi: 10.1007/978-3-030-75308-5.
    [19] L. Gasiński and N. S. Papageorgiou, Constant sign and nodal solutions for superlinear double phase problems, Adv. Calc. Var., 14 (2021), 613-626.  doi: 10.1515/acv-2019-0040.
    [20] L. Gasiński and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Series in Mathematical Analysis and Applications, 8, Chapman & Hall/CRC, Boca Raton, FL, 2005.
    [21] L. Gasiński and N. S. Papageorgiou, Positive solutions for nonlinear elliptic problems with dependence on the gradient, J. Differential Equations, 263 (2017), 1451-1476.  doi: 10.1016/j.jde.2017.03.021.
    [22] L. Gasiński and P. Winkert, Constant sign solutions for double phase problems with superlinear nonlinearity, Nonlinear Anal., 195 (2020), 9 pp. doi: 10.1016/j.na.2019.111739.
    [23] L. Gasiński and P. Winkert, Existence and uniqueness results for double phase problems with convection term, J. Differential Equations, 268 (2020), 4183-4193.  doi: 10.1016/j.jde.2019.10.022.
    [24] L. Gasiński and P. Winkert, Sign changing solution for a double phase problem with nonlinear boundary condition via the Nehari manifold, J. Differential Equations, 274 (2021), 1037-1066.  doi: 10.1016/j.jde.2020.11.014.
    [25] F. Giannessi and A. A. Khan, Regularization of non-coercive quasi variational inequalities, Control Cybernet., 29 (2000), 91-110. 
    [26] A. Lê, Eigenvalue problems for the $p$-Laplacian, Nonlinear Anal., 64 (2006), 1057-1099.  doi: 10.1016/j.na.2005.05.056.
    [27] V. K. Le, A range and existence theorem for pseudomonotone perturbations of maximal monotone operators, Proc. Amer. Math. Soc., 139 (2011), 1645-1658.  doi: 10.1090/S0002-9939-2010-10594-4.
    [28] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris; Gauthier-Villars, Paris, 1969.
    [29] W. Liu and G. Dai, Existence and multiplicity results for double phase problem, J. Differential Equations, 265 (2018), 4311-4334.  doi: 10.1016/j.jde.2018.06.006.
    [30] G. Marino and P. Winkert, Existence and uniqueness of elliptic systems with double phase operators and convection terms, J. Math. Anal. Appl., 492 (2020), 13 pp. doi: 10.1016/j.jmaa.2020.124423.
    [31] S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, 26, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5.
    [32] G. Mingione and V. Rădulescu, Recent developments in problems with nonstandard growth and nonuniform ellipticity, J. Math. Anal. Appl., 501 (2021), 41 pp. doi: 10.1016/j.jmaa.2021.125197.
    [33] N. S. Papageorgiou and S. T. Kyritsi-Yiallourou, Handbook of Applied Analysis, Advances in Mechanics and Mathematics, 19, Springer, New York, 2009. doi: 10.1007/b120946.
    [34] N. S. PapageorgiouV. D. Rădulescu and D. D. Repovš, Double-phase problems and a discontinuity property of the spectrum, Proc. Amer. Math. Soc., 147 (2019), 2899-2910.  doi: 10.1090/proc/14466.
    [35] N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Double-phase problems with reaction of arbitrary growth, Z. Angew. Math. Phys., 69 (2018), 21 pp. doi: 10.1007/s00033-018-1001-2.
    [36] N. S. PapageorgiouV. D. Rădulescu and D. D. Repovš, Existence and multiplicity of solutions for double-phase Robin problems, Bull. Lond. Math. Soc., 52 (2020), 546-560.  doi: 10.1112/blms.12347.
    [37] N. S. PapageorgiouC. Vetro and F. Vetro, Continuous spectrum for a two phase eigenvalue problem with an indefinite and unbounded potential, J. Differential Equations, 268 (2020), 4102-4118.  doi: 10.1016/j.jde.2019.10.026.
    [38] N. S. Papageorgiou, C. Vetro and F. Vetro, Multiple solutions for parametric double phase Dirichlet problems, Commun. Contemp. Math., 23 (2021), 18 pp. doi: 10.1142/S0219199720500066.
    [39] N. S. PapageorgiouC. Vetro and F. Vetro, Nonlinear multivalued Duffing systems, J. Math. Anal. Appl., 468 (2018), 376-390.  doi: 10.1016/j.jmaa.2018.08.024.
    [40] N. S. PapageorgiouC. Vetro and F. Vetro, Relaxation for a class of control systems with unilateral constraints, Acta Appl. Math., 167 (2020), 99-115.  doi: 10.1007/s10440-019-00270-4.
    [41] N. S. Papageorgiou and P. Winkert, Applied Nonlinear Functional Analysis. An Introduction, De Gruyter Graduate, De Gruyter, Berlin, 2018.
    [42] K. Perera and M. Squassina, Existence results for double-phase problems via Morse theory, Commun. Contemp. Math., 20 (2018), 14 pp. doi: 10.1142/S0219199717500237.
    [43] V. D. Rădulescu, Isotropic and anistropic double-phase problems: Old and new, Opuscula Math., 39 (2019), 259-279.  doi: 10.7494/OpMath.2019.39.2.259.
    [44] J. F. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland Mathematics Studies, 134, North-Holland Publishing Co., Amsterdam, 1987.
    [45] C. Vetro, Parametric and nonparametric $A$-Laplace problems: Existence of solutions and asymptotic analysis, Asymptot. Anal., 122 (2021), 105-118.  doi: 10.3233/ASY-201612.
    [46] C. Vetro and F. Vetro, On problems driven by the $(p(\cdot), q(\cdot))$-Laplace operator, Mediterr. J. Math., 17 (2020), 11 pp. doi: 10.1007/s00009-019-1448-1.
    [47] S. ZengY. BaiL. Gasiński and P. Winkert, Convergence analysis for double phase obstacle problems with multivalued convection term, Adv. Nonlinear Anal., 10 (2021), 659-672.  doi: 10.1515/anona-2020-0155.
    [48] S. Zeng, Y. Bai, L. Gasiński and P. Winkert, Existence results for double phase implicit obstacle problems involving multivalued operators, Calc. Var. Partial Differential Equations, 59 (2020), 18 pp. doi: 10.1007/s00526-020-01841-2.
    [49] S. Zeng, L. Gasiński, P. Winkert and Y. Bai, Existence of solutions for double phase obstacle problems with multivalued convection term, J. Math. Anal. Appl., 501 (2021), 12 pp. doi: 10.1016/j.jmaa.2020.123997.
    [50] Q. Zhang and V. D. Rădulescu, Double phase anisotropic variational problems and combined effects of reaction and absorption terms, J. Math. Pures Appl. (9), 118 (2018), 159-203.  doi: 10.1016/j.matpur.2018.06.015.
    [51] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675-710. 
    [52] V. V. Zhikov, On variational problems and nonlinear elliptic equations with nonstandard growth conditions, J. Math. Sci. (N.Y.), 173 (2011), 463-570.  doi: 10.1007/s10958-011-0260-7.
  • 加载中
SHARE

Article Metrics

HTML views(1083) PDF downloads(219) Cited by(0)

Access History

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return