Nonlocal Double Phase Implicit Obstacle Problems with Multivalued Boundary Conditions
Abstract.
In this paper, we consider a mixed boundary value problem with a nonhomogeneous, nonlinear differential operator (called double phase operator), a nonlinear convection term (a reaction term depending on the gradient), three multivalued terms, and an implicit obstacle constraint. Under very general assumptions on the data, we prove that the solution set of such an implicit obstacle problem is nonempty (so there is at least one solution) and weakly compact. The proof of our main result uses the Kakutani–Ky Fan fixed point theorem for multivalued operators along with the theory of nonsmooth analysis and variational methods for pseudomonotone operators.
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Acknowledgment.
The authors wish to thank the two knowledgeable referees for their constructive comments and remarks.
References
1.
B. Alleche and V. D. Rădulescu, Set-valued equilibrium problems with applications to Browder variational inclusions and to fixed point theory, Nonlinear Anal., 28 (2016), pp. 251–268.
2.
D. Aussel, A. Sultana, and V. Vetrivel, On the existence of projected solutions of quasi-variational inequalities and generalized Nash equilibrium problems, J. Optim. Theory Appl., 170 (2016), pp. 818–837.
3.
P. Baroni, M. Colombo, and G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal., 121 (2015), pp. 206–222.
4.
P. Baroni, M. Colombo, and G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations, 57 (2018), 62.
5.
G. Bonanno, D. Motreanu, and P. Winkert, Variational-hemivariational inequalities with small perturbations of nonhomogeneous Neumann boundary conditions, J. Math. Anal. Appl., 381 (2011), pp. 627–637.
6.
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
7.
S. Carl and V. K. Le, Multi-valued Variational Inequalities and Inclusions, Springer Monogr. Math., Springer, Cham, 2021.
8.
S. Carl, V. K. Le, and P. Winkert, Multi-valued variational inequalities for variable exponent double phase problems: Comparison and extremality results, Adv. Differential Equations, to appear.
9.
M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal., 218 (2015), pp. 219–273.
10.
M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal., 215 (2015), pp. 443–496.
11.
Á. Crespo-Blanco, L. Gasiński, P. Harjulehto, and P. Winkert, A new class of double phase variable exponent problems: Existence and uniqueness, J. Differential Equations, 323 (2022), pp. 182–228.
12.
C. De Filippis and G. Mingione, Manifold constrained non-uniformly elliptic problems, J. Geom. Anal., 30 (2020), pp. 1661–1723.
13.
C. De Filippis and G. Mingione, On the regularity of minima of non-autonomous functionals, J. Geom. Anal., 30 (2020), pp. 1584–1626.
14.
C. De Filippis and G. Mingione, Interpolative gap bounds for nonautonomous integrals, Anal. Math. Phys., 11 (2021), 117.
15.
C. De Filippis and G. Mingione, Lipschitz bounds and nonautonomous integrals, Arch. Ration. Mech. Anal., 242 (2021), pp. 973–1057.
16.
F. Faraci, D. Motreanu, and D. Puglisi, Positive solutions of quasi-linear elliptic equations with dependence on the gradient, Calc. Var. Partial Differential Equations, 54 (2015), pp. 525–538.
17.
F. Faraci and D. Puglisi, A singular semilinear problem with dependence on the gradient, J. Differential Equations, 260 (2016), pp. 3327–3349.
18.
G. M. Figueiredo and G. F. Madeira, Positive maximal and minimal solutions for non-homogeneous elliptic equations depending on the gradient, J. Differential Equations, 274 (2021), pp. 857–875.
19.
L. Gasiński and N. S. Papageorgiou, Positive solutions for nonlinear elliptic problems with dependence on the gradient, J. Differential Equations, 263 (2017), pp. 1451–1476.
20.
L. Gasiński and P. Winkert, Existence and uniqueness results for double phase problems with convection term, J. Differential Equations, 268 (2020), pp. 4183–4193.
21.
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), pp. 1037–1066.
22.
A. Granas and J. Dugundji, Fixed Point Theory, Universitext, Springer, New York, 2003.
23.
W. Han, S. Migórski, and M. Sofonea, A class of variational-hemivariational inequalities with applications to frictional contact problems, SIAM J. Math. Anal., 46 (2014), pp. 3891–3912, https://doi.org/10.1137/140963248.
24.
A. Iannizzotto and N. S. Papageorgiou, Existence of three nontrivial solutions for nonlinear Neumann hemivariational inequalities, Nonlinear Anal., 70 (2009), pp. 3285–3297.
25.
M. Kamenskii, V. Obukhovskii, and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter Ser. Nonlinear Anal. Appl. 7, Walter de Gruyter, Berlin, 2001.
26.
A. Lê, Eigenvalue problems for the \(p\)-Laplacian, Nonlinear Anal., 64 (2006), pp. 1057–1099.
27.
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod/Gauthier-Villars, Paris, 1969.
28.
W. Liu and G. Dai, Existence and multiplicity results for double phase problem, J. Differential Equations, 265 (2018), pp. 4311–4334.
29.
Z. Liu, D. Motreanu, and S. Zeng, Positive solutions for nonlinear singular elliptic equations of \(p\)-Laplacian type with dependence on the gradient, Calc. Var. Partial Differential Equations, 58 (2019), 28.
30.
S. A. Marano and P. Winkert, On a quasilinear elliptic problem with convection term and nonlinear boundary condition, Nonlinear Anal., 187 (2019), pp. 159–169.
31.
P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Ration. Mech. Anal., 105 (1989), pp. 267–284.
32.
P. Marcellini, Regularity and existence of solutions of elliptic equations with \(p,q\)-growth conditions, J. Differential Equations, 90 (1991), pp. 1–30.
33.
S. Migórski and A. Ochal, Quasi-static hemivariational inequality via vanishing acceleration approach, SIAM J. Math. Anal., 41 (2009), pp. 1415–1435, https://doi.org/10.1137/080733231.
34.
S. Migórski, A. A. Khan, and S. Zeng, Inverse problems for nonlinear quasi-variational inequalities with an application to implicit obstacle problems of \(p\)-Laplacian type, Inverse Problems, 35 (2019), 035004.
35.
S. Migórski, A. A. Khan, and S. Zeng, Inverse problems for nonlinear quasi-hemivariational inequalities with application to mixed boundary value problems, Inverse Problems, 36 (2020), 024006.
36.
S. Migórski, A. Ochal, and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities, Adv. Mech. Math. 26, Springer, New York, 2013.
37.
G. Mingione and V. Rădulescu, Recent developments in problems with nonstandard growth and nonuniform ellipticity, J. Math. Anal. Appl., 501 (2021), 125197.
38.
U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. Math., 3 (1969), pp. 510–585.
39.
Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Monogr. Textbooks Pure Appl. Math. 188, Marcel Dekker, New York, 1995.
40.
P. D. Panagiotopoulos, Hemivariational Inequalities, Springer, Berlin, Heidelberg, 1993.
41.
P. D. Panagiotopoulos, Nonconvex problems of semipermeable media and related topics, Z. Angew. Math. Mech., 65 (1985), pp. 29–36.
42.
N. S. Papageorgiou, V. D. Rădulescu, and D. D. Repovš, Positive solutions for nonlinear Neumann problems with singular terms and convection, J. Math. Pures Appl. (9), 136 (2020), pp. 1–21.
43.
V. D. Rădulescu, Isotropic and anisotropic double-phase problems: Old and new, Opuscula Math., 39 (2019), pp. 259–279.
44.
M. A. Ragusa and A. Tachikawa, Regularity for minimizers for functionals of double phase with variable exponents, Adv. Nonlinear Anal., 9 (2020), pp. 710–728.
45.
J. Simon, Régularité de la solution d’une équation non linéaire dans RN, in Journées d’Analyse Non Linéaire (Proc. Conf., Besançon, 1977), Lecture Notes in Math. 665, Springer, Berlin, 1978, pp. 205–227.
46.
J. Stefan, Über einige Probleme der Theorie der Wärmeleitung, Akad. Mat. Natur., 98 (1889), pp. 473–484.
47.
S. Zeng, Y. Bai, and L. Gasiński, Nonlinear nonhomogeneous obstacle problems with multivalued convection term, J. Geom. Anal., 32 (2022), 75.
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), 176.
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), 123997.
50.
S. Zeng, V. Rădulescu, and P. Winkert, Double phase obstacle problems with variable exponent, Adv. Differential Equations, 27 (2022), pp. 611–645.
51.
S. Zeng, V. Rădulescu, and P. Winkert, Double phase obstacle problems with multivalued convection and mixed boundary value conditions, Discrete Contin. Dyn. Syst. Ser. B, 28 (2023), pp. 999–1023.
52.
S. Zeng, V. D. Rădulescu, and P. Winkert, Double phase implicit obstacle problems with convection and multivalued mixed boundary value conditions, SIAM J. Math. Anal., 54 (2022), pp. 1898–1926, https://doi.org/10.1137/21M1441195.
53.
V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), pp. 675–710.
Information & Authors
Information
Published In
SIAM Journal on Mathematical Analysis
Pages: 877 - 912
DOI: 10.1137/22M1501040
ISSN (online): 1095-7154
Copyright
© 2024 Society for Industrial and Applied Mathematics.
History
Submitted: 6 June 2022
Accepted: 11 September 2023
Published online: 17 January 2024
Keywords
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Funding Information
Yulin Normal University: G2023ZK13
Romanian Ministry of Research, Innovation and Digitization: PNRR-III-C9-2022-I8
China Postdoctoral Science Foundation: 2022M721560
H2020 Marie Skłodowska-Curie Actions: 823731 CONMECH
National Natural Science Foundation of China: 12001478, 12371312
Natural Science Foundation of Guangxi Province: 2021GXNSFFA196004, GKAD23026237
Funding: This work was supported by Natural Science Foundation of Guangxi grants 2021GXNSFFA196004 and GKAD23026237, NNSF of China grants 12001478 and 12371312, China Postdoctoral Science Foundation funded project 2022M721560, Startup Project of Doctor Scientific Research of Yulin Normal University G2023ZK13, and the European Union’s Horizon 2020 Research and Innovation Programme under Marie Sklodowska-Curie grant agreement 823731 CONMECH. It was also supported by the project cooperation between Guangxi Normal University and Yulin Normal University. The research of the second author was supported by the grant “Nonlinear Differential Systems in Applied Sciences” of the Romanian Ministry of Research, Innovation and Digitization, within PNRR-III-C9-2022-I8 (grant 22).
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