Superlinear elliptic hemivariational inequalities

Year 2023, Volume: 52 Issue: 6 - Special Issue: Nonlinear Evolution Problems with Applications, 1631 - 1657, 03.11.2023

Yunru Bai , Leszek Gasinski , Nikolaos Papageorgiou

https://doi.org/10.15672/hujms.1173649

Abstract

We study a nonlinear nonhomogeneous Dirichlet problem with a nonsmooth potential which is superlinear but without satisfying the Ambrosetti-Rabinowitz condition. Using the nonsmooth critical point theory and critical groups we prove two multiplicity theorems producing three and five solutions respectively. In the second multiplicity theorem, we provide sign information for all the solutions and the solutions are ordered.

Keywords

Hemivariational inequality, Clarke subdifferential, nonsmooth critical point theory, critical groups, nodal solutions

References

• [1] S. Aizicovici, N.S. Papageorgiou and V. Staicu, On a p-superlinear Neumann p- Laplacian equation, Topol. Methods Nonlinear Anal. 34 (1), 111–130, 2009.
• [2] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14, 349–381, 1973.
• [3] T. Bartsch and Z. Liu, On a superlinear elliptic p-Laplacian equation, J. Differential Equations 198 (1), 149–175, 2004.
• [4] K.C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1), 102–129, 1981.
• [5] F.H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, Inc., New York, 1983.
• [6] J.-N. Corvellec, Morse theory for continuous functionals, J. Math. Anal. Appl. 196 (3), 1050–1072, 1995.
• [7] J.-N. Corvellec, On the second deformation lemma, Topol. Methods Nonlinear Anal. 17 (1), 55–66, 2001.
• [8] J.-N. Corvellec, A. Hantoute, Homotopical stability of isolated critical points of continuous functionals, Set-Valued Anal. 10 (2–3), 143–164, 2002.
• [9] M.E. Filippakis and N.S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian, J. Differential Equations 245 (7), 1883–1922, 2008.
• [10] L. Gasiński and N.S. Papageorgiou, Multiple solutions for nonlinear coercive problems with a nonhomogeneous differential operator and a nonsmooth potential, Set-Valued Var. Anal. 20 (3), 417–443, 2012.
• [11] L. Gasiński and N.S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Chapman & Hall/CRC, Boca Raton, FL, 2005.
• [12] L. Gasiński and N.S. Papageorgiou, Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton, FL, 2006.
• [13] L. Gasiński and N.S. Papageorgiou, Exercises in Analysis. Part 2. Nonlinear analysis, Springer, Cham, 2016.
• [14] S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I, Theory, Kluwer Academic Publishers, Dordrecht, 1997.
• [15] G.M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations, Comm. Partial Differential Equations 16 (2–3), 311–361, 1991.
• [16] S. Liu, Existence of solutions to a superlinear p-Laplacian equation, Electron. J. Differential Equations 66 (6), 2001.
• [17] S. Migórski, A. Ochal and M. Sofonea, Nonlinear inclusions and hemivariational inequalities, Springer, New York, 2013.
• [18] D. Mugnai and N.S. Papageorgiou, Resonant nonlinear Neumann problems with indefinite weight, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 (4), 729–788, 2012.
• [19] R.S. Palais, Homotopy theory of infinite dimensional manifolds, Topology 5, 1–16, 1966.
• [20] P.D. Panagiotopoulos, Hemivariational inequalities, Springer-Verlag, Berlin, 1993.
• [21] N.S. Papageorgiou and V.D. Rădulescu, Coercive and noncoercive nonlinear Neumann problems with indefinite potential, Forum Math. 28 (3), 545–571, 2016.
• [22] N.S. Papageorgiou, V.D. Rădulescu and D.D. Repovš, Nonhomogeneous hemivariational inequalities with indefinite potential and Robin boundary condition, J. Optim. Theory Appl. 175 (2), 293–323, 2017.
• [23] N.S. Papageorgiou, V.D. Rădulescu and D.D. Repovš, Nonlinear Analysis - Theory and Methods, Springer, Cham, 2019.
• [24] N.S. Papageorgiou and P. Winkert, Applied Nonlinear Functional Analysis, De Gruyter, Berlin, 2018.
• [25] N.S. Papageorgiou and P. Winkert, Nonlinear nonhomogeneous Dirichlet equations involving a superlinear nonlinearity, Results Math. 70 (1–2), 31–79, 2016.
• [26] N.S. Papageorgiou and P. Winkert, Positive solutions for singular anisotropic (p, q)-equations, J. Geom. Anal. 31 (12), 11849–11877, 2021.
• [27] P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser Verlag, Basel, 2007.
• [28] R.T. Rockafellar and R.J.B. Wets, Variational Analysis, Springer-Verlag, Berlin, 1998.
• [29] T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser/Springer Basel AG, Basel, 2013.
• [30] Z.Q. Wang, On a superlinear elliptic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1), 43–57, 1991.