On a quasilinear elliptic problem involving the 1-biharmonic operator and a Strauss type compactness result

Hurtado, Elard J.; Pimenta, Marcos T.O.; Miyagaki, Olimpio H.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 86.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

In this paper we prove the compactness of the embeddings of the space of radially symmetric functions of BL(ℝ$$) into some Lebesgue spaces. In order to do so we prove a regularity result for solutions of the Poisson equation with measure data in ℝ$$, as well as a version of the Radial Lemma of Strauss to the space BL(ℝ$$). An application is presented involving a quasilinear elliptic problem of higher-order, where variational methods are used to find the solutions.

[1] C.O. Alves and M.T.O. Pimenta, On existence and concentration of solutions to a class of quasilinear problems involving the 1-Laplace operator. Calc. Var. Partial Differ. Equ. 56 (2017) 143. | DOI | MR | Zbl

[2] F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, Minimizing total variation flow. C. R. Acad. Sci., Paria, Sr. I, Math. 331 (2000) 867–872. | DOI | MR | Zbl

[3] F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, Minimizing total variation flow. Differ. Integr. Equ. 14 (2001) 321–360. | MR | Zbl

[4] F. Andreu, C. Ballester, V. Caselles and J.M. Mazón, The Dirichlet problem for the total variation flow. J. Funct. Anal. 180 (2001) 347–403. | DOI | MR | Zbl

[5] F. Andreu, V. Caselles and J.M. Mazón, Parabolic quasilinear equations minimizing linear growth functionals. In Vol. 233 of Progress in Mathematics. Birkhäuser Verlag, Basel. (2004). | MR | Zbl

[6] G. Anzellotti, The Euler equation for functionals with linear growth. Trans. Am. Math. Soc. 290 (1985) 483–501. | DOI | MR | Zbl

[7] H. Attouch, G. Buttazzo and G. Michaille, Variational analysis in Sobolev and BV spaces: applications to PDEs and optimization. MPS-SIAM, Philadelphia (2006). | MR | Zbl

[8] Z. Balogh and A. Kristály, Lions-type compactness and Rubik actions on the Heisenberg group. Calc. Var. Partial Differ. Equ. 48 (2013) 89–109. | DOI | MR | Zbl

[9] S. Barile and M.T.O. Pimenta, Some existence results of bounded variation solutions to 1-biharmonic problems. J. Math. Anal. Appl. 463 (2018) 726–743. | DOI | MR | Zbl

[10] H. Brezis and A. Ponce, Kato’s inequality up to the boundary. Commun. Contemp. Math. 10 (2008) 1217–1241. | DOI | MR | Zbl

[11] D. Cassani, B. Ruf and C. Tarsi, Best constants in a borderline case of second-order Moser type inequalities. Ann. Inst. Henri Poincaré - AN 27 (2010) 73–93. | Numdam | MR | Zbl

[12] K. Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl. 80 (1981) 102–129. | DOI | MR | Zbl

[13] F. Clarke, Generalized gradients and applications. Trans. Am. Math. Soc. 205 (1975) 247–262. | DOI | MR | Zbl

[14] F. Demengel, Théorèmes d’existence pour des équations avec l’opérateur ”1-laplacien”, première valeur propre pour $-{\Delta }_1$. C. R. Math. Acad. Sci. Paris 334 (2002) 1071–1076. | DOI | MR | Zbl

[15] I. Ekeland and R. Teman, Convex analysis and variational problems. North-, Amsterdam (1976). | MR | Zbl

[16] G.M. Figueiredo and M.T.O. Pimenta, Nehari method for locally Lipschitz functionals with examples in problems in the space of bounded variation functions. NoDEA Nonlinear Differ. Equ. Appl. 25 (2018) 47. | DOI | MR | Zbl

[17] G.M. Figueiredo and M.T.O. Pimenta, Existence of bounded variation solution for a 1-Laplacian problem with vanishing potentials. J. Math. Anal. Appl. 459 (2018) 861–878. | DOI | MR | Zbl

[18] G.M. Figueiredo and M.T.O. Pimenta, Strauss’ and Lions’ type results in $BV\left({ℝ}^N\right)$ with an application to an 1-Laplacian problem. Milan J. Math. 86 (2018) 15–30. | DOI | MR | Zbl

[19] B. Kawohl and F. Schuricht, Dirichlet problems for the 1-Laplace operator, including the eigenvalue problem. Commun. Contemp. Math. 9 (2007) 515–543. | DOI | MR | Zbl

[20] J. Kobayashi and M. Ôtani, The principle of symmetric criticality for non-differentiable mappings. J. Funct. Anal. 214 (2004) 428–449. | DOI | MR | Zbl

[21] J.M. Mazón and S. Segura De León, The Dirichlet problem of a singular elliptic equation arising in the level set formulation of the inverse mean curvature flow. Adv. Calc. Var. 6 (2013) 123–164. | DOI | MR | Zbl

[22] J.M. Mazón, J. Rossi and S. Segura De León, Functions of least gradient and 1-harmonic functions. Indiana Univ. Math. J. 63 (2014) 1067–1084. | DOI | MR | Zbl

[23] A. Mercaldo, J. Rossi, S. Segura De León and C. Trombetti, Anisotropic $pq$−Laplacian equations when $p$ goes to $1$. Nonlinear Anal. 73 (2010) 3546–3560. | DOI | MR | Zbl

[24] A. Mercaldo, S. Segura De León and C. Trombetti, On the solutions to 1-Laplacian equation with $L^1$ data. J. Funct. Anal. 256 (2009) 2387–2416. | DOI | MR | Zbl

[25] G. Mingione, The Calderón-Zygmund theory for elliptic problems with measure data. Ann. Sci. Norm. Super. Pisa C1 6 (2007) 195–261. | Numdam | MR | Zbl

[26] A. Obereder, S. Osher and O. Scherzer, On the use of dual norms in bounded variation type regularization. Geometr. Prop. Incomplete data Comput. Imag. Vis. 31 (2006) 373–390.

[27] E. Parini, B. Ruf and C. Tarsi, The eigenvalue problem for the 1-biharmonic problem. Ann. Sc. Norm. Super. Pisa C1 13 (2014) 307–322. | MR | Zbl

[28] E. Parini, B. Ruf and C. Tarsi, Limiting Sobolev inequalities and the 1-biharmonic operator. Adv. Nonlinear Anal. 3 (2014) s19–s36. | DOI | MR | Zbl

[29] E. Parini, B. Ruf and C. Tarsi, Higher-order functional inequalities related to the clamped 1-biharmonic operator. Ann. Matemat. 194 (2015) 1835–1858. | MR | Zbl

[30] A. Ponce, Elliptic PDEs, measures and capacities. From the Poisson equations to nonlinear Thomas-Fermi problems. EMS Tracts Math. Zürich 23 (2016). | MR | Zbl

[31] P. Rabinowitz, On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43 (1992) 270–291. | DOI | MR | Zbl

[32] M. Squassina, On Palais’ principle for non-smooth functionals. Nonlinear Anal. 74 (2011) 3786–3804. | DOI | MR | Zbl

[33] W.A. Strauss, Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55 (1977) 149–162. | DOI | MR | Zbl