On the character of nonlinearity discontinuities in eigenvalue problems for elliptic equations

D. K. Potapov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 128 (2012) no. 3, pp. 188-190

Résumé

The eigenvalue problems for equations of elliptic type with discontinuous by the phase variable nonlinearities are considered. The character of nonlinearity discontinuities is investigated. In this paper the restrictions on discontinuity points of nonlinearity are weaker than in works of other authors.

Détail

Keywords: boundary value problems, eigenvalue problems, discontinuous nonlinearity, character of discontinuities.
Mots-clés : elliptic equations

Bibliographie

[1] Pavlenko V. N., Potapov D. K., “Existence of a ray of eigenvalues for equations with discontinuous operators”, Siberian Math. J., 42:4 (2001), 766–773 | DOI | MR | Zbl

[2] Pavlenko V. N., Potapov D. K., “Approximation of boundary-value problems of elliptic type with a spectral parameter and discontinuous nonlinearity”, Russian Math. (Iz. VUZ), 49:4 (2005), 46–52 | MR | Zbl

[3] Potapov D. K., “On an upper bound for the value of the bifurcation parameter in eigenvalue problems for elliptic equations with discontinuous nonlinearities”, Differ. Equ., 44:5 (2008), 737–739 | DOI | MR | Zbl

[4] Potapov D. K., “On the eigenvalue set structure for higher-order equations of elliptic type with discontinuous nonlinearities”, Differ. Equ., 46:1 (2010), 155–157 | DOI | MR | Zbl

[5] Potapov D. K., “Estimations of a differential operator in spectral parameter problems for elliptic equations with discontinuous nonlinearities”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2010, no. 5(21), 268–271 | DOI

[6] Potapov D. K., “Bifurcation problems for equations of elliptic type with discontinuous nonlinearities”, Math. Notes, 90:2 (2011), 260–264 | DOI | DOI | MR | Zbl

[7] Potapov D. K., “On number of solutions in eigenvalue problems for elliptic equations with discontinuous nonlinearities”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2012, no. 1(26), 251–255 | DOI | MR

[8] Marano S. A., “Elliptic eigenvalue problems with highly discontinuous nonlinearities”, Proc. Amer. Math. Soc., 125:10 (1997), 2953–2961 | DOI | MR | Zbl

[9] Marano S. A., Motreanu D., “On a three critical points theorem for non-differentiable functions and applications to nonlinear boundary value problems”, Nonlinear Anal., 48:1 (2002), 37–52 | DOI | MR | Zbl

[10] Bonanno G., “Some remarks on a three critical points theorem”, Nonlinear Anal., 54:4 (2003), 651–665 | DOI | MR | Zbl

[11] Bonanno G., Giovannelli N., “An eigenvalue Dirichlet problem involving the $p$-Laplacian with discontinuous nonlinearities”, J. Math. Anal. Appl., 308:2 (2005), 596–604 | DOI | MR | Zbl

[12] Zhang G., Liu S., “Three symmetric solutions for a class of elliptic involving the $p$-Laplacian with discontinuous nonlinearities in $\mathbb R^n$”, Nonlinear Anal., 67:7 (2007), 2232–2239 | DOI | MR | Zbl

[13] Bonanno G., Candito P., “Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities”, J. Differential Eq., 244:12 (2008), 3031–3059 | DOI | MR | Zbl

[14] Bonanno G., Chinni A., “Discontinuous elliptic problems involving the $p(x)$-Laplacian”, Math. Nachr., 284:5–6 (2011), 639–652 | DOI | MR | Zbl

[15] Gol'dshtik M. A., “A mathematical model of separated flows in an incompressible liquid”, Soviet. Math. Dokl., 7 (1963), 1090–1093

[16] Potapov D. K., “A mathematical model of separated flows in an incompressible fluid”, Izv. RAEN. Ser. MMMIU, 8:3–4 (2004), 163–170

[17] Potapov D. K., “Continuous approximations of Gol'dshtik's model”, Math. Notes, 87:2 (2010), 244–247 | DOI | DOI | MR | Zbl

[18] Potapov D. K., “Approximation to the Dirichlet problem for a higher-order elliptic equations with a spectral parameter and a discontinuous nonlinearity”, Differ. Equ., 43:7 (2007), 1031–1032 | DOI | MR | Zbl