Previous |  Up |  Next

Article

Keywords:
extrinsic mean curvature operator; Dirichlet problem; radial solution; positive solution; Leray-Schauder degree; critical point theory
Summary:
The paper surveys recent results obtained for the existence and multiplicity of radial solutions of Dirichlet problems of the type $$ \nabla \cdot \bigg (\frac {\nabla v}{\sqrt {1 - |\nabla v|^2}}\bigg ) = f(|x|,v) \quad \text {in} \ B_R,\quad u = 0 \quad \text {on} \ \partial B_R , $$ where $B_R$ is the open ball of center $0$ and radius $R$ in $\mathbb R^n$, and $f$ is continuous. Comparison is made with similar results for the Laplacian. Topological and variational methods are used and the case of positive solutions is emphasized. The paper ends with the case of a general domain.
References:
[1] Ambrosetti, A., Brézis, H., Cerami, G.: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122 (1994), 519-543. DOI 10.1006/jfan.1994.1078 | MR 1276168 | Zbl 0805.35028
[2] Ambrosetti, A., Rabinowitz, P. H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973), 349-381. DOI 10.1016/0022-1236(73)90051-7 | MR 0370183 | Zbl 0273.49063
[3] Bartnik, R., Simon, L.: Spacelike hypersurfaces with prescribed boundary values and mean curvature. Commun. Math. Phys. 87 (1982), 131-152. DOI 10.1007/BF01211061 | MR 0680653 | Zbl 0512.53055
[4] Bereanu, C., Jebelean, P.: Multiple critical points for a class of periodic lower semicontinuous functionals. Discrete Contin. Dyn. Syst. 33 (2013), 47-66. DOI 10.3934/dcds.2013.33.47 | MR 2972945 | Zbl 1281.34024
[5] Bereanu, C., Jebelean, P., Mawhin, J.: Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces. Proc. Am. Math. Soc. 137 (2009), 161-169. DOI 10.1090/S0002-9939-08-09612-3 | MR 2439437 | Zbl 1161.35024
[6] Bereanu, C., Jebelean, P., Mawhin, J.: Radial solutions for Neumann problems involving mean curvature operators in Euclidean and Minkowski spaces. Math. Nachr. 283 (2010), 379-391. DOI 10.1002/mana.200910083 | MR 2643019 | Zbl 1185.35113
[7] Bereanu, C., Jebelean, P., Mawhin, J.: Radial solutions for Neumann problems with $\phi$-Laplacians and pendulum-like nonlinearities. Discrete Contin. Dyn. Syst. 28 (2010), 637-648. DOI 10.3934/dcds.2010.28.637 | MR 2644761 | Zbl 1193.35083
[8] Bereanu, C., Jebelean, P., Mawhin, J.: Variational methods for nonlinear perturbations of singular $\phi$-Laplacians. Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 22 (2011), 89-111. DOI 10.4171/RLM/589 | MR 2799910
[9] Bereanu, C., Jebelean, P., Mawhin, J.: Multiple solutions for Neumann and periodic problems with singular $\phi$-Laplacian. J. Funct. Anal. 261 (2011), 3226-3246. DOI 10.1016/j.jfa.2011.07.027 | MR 2835997 | Zbl 1241.35076
[10] Bereanu, C., Jebelean, P., Mawhin, J.: Radial solutions of Neumann problems involving mean extrinsic curvature and periodic nonlinearities. Calc. Var. Partial Differ. Equ. 46 (2013), 113-122. DOI 10.1007/s00526-011-0476-x | MR 3016504 | Zbl 1262.35088
[11] Bereanu, C., Jebelean, P., Mawhin, J.: The Dirichlet problem with mean curvature operator in Minkowski space---a variational approach. Adv. Nonlinear Stud 14 (2014), 315-326. MR 3194356 | Zbl 1305.35030
[12] Bereanu, C., Jebelean, P., Şerban, C.: Nontrivial solutions for a class of one-parameter problems with singular $\phi$-Laplacian. Ann. Univ. Buchar., Math. Ser. 3(61) (2012), 155-162. MR 3034970 | Zbl 1274.35078
[13] Bereanu, C., Jebelean, P., Torres, P. J.: Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space. J. Funct. Anal. 264 (2013), 270-287. DOI 10.1016/j.jfa.2012.10.010 | MR 2995707
[14] Bereanu, C., Jebelean, P., Torres, P. J.: Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space. J. Funct. Anal. 265 (2013), 644-659. DOI 10.1016/j.jfa.2013.04.006 | MR 3062540 | Zbl 1285.35051
[15] Bereanu, C., Torres, P. J.: Existence of at least two periodic solutions of the forced relativistic pendulum. Proc. Am. Math. Soc. 140 (2012), 2713-2719. DOI 10.1090/S0002-9939-2011-11101-8 | MR 2910759 | Zbl 1275.34057
[16] Brézis, H.: Positive solutions of nonlinear elliptic equations in the case of critical Sobolev exponent. Nonlinear Partial Differential Equations and Their Applications, Collége de France Seminar, Vol. III, 129-146, Res. Notes Math. 70, Pitman, Boston, 1982. MR 0670270 | Zbl 0514.35031
[17] Brézis, H., Mawhin, J.: Periodic solutions of the forced relativistic pendulum. Differ. Integral Equ. 23 (2010), 801-810. MR 2675583 | Zbl 1240.34207
[18] Brézis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36 (1983), 437-477. DOI 10.1002/cpa.3160360405 | MR 0709644 | Zbl 0541.35029
[19] Coelho, I., Corsato, C., Obersnel, F., Omari, P.: Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation. Adv. Nonlinear Stud. 12 (2012), 621-638. MR 2976056 | Zbl 1263.34028
[20] Coelho, I., Corsato, C., Rivetti, S.: Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball. Topol. Methods Nonlinear Anal (to appear).
[21] Corsato, C., Obersnel, F., Omari, P., Rivetti, S.: Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space. J. Math. Anal. Appl. 405 (2013), 227-239. DOI 10.1016/j.jmaa.2013.04.003 | MR 3053503
[22] Hammerstein, A.: Nichtlineare Integralgleichungen nebst Anwendungen. Acta Math. 54 (1930), 117-176 German. DOI 10.1007/BF02547519 | MR 1555304
[23] Mawhin, J.: Semicoercive monotone variational problems. Bull. Cl. Sci., V. Sér., Acad. R. Belg. 73 (1987), 118-130. MR 0938142
[24] Mawhin, J.: Multiplicity of solutions of variational systems involving $\phi$-Laplacians with singular $\phi$ and periodic nonlinearities. Discrete Contin. Dyn. Syst. 32 (2012), 4015-4026. DOI 10.3934/dcds.2012.32.4015 | MR 2945817 | Zbl 1260.34076
[25] Mawhin, J., Jr., J. R. Ward, Willem, M.: Variational methods and semilinear elliptic equations. Arch. Ration. Mech. Anal. 95 (1986), 269-277. DOI 10.1007/BF00251362 | MR 0853968 | Zbl 0656.35044
[26] Mawhin, J., Willem, M.: Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations. J. Differ. Equations 52 (1984), 264-287. DOI 10.1016/0022-0396(84)90180-3 | MR 0741271 | Zbl 0557.34036
[27] Pohožaev, S. I.: On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$. Russian Dokl. Akad. Nauk SSSR 165 (1965), 36-39. MR 0192184
[28] Rabinowitz, P. H.: On a class of functionals invariant under a $\mathbb Z^n$ action. Trans. Am. Math. Soc. 310 (1988), 303-311. MR 0965755
[29] Szulkin, A.: Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3 (1986), 77-109. MR 0837231 | Zbl 0612.58011
[30] Willem, M.: Minimax Theorems. Progress in Nonlinear Differential Equations and Their Applications 24 Birkhäuser, Boston (1996). MR 1400007 | Zbl 0856.49001
Partner of
EuDML logo