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Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space. (English) Zbl 1285.35051
Summary: We study the Dirichlet problem with mean curvature operator in Minkowski space $\text{div}\left( \frac{\nabla v}{\sqrt{1 - |\nabla v|^2}}\right) + {\lambda}[{\mu}(|x|)v^q] = 0 \quad \text{in } \mathcal B(R), \quad v = 0 \quad \text{on } \partial \mathcal B(R),$ where $${\lambda} > 0$$ is a parameter, $$q > 1$$, $$R > 0$$, $${\mu} : [0, \infty) \to \mathbb R$$ is continuous, strictly positive on $$(0, \infty)$$ and $$\mathcal B(R) = \{x \in \mathbb R^N : |x| < R\}$$. Using upper and lower solutions and Leray-Schauder degree type arguments, we prove that there exists $${\Lambda} > 0$$ such that the problem has zero, at least one or at least two positive radial solutions according to $${\lambda} \in (0, {\Lambda}), {\lambda} = {\Lambda}$$ or $${\lambda} > {\Lambda}$$. Moreover, $${\Lambda}$$ is strictly decreasing with respect to $$R$$.

##### MSC:
 35J93 Quasilinear elliptic equations with mean curvature operator
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##### References:
 [1] Alías, L. J.; Palmer, B., On the Gaussian curvature of maximal surfaces and the Calabi-Bernstein theorem, Bull. London Math. Soc., 33, 454-458, (2001) · Zbl 1041.53038 [2] Ambrosetti, A.; Garcia Azorero, J.; Peral, I., Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal., 137, 219-242, (1996) · Zbl 0852.35045 [3] Bartnik, R.; Simon, L., Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys., 87, 131-152, (1982-1983) · Zbl 0512.53055 [4] Bereanu, C.; Jebelean, P.; Mawhin, J., Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces, Proc. Amer. Math. Soc., 137, 161-169, (2009) · Zbl 1161.35024 [5] Bereanu, C.; Jebelean, P.; Mawhin, J., Radial solutions for Neumann problems involving mean curvature operators in Euclidean and Minkowski spaces, Math. Nachr., 283, 379-391, (2010) · Zbl 1185.35113 [6] Bereanu, C.; Jebelean, P.; Mawhin, J., Multiple solutions for Neumann and periodic problems with singular ϕ-Laplacian, J. Funct. Anal., 261, 3226-3246, (2011) · Zbl 1241.35076 [7] Bereanu, C.; Jebelean, P.; Mawhin, J., Radial solutions of Neumann problems involving mean extrinsic curvature and periodic nonlinearities, Calc. Var. Partial Differential Equations, 46, 113-122, (2013) · Zbl 1262.35088 [8] Bereanu, C.; Jebelean, P.; Torres, P. J., Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space, J. Funct. Anal., 264, 270-287, (2013) · Zbl 1336.35174 [9] Cheng, S.-Y.; Yau, S.-T., Maximal spacelike hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math., 104, 407-419, (1976) · Zbl 0352.53021 [10] Clément, P.; Manásevich, R.; Mitidieri, E., On a modified capillary equation, J. Differential Equations, 124, 343-358, (1996) · Zbl 0841.34038 [11] Coffman, C. V.; Ziemer, W. K., A prescribed mean curvature problem on domains without radial symmetry, SIAM J. Math. Anal., 22, 982-990, (1991) · Zbl 0741.35010 [12] Franchi, B.; Lanconelli, E.; Serrin, J., Existence and uniqueness of nonnegative solutions of quasilinear equations in $$\mathbb{R}^n$$, Adv. Math., 118, 177-243, (1996) · Zbl 0853.35035 [13] Gidas, B.; Ni, W. M.; Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68, 209-243, (1979) · Zbl 0425.35020 [14] Habets, P.; Omari, P., Multiple positive solutions of a one-dimensional prescribed mean curvature problem, Commun. Contemp. Math., 9, 701-730, (2007) · Zbl 1153.34015 [15] Krasnoselʼskii, M. A., Positive solutions of operator equations, (1964), Noordhoff Groningen [16] López, R., Stationary surfaces in Lorentz-Minkowski space, Proc. Roy. Soc. Edinburgh Sect. A, 138, 1067-1096, (2008) · Zbl 1175.53030 [17] Nakao, M., A bifurcation problem for a quasi-linear elliptic boundary value problem, Nonlinear Anal., 14, 251-262, (1990) · Zbl 0693.35005 [18] Ni, W. M., A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., 31, 801-807, (1982) · Zbl 0515.35033 [19] Ni, W. M.; Serrin, J., Non-existence theorems for quasilinear partial differential equations, Rend. Circ. Mat. Palermo Suppl., 8, 171-185, (1985) · Zbl 0625.35028 [20] Serrin, J., Positive solutions of a prescribed mean curvature problem, (Calculus of Variations and Differential Equations, Lecture Notes in Math., vol. 1340, (1988), Springer-Verlag New York) · Zbl 0673.35026 [21] Treibergs, A. E., Entire spacelike hypersurfaces of constant mean curvature in Minkowski space, Invent. Math., 66, 39-56, (1982) · Zbl 0483.53055
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