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On the eigenvalue problem involving the weighted $$p$$-Laplacian in radially symmetric domains. (English) Zbl 1401.35233
Summary: We investigate the following eigenvalue problem $\begin{cases} -\operatorname{div}(L(x) |\nabla u |^{p - 2}\nabla u) = \lambda K(x) | u |^{p - 2} u \quad &\text{in } A_{R_1}^{R_2}, \\ u = 0 & \text{on } \partial A_{R_1}^{R_2}, \end{cases}$ where $$A_{R_1}^{R_2} : = \{x \in \mathbb R^N : R_1 < | x | < R_2 \}$$ $$(0 < R_1 < R_2 \leq \infty)$$, $$\lambda > 0$$ is a parameter, the weights $$L$$ and $$K$$ are measurable with $$L$$ positive a.e. in $$A_{R_1}^{R_2}$$ and $$K$$ possibly sign-changing in $$A_{R_1}^{R_2}$$. We prove the existence of the first eigenpair and discuss the regularity and positiveness of eigenfunctions. The asymptotic estimates for $$u(x)$$ and $$\nabla u(x)$$ as $$| x | \rightarrow R_1^+$$ or $$R_2^-$$ are also investigated.
##### MSC:
 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 35J62 Quasilinear elliptic equations 35B40 Asymptotic behavior of solutions to PDEs 35B50 Maximum principles in context of PDEs 35J20 Variational methods for second-order elliptic equations 35J25 Boundary value problems for second-order elliptic equations 35P15 Estimates of eigenvalues in context of PDEs
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##### References:
 [1] Agudelo, O.; Drábek, P., Anisotropic semipositone quasilinear problems, J. Math. Anal. Appl., 452, 2, 1145-1167, (2017) · Zbl 1373.35147 [2] Anoop, T. V.; Drábek, P.; Sankar, L.; Sasi, S., Antimaximum principle in exterior domains, Nonlinear Anal., 130, 241-254, (2016) · Zbl 1329.35158 [3] Anoop, T. V.; Drábek, P.; Sasi, S., Weighted quasilinear eigenvalue problems in exterior domains, Calc. Var. Partial Differential Equations, 53, 3-4, 961-975, (2015) · Zbl 1333.35138 [4] Autuori, G.; Colasuonno, F.; Pucci, P., On the existence of stationary solutions for higher-order p-Kirchhoff problems, Commun. Contemp. Math., 16, (2014) · Zbl 1325.35129 [5] Chhetri, M.; Drábek, P., Principal eigenvalue of p-Laplacian operator in exterior domain, Results Math., 66, 3-4, 461-468, (2014) · Zbl 1327.35285 [6] Colasuonnoa, F.; Pucci, P.; Varga, C., Multiple solutions for an eigenvalue problem involving p-Laplacian type operators, Nonlinear Anal., 75, 4496-4512, (2012) · Zbl 1251.35059 [7] DiBenedetto, E., $$C^{1 + \alpha}$$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7, 8, 827-850, (1983) · Zbl 0539.35027 [8] Drábek, P.; Kufner, A.; Kuliev, K., Half-linear Sturm-Liouville problem with weights: asymptotic behavior of eigenfunctions, Proc. Steklov Inst. Math., 284, 148-154, (2014) · Zbl 1319.34151 [9] Drábek, P.; Kufner, A.; Nicolosi, F., Quasilinear elliptic equations with degenerations and singularities, De Gruyter Series in Nonlinear Analysis and Applications, vol. 5, (1997), Walter de Gruyter and Co. Berlin · Zbl 0894.35002 [10] Drábek, P.; Kuliev, K., Half-linear Sturm-Liouville problem with weights, Bull. Belg. Math. Soc. Simon Stevin, 19, 107-119, (2012) · Zbl 1252.34034 [11] Evans, L. C.; Gariepy, R. F., Measure theory and fine properties of functions, (1992), CRC Press · Zbl 0804.28001 [12] Ho, K.; Sim, I., Corrigendum to “existence and some properties of solutions for degenerate elliptic equations with exponent variable” [nonlinear anal. 98 (2014) 146-164], Nonlinear Anal., 128, 423-426, (2015) [13] Kawohl, B.; Lucia, M.; Prashanth, S., Simplicity of the principal eigenvalue for indefinite quasilinear problems, Adv. Difference Equ., 12, 4, 407-434, (2007) · Zbl 1158.35069 [14] Ladyzhenskaya, O. A.; Ural’tseva, N. N., Linear and quasilinear elliptic equations, (1968), Acad. Press · Zbl 0164.13002 [15] Le, V.; Schmitt, K., On boundary value problems for degenerate quasilinear elliptic equations and inequalities, J. Differential Equations, 144, 170-218, (1998) · Zbl 0912.35069 [16] Lê, A.; Schmitt, K., Variational eigenvalues of degenerate eigenvalue problems for the weighted p-Laplacian, Adv. Nonlinear Stud., 5, 4, 573-585, (2005) · Zbl 1210.35175 [17] Lieberman, G. M., Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12, 11, 1203-1219, (1988) · Zbl 0675.35042 [18] Mitidieri, E.; Pohozaev, S. I., A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova, Proc. Steklov Inst. Math., 234, 1-362, (2001), (in Russian). Translation in · Zbl 1074.35500 [19] Montefusco, E.; Radulescu, V., Nonlinear eigenvalue problems for quasilinear operators on unbounded domains, NoDEA Nonlinear Differential Equations Appl., 8, 481-497, (2001) · Zbl 1001.35094 [20] Murthy, V.; Stampacchia, G., Boundary value problems for some degenerate-elliptic operators, Ann. Mat. Pura Appl., 80, 1-122, (1968) · Zbl 0185.19201 [21] Opic, B.; Kufner, A., Hardy-type inequalities, Pitman Research Notes in Mathematics Series, vol. 279, (1990), Longman Scientific and Technical Harlow · Zbl 0698.26007 [22] Perera, K.; Pucci, P.; Varga, C., An existence result for a class of quasilinear elliptic eigenvalue problems in unbounded domains, NoDEA Nonlinear Differential Equations Appl., 21, 3, 441-451, (2014) · Zbl 1296.35108 [23] Pucci, P.; Serrin, J., The strong maximum principle revisited, J. Differential Equations, 196, 1, 1-66, (2004) · Zbl 1109.35022 [24] Wang, Y.-Z.; Li, H.-Q., Lower bound estimates for the first eigenvalue of the weighted p-Laplacian on smooth metric measure spaces, Differential Geom. Appl., 45, 23-42, (2016) · Zbl 1334.58020
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