Abstract
We investigate the stability of the Dirichlet eigenfunctions for the \({p(\cdot)}\)-Laplacian under perturbations of the variable-exponent \({p(\cdot)}\).
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Adams R.: Sobolev Spaces. Academic Press, New York (1975)
Arendt W., Monniaux S.: Domain perturbation for the first eigenvalue of the Dirichlet Schrödinger operator. Oper. Theory Adv. Appl. 78, 9–19 (1995)
Bennewitz C.: Approximation numbers=singular values. J. Comp. Appl. Math. 208(1), 102–110 (2007)
Colasuonno F., Squassina M.: Stability of eigenvalues for variable exponent problems. Nonlinear Anal. 123-124, 56–67 (2015)
Diening L., Harjulehto P., Hästö P., Růžička M.: Lebesgue and Sobolev spaces with variable exponents, Lecture notes in Mathematics, vol. 2017. Springer, Heidelberg (2011)
Dinca G., Matei P.: Geometry of Sobolev spaces with variable exponent: Smoothness and uniform convexity. C. R. Math. Acad. Sci. Paris Ser. I 347, 885–889 (2009)
Dinca G., Matei P.: Geometry of Sobolev spaces with variable exponent and a generalization of the p-Laplacian. Anal. Appl. 7(4), 373–390 (2009)
Edmunds D., Lang J., Méndez O.: Differential operators on spaces of variable integrability. World Scientific, Singapore (2015)
Edmunds D., Lang J., Nekvinda A.: Some s-numbers of an integral operator of Hardy type on \({L^{p(\cdot)}}\) spaces. J. Funct. Anal. 257, 219–242 (2009)
Franzina G., Lindqvist P.: An eigenvalue problem with variable exponents. Nonlinear Anal. Appl. 85, 1–16 (2013)
Kovác̆ik O., Rákosník J.: On spaces \({L^{p(x)}({\rm \Omega})}\) and \({W^{k,p(x)}({\rm \Omega})}\). Czechoslovak Math. J. 41, 592–618 (1991)
Lang, J., Méndez, O.: Modular eigenvalues of the Dirichlet \({p(\cdot)}\)-Laplacian and their stability. In: Spectral Theory, Function Spaces and Inequalities, pp. 125–137. Oper. Theory Adv. Appl. 219. Birkhäuser/Springer Basel AG, Basel (2012)
Lang J., Méndez O.: Convergence properties of modular eigenfunctions for the \({p(\cdot)}\)-Laplacian. Nonlinear Anal. 104C, 156–170 (2014)
Lang J., Méndez O.: An extension of a result by Lindqvist to the case of variable integrability. J. Differ. Equ. 259(2), 562–595 (2015)
Lindqvist P.: Stability for the solutions of \({div(|\nabla u|^{p-2}\nabla u) = f}\) with varying p. J. Math. Anal. Appl. 127(1), 93–102 (1987)
Lindqvist P.: On non-linear Rayleigh quotients. Pot. Anal. 2, 199–218 (1993)
Lukeš J., Pick L., Pokorný D.: On geometric properties of the spaces \({L^{p(x)}({\rm \Omega})}\). Rev. Math. Complut. 24, 115–130 (2011)
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Lang, J., Méndez, O. Stability of the Norm-Eigenfunctions of the \({p(\cdot)}\)-Laplacian. Integr. Equ. Oper. Theory 85, 245–257 (2016). https://doi.org/10.1007/s00020-015-2275-9
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DOI: https://doi.org/10.1007/s00020-015-2275-9