A multiplicity results for a singular equation involving the p(x)-Laplace operator

References

• Acerbi E, Mingione G. Regularity results for a class of functionals with nonstandard growth. Arch. Ration. Mech. Anal. 2001;156:121–140.
   Web of Science ®Google Scholar
• Diening L. Theorical and numerical results for electrorheological fluids [PhD thesis]. Germany: University of Freiburg; 2002.
   Google Scholar
• Zhang Q. Existence and asymptotic behavior of positive solutions to p(x)-Laplacian equations with singular nonlinearities. J. Inequal. Appl. 2007. 9 p. Art. ID 19349.
   Web of Science ®Google Scholar
• Saoudi K. Existence and non-existence of solution for a singular nonlinear Dirichlet problem involving the p(x)-Laplace operator. J. Adv. Math. Stud. 2016;9:292–303.
   Google Scholar
• Fan XL. Solutions for p(x)-Laplacian Dirichlet problems with singular coefficients. J. Math. Anal. Appl. 2005;312:464–477.
   Web of Science ®Google Scholar
• Liu JJ. Positive solutions of the p(x)-Laplace equation with singular nonlinearity. Nonlinear Anal. 2010;72:4428–4437.
   Web of Science ®Google Scholar
• Mohammed A. Positive solutions of the p-Laplace equation with singular nonlinearity. J. Math. Anal. Appl. 2009;352:234–245.
   Web of Science ®Google Scholar
• Ghergu M, Radulescu V. Singular elliptic problems: bifurcation and asymptotic analysis. Vol. 37, Oxford lecture series in mathematics and its applications. Oxford: The Clarendon Press, Oxford University Press; 2008.
   Google Scholar
• Crandall MG, Rabinowitz PH, Tartar L. On a Dirichlet problem with a singular nonlinearity. Comm. Partial Differ. Equ. 1977;2:193–222.
   Google Scholar
• Coclite MM, Palmieri G. On a singular nonlinear Dirichlet problem. Comm. Partial Differ. Equ. 1989;14:1315–1327.
   Web of Science ®Google Scholar
• Giacomoni J, Saoudi K. Multiplicity of positive solutions for a singular and critical problem. Nonlinear Anal. 2009;71:4060–4077.
   Web of Science ®Google Scholar
• Saoudi K. Existence and non-existence for a singular problem with variables potentials. Forthcoming.
   Google Scholar
• Saoudi K, Kratou M. Existence of multiple solutions for a singular and quasilinear equation. Complex Var. Elliptic Equ. 2015;60:893–925.
   Web of Science ®Google Scholar
• Ghanmi A, Saoudi K. The Nehari manifold for a singular elliptic equation involving the fractional Laplace operator. Fractional Differ. Calculus. 2016;6:201–217.
   Google Scholar
• Ghanmi A, Saoudi K. A multiplicity results for a singular problem involving the fractional p-Laplacian operator. Complex Var. Elliptic Equ. 2016;61:1199–1216.
   Web of Science ®Google Scholar
• Fan XL, Zhao D. On spaces Lp(x) and Wm,p(x). J. Math. Anal. Appl. bf. 2001;263:424–446.
   Web of Science ®Google Scholar
• Kovăčik O, Răkosnik J. On spaces Lp(x) and Wk, p(x). Czechoslovak Math. J. 1991;41:592–618.
   Web of Science ®Google Scholar
• Zhang QH. A strong maximum principle for differential equations with nonstandard p(x)-growth conditions. J. Math. Anal. Appl. 2005;312:24–32.
   Web of Science ®Google Scholar
• Fan XL, Zhao YZ, Zhang QH. A strong maximum principle for p(x)-Laplace equations. Chinese J. Contemp. Math. 2003;24:277–282.
   Google Scholar
• Hernández J, Mancebo F, Vega JM. On the linearization of some singular, nonlinear elliptic problems and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire. 2002;19:777–813.
   Web of Science ®Google Scholar
• Fan XL. On the sub-supersolution method for p(x)-Laplacian equations. J. Math. Anal. Appl. 2007;330:665–682.
   Web of Science ®Google Scholar
• Brezis H, Lieb E. A relation between point convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 1983;88:486–490.
   Web of Science ®Google Scholar
• Boccardo L, Murat F. Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal. 1992;19:581–597.
   Web of Science ®Google Scholar
• Brezis H, Nirenberg L. Minima locaux relatifs à C1 et H1. C.R. Acad. Sci. Paris sér. I Math. 1993;317:465–472.
   Google Scholar
• García Azorero JP, Peral Alonso I, Manfredi JJ. Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations. Commun. Contemp. Math. 2000;2:385–404.
   Web of Science ®Google Scholar
• Brock F, Iturraga L, Ubilla P. A multiplicity result for the p-Laplacian involving a parameter. Ann. Henri Poincaré. 2008;9:1371–1386.
   Web of Science ®Google Scholar
• De Figueiredo DG, Gossez JP, Ubilla P. Local “superlinearity” and “sublinearity” for the p-Laplacian. J. Funct. Anal. 2009;257:721–752.
   Web of Science ®Google Scholar
• Giacomoni J, Saoudi K. W1, p0 versus C1 local minimizers for a singular and critical functional. J. Math. Anal. Appl. 2010;363:697–710.
   Web of Science ®Google Scholar
• Saoudi K. W01, p(x) versus C1 local minimizers for a functional with critical growth. J. Partial Differ. Equ. 2014;27:115–124.
   Google Scholar
• Saoudi K. W1, N versus C1 local minimizer for a singular functional with Neumann boundary condition. Forthcoming.
   Google Scholar
• Takáč P. On the Fredholm alternative for the p-Laplacian at the first eigenvalue. Indiana Univ. Math. J. 2002;51:187–237.
   Web of Science ®Google Scholar
• Fan XL, Zhao D. A class of De Giorgi type and Holder continuity. Nonlinear Anal. 1996;36:295–318.
   Web of Science ®Google Scholar
• Fan XL. Global C1,α regularity for variable exponent elliptic equations in divergence form. J. Differ. Equ. 2007;235:397–417.
   Web of Science ®Google Scholar
• Gilbarg D, Trudinger NS. Elliptic partial differential equations of second order. New York (NY): Springer Verlag; 1983.
   Google Scholar
• Tolksdorf P. Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 1984;51:126–150.
   Web of Science ®Google Scholar
• Coscia A, Mingione G. Hölder continuity of the gradient of p(x)-harmonic mappings. C. R. Acad. Sci. Paris Ser. I. 1999;328:363–368.
   Web of Science ®Google Scholar
• Giaquinta M. Multiple integrals in the calculus of variations and nonlinear elliptic systems. Princeton (NJ): Princeton University Press; 1983.
   Google Scholar