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On a Class of Weighted p-Laplace Equation with Singular Nonlinearity
P. Garain,
Published in
Volume: 17
Issue: 4
This article deals with the existence of the following quasilinear degenerate singular elliptic equation: (Pλ){-div(w(x)|∇u|p-2∇u)=gλ(u),u>0inΩ,u=0on∂Ω,where Ω ⊂ Rn is a smooth bounded domain, n≥ 3 , λ> 0 , p> 1 , and w is a Muckenhoupt weight. Using variational techniques, for gλ(u) = λf(u) u-q and certain assumptions on f, we show existence of a solution to (Pλ) for each λ> 0. Moreover, when gλ(u) = λu-q+ ur, we establish existence of at least two solutions to (Pλ) in a suitable range of the parameter λ. Here, we assume q∈ (0 , 1) and r∈(p-1,ps∗-1). © 2020, Springer Nature Switzerland AG.}, author_keywords={Multiple weak solutions; Singular nonlinearity; Variational method; Weighted p-Laplacian}, references={Arcoya, D., Boccardo, L., Multiplicity of solutions for a Dirichlet problem with a singular and a supercritical nonlinearities (2013) Differ. Integral Equ., 26 (1-2), pp. 119-128; Arcoya, D., Moreno-Mérida, L., Multiplicity of solutions for a Dirichlet problem with a strongly singular nonlinearity (2014) Nonlinear Anal., 95, pp. 281-291; Bal, K., Garain, P., (2017) Multiplicity Results for a Quasilinear Equation with Singular Nonlinearity; Boccardo, L., A Dirichlet problem with singular and supercritical nonlinearities (2012) Nonlinear Anal., 75 (12), pp. 4436-4440; Boccardo, L., Orsina, L., Semilinear elliptic equations with singular nonlinearities (2010) Calc. Var. Partial Differ. Equ., 37 (3-4), pp. 363-380; Canino, A., Sciunzi, B., Trombetta, A., Existence and uniqueness for p -Laplace equations involving singular nonlinearities (2016) NoDEA Nonlinear Differ. Equ. Appl., 23 (2), p. 18. , Art. 8; Crandall, M.G., Rabinowitz, P.H., Tartar, L., On a Dirichlet problem with a singular nonlinearity (1977) Commun. Partial Differ. Equ., 2 (2), pp. 193-222; Quasilinear elliptic equations with degenerations and singularities (1997) De Gruyter Series in Nonlinear Analysis and Applications, 5. , Walter de Gruyter & Co., Berlin; Fabes, E.B., Kenig, C.E., Serapioni, R.P., The local regularity of solutions of degenerate elliptic equations (1982) Commun. Partial Differ. Equ., 7 (1), pp. 77-116; Garain, P., On a degenerate singular elliptic problem (2018) Arxiv; Ghergu, M., Rădulescu, V.D., Singular elliptic problems: Bifurcation and asymptotic analysis (2008) Oxford Lecture Series in Mathematics and Its Applications, 37. , The Clarendon Press, Oxford University Press, Oxford; Giacomoni, J., Schindler, I., Takác, P., Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation (2007) Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 6 (1), pp. 117-158; Haitao, Y., Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem (2003) J. Differ. Equ., 189 (2), pp. 487-512; Heinonen, J., Kilpeläinen, T., Martio, O., (1993) Nonlinear Potential Theory of Degenerate Elliptic Equations, , Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York; Hirano, N., Saccon, C., Shioji, N., Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities (2004) Adv. Differ. Equ., 9 (1-2), pp. 197-220; Kilpeläinen, T., Weighted Sobolev spaces and capacity (1994) Ann. Acad. Sci. Fenn. Ser. A I Math, 19 (1), pp. 95-113; Eunkyung, K., Eun, K.L., Shivaji, R., Multiplicity results for classes of infinite positone problems (2011) Z. Anal. Anwend., 30 (3), pp. 305-318; Lazer, A.C., McKenna, P.J., On a singular nonlinear elliptic boundary-value problem (1991) Proc. Am. Math. Soc., 111 (3), pp. 721-730; Lindqvist, P., (2006) Notes on the P -Laplace Equation, 102. , Report, University of Jyväskylä Department of Mathematics and Statistics. University of Jyväskylä, Jyväskylä; Muckenhoupt, B., Weighted norm inequalities for the Hardy maximal function (1972) Trans. Am. Math. Soc., 165, pp. 207-226; Peral, I., Multiplicity of Solution for P-Laplacian, , ICTP lecture notes; Wu, L., Niu, P., Harnack inequalities for weighted subelliptic p -Laplace equations constructed by Hörmander vector fields (2017) Math. Rep. (Bucur.), 19 (69), pp. 313-337. , 3}, correspondence_address1={Garain, P.; Department of Mathematics and Systems Analysis, Otakaari 1, Finland; email: pgarain92@gmail.com}, publisher={Birkhauser}, issn={16605446}, language={English}, abbrev_source_title={Mediterr. J. Math.}, document_type={Article}, source={Scopus},
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JournalMediterranean Journal of Mathematics