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In this paper, we look for the weight functions (say g) that admit the following generalized Hardy-Rellich type inequality: for some constant C &gt; 0, where Ω is an open set in ℝN with N ≥ 1. We find various classes of such weight functions, depending on the dimension N and the geometry of Ω. Firstly, we use the Muckenhoupt condition for the one-dimensional weighted Hardy inequalities and a symmetrization inequality to obtain admissible weights in certain Lorentz-Zygmund spaces. Secondly, using the fundamental theorem of integration we obtain the weight functions in certain weighted Lebesgue spaces. As a consequence of our results, we obtain simple proofs for the embeddings of into certain Lorentz-Zygmund spaces proved by Hansson and later by Brezis and Wainger. © Copyright Royal Society of Edinburgh 2019.

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics

On the generalized Hardy-Rellich inequalities

We investigate the following eigenvalue problem {−div(L(x)|∇u|p−2∇u)=λK(x)|u|p−2uin AR1 R2,u=0on ∂AR1 R2, where AR1 R2:={x∈RN:R1&lt;|x|&lt;R2} (0&lt;R1&lt;R2≤∞), λ&gt;0 is a parameter, the weights L and K are measurable with L positive a.e. in AR1 R2 and K possibly sign-changing in AR1 R2. We prove the existence of the first eigenpair and discuss the regularity and positiveness of eigenfunctions. The asymptotic estimates for u(x) and ∇u(x) as |x|→R1 + or R2 − are also investigated. © 2018 Elsevier Inc.}, author_keywords={Asymptotic behavior; Exterior domain; Maximum principles; Regularity; The first eigenvalue; The weighted p-Laplacian; Variational method}, funding_details={Ministerstvo Školství, Mládeže a TělovýchovyMinisterstvo Školství, Mládeže a Tělovýchovy, MŠMT, LO1506}, funding_details={Grantová Agentura České RepublikyGrantová Agentura České Republiky, GA ČR, 18-032523S}, funding_text_1={P. Drábek and A. Sarkar were partly supported by the Grant Agency of the Czech Republic , project no. 18-032523S . K. Ho and A. Sarkar were supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports .}, funding_text_2={P. Dr?bek and A. Sarkar were partly supported by the Grant Agency of the Czech Republic, project no. 18-032523S. K. Ho and A. Sarkar were supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports.}, references={Agudelo, O., Drábek, P., Anisotropic semipositone quasilinear problems (2017) J. Math. Anal. Appl., 452 (2), pp. 1145-1167; Anoop, T.V., Drábek, P., Sankar, L., Sasi, S., Antimaximum principle in exterior domains (2016) Nonlinear Anal., 130, pp. 241-254; Anoop, T.V., Drábek, P., Sasi, S., Weighted quasilinear eigenvalue problems in exterior domains (2015) Calc. Var. Partial Differential Equations, 53 (3-4), pp. 961-975; Autuori, G., Colasuonno, F., Pucci, P., On the existence of stationary solutions for higher-order p-Kirchhoff problems (2014) Commun. Contemp. Math., 16; Chhetri, M., Drábek, P., Principal eigenvalue of p-Laplacian operator in exterior domain (2014) Results Math., 66 (3-4), pp. 461-468; Colasuonnoa, F., Pucci, P., Varga, C., Multiple solutions for an eigenvalue problem involving p-Laplacian type operators (2012) Nonlinear Anal., 75, pp. 4496-4512; DiBenedetto, E., C1+α local regularity of weak solutions of degenerate elliptic equations (1983) Nonlinear Anal., 7 (8), pp. 827-850; Drábek, P., Kufner, A., Kuliev, K., Half-linear Sturm–Liouville problem with weights: asymptotic behavior of eigenfunctions (2014) Proc. Steklov Inst. Math., 284, pp. 148-154; Drábek, P., Kufner, A., Nicolosi, F., Quasilinear Elliptic Equations with Degenerations and Singularities (1997) De Gruyter Series in Nonlinear Analysis and Applications, 5. , Walter de Gruyter and Co. Berlin; Drábek, P., Kuliev, K., Half-linear Sturm–Liouville problem with weights (2012) Bull. Belg. Math. Soc. Simon Stevin, 19, pp. 107-119; Evans, L.C., Gariepy, R.F., Measure Theory and Fine Properties of Functions (1992), CRC Press; 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] (2015) Nonlinear Anal., 128, pp. 423-426; Kawohl, B., Lucia, M., Prashanth, S., Simplicity of the principal eigenvalue for indefinite quasilinear problems (2007) Adv. Difference Equ., 12 (4), pp. 407-434; Ladyzhenskaya, O.A., Ural'tseva, N.N., Linear and Quasilinear Elliptic Equations (1968), Acad. Press; Le, V., Schmitt, K., On boundary value problems for degenerate quasilinear elliptic equations and inequalities (1998) J. Differential Equations, 144, pp. 170-218; Lê, A., Schmitt, K., Variational eigenvalues of degenerate eigenvalue problems for the weighted p-Laplacian (2005) Adv. Nonlinear Stud., 5 (4), pp. 573-585; Lieberman, G.M., Boundary regularity for solutions of degenerate elliptic equations (1988) Nonlinear Anal., 12 (11), pp. 1203-1219; Mitidieri, E., Pohozaev, S.I., A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities (2001) Proc. Steklov Inst. Math., Tr. Mat. Inst. Steklova, 234, pp. 1-384. , (in Russian). Translation in; Montefusco, E., Radulescu, V., Nonlinear eigenvalue problems for quasilinear operators on unbounded domains (2001) NoDEA Nonlinear Differential Equations Appl., 8, pp. 481-497; Murthy, V., Stampacchia, G., Boundary value problems for some degenerate-elliptic operators (1968) Ann. Mat. Pura Appl., 80, pp. 1-122; Opic, B., Kufner, A., Hardy-Type Inequalities (1990) Pitman Research Notes in Mathematics Series, 279. , Longman Scientific and Technical Harlow; Perera, K., Pucci, P., Varga, C., An existence result for a class of quasilinear elliptic eigenvalue problems in unbounded domains (2014) NoDEA Nonlinear Differential Equations Appl., 21 (3), pp. 441-451; Pucci, P., Serrin, J., The strong maximum principle revisited (2004) J. Differential Equations, 196 (1), pp. 1-66; Wang, Y.-Z., Li, H.-Q., Lower bound estimates for the first eigenvalue of the weighted p-Laplacian on smooth metric measure spaces (2016) Differential Geom. Appl., 45, pp. 23-42}, correspondence_address1={Sarkar, A.; NTIS, Technická 8, Czech Republic; email: sarkara@ntis.zcu.cz}, publisher={Academic Press Inc.}, issn={0022247X}, language={English}, abbrev_source_title={J. Math. Anal. Appl.}, document_type={Article}, source={Scopus},

Journal of Mathematical Analysis and Applications

On the eigenvalue problem involving the weighted p-Laplacian in radially symmetric domains

We investigate the Fredholm alternative for the p-Laplacian in an exterior domain which is the complement of the closed unit ball in RN (N≥2). By employing techniques of Calculus of Variations we obtain the multiplicity of solutions. The striking difference between our case and the entire space case is also discussed. © 2018 Elsevier Ltd

Nonlinear Analysis, Theory, Methods and Applications

The Fredholm alternative for the p-Laplacian in exterior domains

In this article we study the existence of at least two positive weak solutions of an nonhomogeneous fourth-order Navier boundary-value problem involving critical exponential growth on a bounded domain in ℝ4, with a parameter λ &gt; 0. We establish upper and lower bounds for λ, which determine multiplicity and non-existence of solutions. © 2017 Texas State University.

Electronic Journal of Differential Equations

Multiple solutions of a fourth-order nonhomogeneous equation with critical growth in ℝ4

We consider the equation (Formula presented.) in the sense of distribution in (Formula presented.) where (Formula presented.) and (Formula presented.). Then it is known that (Formula presented.) solves (Formula presented.) , for some nonnegative constants (Formula presented.) and (Formula presented.). In this paper, we study the existence of singular solutions to (Formula presented.) in a domain (Formula presented.) , (Formula presented.) is a nonnegative measurable function in some Lebesgue space. If (Formula presented.) in (Formula presented.) , then we find the growth of the nonlinearity (Formula presented.) that determines (Formula presented.) and (Formula presented.) to be (Formula presented.). In case when (Formula presented.) , we will establish regularity results when (Formula presented.) , for some (Formula presented.). This paper extends the work of Soranzo (1997) where the author finds the barrier function in higher dimensions (Formula presented.) with a specific weight function (Formula presented.). Later, we discuss its analogous generalization for the polyharmonic operator. © 2015 Taylor & Francis.

Complex Variables and Elliptic Equations

Isolated singularities of polyharmonic operator in even dimension

Let N ≥ 5 and D2,2(ℝN) denote the closure of C0∞ (ℝN) in the norm (Formula presented). Let K ∈ C2(ℝN). We consider the following problem for e{open} ≥ 0: (Formula presented). We show an exact multiplicity result for (Pe{open}) for all small e{open} &gt; 0. © 2014 Springer Basel.

Archiv der Mathematik

Exact multiplicity for the perturbed Q-curvature problem in ℝN, N≥5

Let g0 denote the standard metric on S4 and Pg0=δg02-2δg0 denote the corresponding Paneitz operator. In this work, we study the following fourth order elliptic problem with exponential nonlinearityPg0u+6=2Q(x)e4uon S4. Here Q is a prescribed smooth function on S4 which is assumed to be a perturbation of a constant. We prove existence results to the above problem under assumptions only on the "shape" of Q near its critical points. These are more general than the non-degeneracy conditions assumed so far. We also show local uniqueness and exact multiplicity results for this problem. The main tool used is the Lyapunov-Schmidt reduction. © 2013.