Existence of positive continuous weak solutions for some semilinear elliptic eigenvalue problems

• Autorzy: Zeddini Noureddine , Sari Rehab Saeed

Identyfikatory

DOI

10.7494/OpMath.2022.42.3.489

• Języki publikacji: EN

Abstrakty

EN

Let D be a bounded C1,1-domain in Rd, d ≥ 2. The aim of this article is twofold. The first goal is to give a new characterization of the Kato class of functions K(D) that was defined by N. Zeddini for d = 2 and by H. Mâagli and M. Zribi for d ≥ 3 and adapted to study some nonlinear elliptic problems in D. The second goal is to prove the existence of positive continuous weak solutions, having the global behavior of the associated homogeneous problem, for sufficiently small values of the nonnegative constants λ and μ to the following system Δu = λf(x, u, v), Δv = μg(x, u, v) in D, u = ϕ1 and v = ϕ2 on ∂D, where ϕ1 and ϕ2 are nontrivial nonnegative continuous functions on ∂D. The functions f and g are nonnegative and belong to a class of functions containing in particular all functions of the type f(x, u, v) = p(x)uαh1(v) and g(x, u, v) = q(x)h2(u)vβ with α ≥ 1, β ≥ 1, h1, h2 are continuous on [0,∞) and p, q are nonnegative functions in K(D).

Słowa kluczowe

EN

Green function; Kato class; nonlinear elliptic systems; positive solution; maximum principle; Schauder fixed point theorem

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2022
• Tom: Vol. 42, no. 3
• Strony: 489--519
• Opis fizyczny: Bibliogr. 21 poz.

Twórcy

autor

Zeddini Noureddine

• Taibah University, Madinah, College of Science, Department of Mathematics, Kingdom of Saudi Arabia

autor

Sari Rehab Saeed

• Taibah University, Madinah, College of Science, Department of Mathematics, Kingdom of Saudi Arabia

Bibliografia

• [1] H. Aikawa, T. Kilpeläinen, N. Shanmugalingam, X. Zhong, Boundary Harnack principle for p-harmonic functions in smooth Euclidean domains, Potential Anal. 26 (2007), no. 3, 281–301.
• [2] R. Alsaedi, H. Mâagli, N. Zeddini, Positive solutions for some competitive elliptic systems, Math. Slovaca 64 (2014), no. 1, 61–72.
• [3] I. Bachar, H. Mâagli, N. Zeddini, Estimates on the Green function and existence of positive solutions of nonlinear singular elliptic equations, Commun. Contemp. Math. 5 (2003), no. 3, 401–434.
• [4] M. Bełdziński, M. Galewski, On solvability of elliptic boundary value problems via global invertibility, Opuscula Math. 20 (2020), no. 1, 37–47.
• [5] H. Ben Sâad, Généralisation des noyaux Vh et applications, Séminaire de Théorie du Potentiel de Paris, Lecture Notes in Math., Springer-Verlag 1061 (1984), 14–39.
• [6] Z.Q. Chen, R.J. Williams, Z. Zhao, On the existence of positive solutions of semilinear elliptic equations with Dirichlet boundary conditions, Math. Ann. 298 (1994), no. 3, 543–556.
• [7] K.L. Chung, J.B. Walsh, Markov processes, Brownian motion, and time symmetry, 2nd ed., Springer, 2005.
• [8] K.L. Chung, Z. Zhao, From Brownian motion to Schrödinger’s equation, Springer-Verlag, 1995.
• [9] R. Dautray, J.L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. 1, Physical Origins and Classical Methods, Springer-Verlag 1990.
• [10] G. Figueiredo, V.D. Radulescu, Nonhomogeneous equations with critical exponential growth and lack of compactness, Opuscula Math. 40 (2020), no. 1, 71–92.
• [11] A. Ghanmi, H. Mâagli, S. Turki, N. Zeddini, Existence of positive bounded solutions for some nonlinear elliptic systems, J. Math. Anal. Appl. 352 (2009), 440–448.
• [12] B. Hamdi, S. Maingot, A. Medeghri, On generalized Bitsadze–Samarskii problems of elliptic type in Lp cases, Rend. Circ. Mat. Palermo (2) 70 (2021), no. 3, 1685–1708.
• [13] N.J. Kalton, I.E. Verbitsky, Nonlinear equations and weighted norm inequalities, Trans. Amer. Math. Soc. 351 (1999), 3441–3497.
• [14] H. Mâagli, Perturbation semi-linéaire des résolvantes et des semi-groupes, Potential Anal. 3 (1994), 61–87.
• [15] H. Mâagli, M. Zribi, On a new Kato class and singular solutions of a nonlinear elliptic equation in bounded domains, Positivity 9 (2005), 667–686.
• [16] N.S. Papageorgiou, V.D. Rădulescu, D. Repovs, Nonlinear Analysis – Theory and Methods, Springer Monographs in Mathematics, Springer, Cham, 2019.
• [17] M. Selmi, Inequalities for Green functions in a Dini–Jordan domain in R2, Potential Anal. 13 (2000), 81–102.
• [18] F. Toumi, Existence of positive bounded solutions of semilinear elliptic problems, Int. J. Differ. Equ. 2010 (2010), 10 pp.
• [19] F. Toumi, Existence of positive bounded solutions for nonlinear elliptic systems, Electron. J. Differ. Equ. Conf. 2013 (2013), no. 175, 1–11.
• [20] N. Zeddini, Positive solutions for a singular nonlinear problem on a bounded domain in R2, Potential Anal. 18 (2003), 97–118.
• [21] Z. Zhao, Green function for Schrödinger operator and conditional Feynman–Kac gauge, J. Math. Anal. Appl. 116 (1986), 309–334.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).