1. MMC
  2. All Volumes and Issues
  3. Volume 10, Number 4, 2023
  4. Positive solutions of an elliptic equation involving a sign-changing potential and a gradient term

Positive solutions of an elliptic equation involving a sign-changing potential and a gradient term

MMC.
2023;
Volume 10, Number 4
: pp. 1109–1118
https://doi.org/10.23939/mmc2023.04.1109
Received: August 20, 2023
Accepted: October 28, 2023

Mathematical Modeling and Computing, Vol. 10, No. 4, pp. 1109–1118 (2023)

Authors:
  1. A. Bouzelmate
  2. H. El Baghouri
  3. A. Gmira
1
LaR2A Laboratory, Faculty of Sciences, Abdelmalek Essaadi University
2
LaR2A Laboratory, Faculty of Sciences, Abdelmalek Essaadi University
3
LaR2A Laboratory, Faculty of Sciences, Abdelmalek Essaadi University

The objective of this paper is to investigate the elliptic singular Laplacian equation $\Delta u -|\nabla\,u|^{q}+u^{p}-u^{-\delta}=0$ in $\mathbb{R}^{N}$, where $N\geq1$, $1<q<p$ and $\delta>2$.  Our main contributions consist of establishing the existence of an entire strictly positive solution and analyzing certain properties of its asymptotic behavior, particularly when it exhibits monotonicity.

elliptic equation
sign-changing potential
gradient term
radial solution
Banach fixed point theorem
energy function
oscillation methods
  1. Serrin J., Zou H.  Existence and nonexistence results for ground states of quasilinear elliptic equations.  Archive for Rational Mechanics and Analysis.  121, 101–130 (1992).
  2. Quittner Q.  On global existence and stationary solutions for two classes of semilinear parabolic equations.  Commentationes Mathematicae Universitatis Carolinae.  34 (1), 105–124 (1993).
  3. Souplet P., Tayachi S., Weissler F. B.  Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term.  Indiana University Mathematics Journal.  45 (3), 655–682  (1996).
  4. Souplet P.  Recent results and open problems on parabolic.  Electronic Journal of Differential Equations.  34, 105–124 (1993).
  5. Bidaut-Vèron M.-F., Vèron L.  Local behaviour of the solutions of the Chipot-Weissler equation.  Preprint arXiv:2303.08074 (2023).
  6. Gkikas K. T., Nguyen P.-T.  Elliptic equations with Hardy potential and gradient-dependent nonlinearity.  Advanced Nonlinear Studies.  20 (2), 399–435 (2020).
  7. Gkikas K., Nguyen P.-T.  Semilinear elliptic equations with Hardy potential and gradient nonlinearity.  Revista Matemática Iberoamericana.  36 (4), 1207–1256 (2020).
  8. Fowler R. H.  The form near infinity of real continuos solutions of a certain differential equation of second order.  Quarterly Journal of Mathematics.  45, 289–350 (1914).
  9. Fowler R. H.  The solutions of Emden's and similar differential equation.  Monthly Notices of the Royal Astronomical Society.  91 (1), 63–91 (1920).
  10. Fowler R. H.  Further studies on Emden's and similar differential equation.  Quarterly Journal of Mathematics.  2 (1), 259–288 (1931).
  11. Lions P. L.  Isolated singularities in semilinear problems.  Journal of Differential Equations.  38 (3), 441–450 (1980).
  12. Aviles P.  Local behavior of positive solutions of some elliptic equation.  Communications in Mathematical Physics.  108, 177–192 (1987).
  13. Gidas B., Spruck J.  Global and local behavior of positive solutions of nonlinear elliptic equations.  Communications on Pure and Applied Mathematics.  34 (4), 525–598 (1980).
  14. Caffarelli L. A., Gidas B., Spruck J.  Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth.  Communications on Pure and Applied Mathematics.  42, 271–297 (1989).
  15. Dàvila J., Montenegro M.  Radial solutions of an elliptic equation with singular nonlinearity.  Journal of Mathematical Analysis and Applications.  352 (1), 360–379 (2009).
  16. Junping S., Miaoxin Y.  On a singular nonlinear semilinear elliptic problem.  Proceedings of the Royal Society of Edinburgh. Section A: Mathematics.  128 (6), 1389–1401 (1998).
  17. Ouyang T., Shi J., Yao M.  Exact multiplicity and bifurcation of solutions of a singular equation. Preprint (1996).
  18. Amann H.  Ordinary Differential Equations.  Walter de Gruyter, Belin, New York (1996).
  19. Bouzelmate A., Gmira A., Reys G.  On the Radial Solutions of a Degenerate Elliptic Equation with Convection Term.  International Journal of Mathematical Analysis.  1 (20), 975–993 (2007).
  20. Bouzelmate A., Gmira A.  Singular solutions of an inhomogeneous elliptic equation.  Nonlinear Functional Analysis and Applications.  26 (2), 237–272 (2021).
  21. Ni W.-M., Serrin J.  Nonexistence theorems for singular solutions of quasilinear partial differential equations.  Communications on Pure and Applied Mathematics.  39 (3), 379–399 (1986).