In this paper, we study the action of the orthogonal group O(n) on the set of solutions of some sub-linear elliptic problem of the form \(\Delta u\) + \(u\) − | \(u\) |−2\(\theta\) \(u\) = 0 on the unit ball B of \(\mathbb{R}\)n, n ≥ 2 and 0 < 2θ < 1. For suitable subgroups G of the orthogonal group O(n), we show the existence of non-radially symmetric solutions, which are G-invariant. We precisely give a necessary and sufficient condition on G for the existence of non-radially symmetric but G-invariant solutions. Besides, we develop complete proofs for the existence and uniqueness of radial solutions. Recall that a majority of studies of this type of problem were focusing on radial solutions which are obviously invariant by the group O(n). On the other hand, the question of the existence of non-radial solutions remaines somehow abandoned. In this paper, we studied this question and provided computer numerical simulations for radial solutions and the estimate of the Sobolev norm of the new non radial solutions.
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Non-Radially Symmetric Solutions for Some Sublinear Defocusing Elliptic Problems
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Received: 06 Nov 2025 | Revised: 10 Dec 2025 | Accepted: 18 Nov 2025 | Published: 09 Jan 2026
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References
- 1.
Aftalion, A.; Pacela, F. Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains. Comptes Rendus Math. 2004, 339, 339–344.
- 2.
Atkinson, F.V.; Brezis, H.; Peletier, L.A. Nodal solutions of elliptic equations with critical Sobolev exponents. J. Differ. Equ. 1990, 85, 151–170.
- 3.
Ambrosetti, A.; Rabinowitz, P.H. Dual variational methods in critical point theory and applications. J. Funct. Anal. 1973, 14, 349–381.
- 4.
Rabinowitz, P.H. Minimax Methods in Critical Point Theory with Applications to Differential Equations; American Mathematical Soc.: Providence, RI, USA, 1986.
- 5.
Struwe, M. Variational Methods, 2nd ed.; Springer: Berlin/Heidelberg, Germany, 1996.
- 6.
Balabane, B.; Dolbeault, J.; Ounaies, H. Nodal solutions for a sublinear elliptic equation. Nonlinear Anal. 2003, 52, 219–237.
- 7.
Ounaies, H. Study of an elliptic equation with a singular potential. Indian J. Pure appl. Math. 2003, 34, 111–131.
- 8.
Chteoui, R.; Ben Mabrouk, A.; Ounaies, H. Existence and Properties of Radial Solutions of a Sub-linear Elliptic Equation. J.
Part. Diff. Eq. 2015, 28, 30–38. - 9.
Chteoui, R.; Ben Mabrouk, A. A Generalized Lyapunov-Sylvester Computational Method for Numerical Solutions of NLS Equation with Singular Potential. Anal. Theory Appl. 2017, 33, 333–354.
- 10.
Cortazar, C.; Elgueta, M. On a semilinear problem in RN with non-lipschitzian nonlinearity. Adv. Diff. Equ. 1996, 1, 199–218.
- 11.
Ben Mabrouk, A.; Ben Mohamed, M.L. Phase plane analysis and classification of solutions of a mixed sublinear-superlinear elliptic problem. Nonlinear Anal. Theory Methods Appl. 2009, 70, 1–15.
- 12.
Ben Mabrouk, A.; Ben Mohamed, M.L. Nodal solutions for some nonlinear elliptic equations. Appl. Math. Comput. 2007, 186, 589–597.
- 13.
Ben Mabrouk, A.; Bezia, A.; Souissi, C. Numerical approximations and asymptotic limits of some nonlinear problems. Math. Comput. Sci. 2023, 3, 53–70
- 14.
Ben Mabrouk, A.; Ben Mohamed, M.L.; Omrani, K. Finite difference approximate solutions for a mixed sub-superlinear equation. Appl. Math. Comput. 2007, 187, 1007–1016.
- 15.
Pao, C.V. Nonlinear Parabolic and Elliptic Equations; Plenum Press: New York, NY. USA, 1992.
- 16.
Bratsos, A.G. A linearized finite-difference method for the solution of the nonlinear cubic Schr¨odinger equation. Comm. Appl. Anal. 2000, 4, 133–139.
- 17.
Bratsos, A.G. A linearized finite-difference scheme for the numerical solution of the nonlinear cubic Schrodinger equation. Korean Comput. Appl. Math. 2001, 8, 459–467.
- 18.
Delfour, M.; Fortin, M.; Payre, G. Finite difference solutions of a non-linear Schr¨odinger equation. J. Comput. Phys. 1981, 44, 277–288.
- 19.
Ben Mabrouk, A.; Ben Mohamed, M.L. Nonradial solutions of a mixed concave-convex elliptic problem. J. Part. Diff. Eq. 2011, 24, 313–323.
- 20.
Bartsch, T.; Willem, M. On an elliptic equation with concave and convex nonlinearities. Proc. Amer. Math. Soc. 1995, 123, 3555–3561.
- 21.
Kajikiya, R. Orthogonal group invariant solutions of the Emden-Fowler equation. Nonlinear Anal. 2001, 44, 845–896.
- 22.
Kajikiya, R. Multiple existence of non radial solutions with group invariance for sublinear elliptic equations. J. Differ. Equ. 2002, 186, 299–343.
- 23.
Onishchik, A.L. Topology of Transitive Group; Johann Ambrozius Barth: Leipzig, Germany, 1994.
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