Opuscula Math. 45, no. 5 (2025), 647-655
https://doi.org/10.7494/OpMath.2025.45.5.647
Opuscula Mathematica
A general elliptic equation with intrinsic operator
Abstract. Existence and bound of a solution is established for a general elliptic equation with intrinsic operator subject to Dirichlet boundary condition. This provides a sufficient condition to the fundamental question if there is a solution belonging to a prescribed ball in the function space. An application deals with an equation involving a convolution product.
Keywords: nonlinear elliptic equation, intrinsic operator, convolution.
Mathematics Subject Classification: 35J92, 47F10.
- H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. https://doi.org/10.1007/978-0-387-70914-7
- S. Carl, V.K. Le, D. Motreanu, Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications, Springer Monographs in Mathematics, Springer, New York, 2007. https://doi.org/10.1007/978-0-387-46252-3_5
- M. Galewski, Basic Monotonicity Methods with Some Applications, Compact Textbooks in Mathematics, Birkhäuser/Springer, Cham, 2021. https://doi.org/10.1007/978-3-030-75308-5
- R. Livrea, D. Motreanu, A. Sciammetta, Quasi-linear elliptic systems with intrinsic operators, Discrete and Continuous Dynamical Systems - Series S (2025). https://doi.org/10.3934/dcdss.2025073
- G. Marino, D. Motreanu, Existence and L-estimates for elliptic equations involving convolution, Comput. Math. Methods 2 (2020), no. 5, e1103. https://doi.org/10.1002/cmm4.1103
- A.H.S. Medeiros, D. Motreanu, A problem involving competing and intrinsic operators, Sao Paulo J. Math. Sci. 18 (2024), no. 1, 300-311. https://doi.org/10.1007/s40863-024-00405-y
- D. Motreanu, Nonlinear Differential Problems with Smooth and Nonsmooth Constraints, Mathematical Analysis and Its Applications, Academic Press, London, 2018, https://doi.org/10.1016/C2016--0-01433-X
- D. Motreanu, V.V. Motreanu, \((p,q)\)-Laplacian equations with convection term and a intrinsic operator, [in:] Differential and integral inequalities, Springer Optim. Appl., 151, Springer, Cham, 2019, 589-601. https://doi.org/10.1007/978-3-030-27407-8_22
- D. Motreanu, V.V. Motreanu, Non-variational elliptic equations involving \((p,q)\)-Laplacian, convection and convolution, Pure Appl. Funct. Anal. 5 (2020), no. 5, 1205-1215.
- D. Motreanu, A. Sciammetta, On a Neumann problem with an intrinsic operator, Axioms 2024, 13, 497. https://doi.org/10.3390/axioms13080497
- D. Motreanu, E. Tornatore, Elliptic equations with unbounded coefficient, convection term and intrinsic operator, Math. Z. 308 (2024), no. 2, Paper no. 38. https://doi.org/10.1007/s00209-024-03591-9
- D. Motreanu, C. Vetro, F. Vetro, The effects of convolution and gradient dependence on a parametric Dirichlet problem, Partial Differ. Equ. Appl. 1 (2020), no. 1, Paper no. 3. https://doi.org/10.1007/s42985-019-0004-y
- Dumitru Motreanu
https://orcid.org/0000-0001-7391-9534- University of Perpignan, Department of Mathematics, 66860 Perpignan, France
- Communicated by Marek Galewski.
- Received: 2025-07-13.
- Revised: 2025-08-25.
- Accepted: 2025-08-26.
- Published online: 2025-09-13.
