Boundary Concentrated Solutions for an Elliptic Equation with Subcritical Nonlinearity
Abstract
:1. Introduction and Main Results
2. The Approximate Solution
- (i)
- ;
- (ii)
- ;
- (iii)
- .
3. Optimization with Respect to the Infinite Dimensional Part
4. Expansion of the Gradient of the Associated Functional
- (i)
- In , it holds that
- (ii)
- In the set , it holds that
5. Proof of Theorems 1 and 2
- ()
- ()
- ()
- ()
- .
- Now, it remains to study the other equations. To this aim, we start by estimating the parameters .
6. Conclusions
- (i)
- Effect of the type of critical points of the potential: The solutions constructed in this paper are based on the assumption that the critical points of the restriction of the potential V on the boundary are non-degenerate. What happens if this assumption is not met, especially when V satisfies a flatness condition?
- (ii)
- Type of blow-up points: This paper concentrates on constructing solutions that localize at isolated boundary points. An intriguing extension would be to explore the existence of solutions that concentrate at non-isolated boundary points.
- (iii)
- Effect of the nonlinear exponent: This work focuses on a slightly subcritical exponent for Sobolev embedding. Future studies could investigate the problem with exponents that are slightly supercritical., i.e., when but close to zero.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
- (i)
- (ii)
- (iii)
- (i)
- ;
- (ii)
- ;
- (iii)
- .
- (i)
- ;
- (ii)
- ;
- (iii)
- (i)
- ;
- (ii)
- ;
- (iii)
- .
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Al-Harbi, S.; Ben Ayed, M. Boundary Concentrated Solutions for an Elliptic Equation with Subcritical Nonlinearity. Axioms 2025, 14, 346. https://doi.org/10.3390/axioms14050346
Al-Harbi S, Ben Ayed M. Boundary Concentrated Solutions for an Elliptic Equation with Subcritical Nonlinearity. Axioms. 2025; 14(5):346. https://doi.org/10.3390/axioms14050346
Chicago/Turabian StyleAl-Harbi, Sadeem, and Mohamed Ben Ayed. 2025. "Boundary Concentrated Solutions for an Elliptic Equation with Subcritical Nonlinearity" Axioms 14, no. 5: 346. https://doi.org/10.3390/axioms14050346
APA StyleAl-Harbi, S., & Ben Ayed, M. (2025). Boundary Concentrated Solutions for an Elliptic Equation with Subcritical Nonlinearity. Axioms, 14(5), 346. https://doi.org/10.3390/axioms14050346