Blowing-up Solutions with Residual Mass in a Slightly Subcritical Dirichlet Problem

Abstract: In this paper, we study the Dirichlet elliptic problem $(\mathcal{P}_\varepsilon)$: $-Δu +V\,u = u^{p-\varepsilon}$, $u>0$ in $Ω$, $u=0$ on $\partialΩ$, where $Ω\subset \R^n$ ( $n\geq 3$) is a bounded domain, $V$ is a smooth positive function on $\overlineΩ$, $p+1= 2n/(n-2)$ is the critical Sobolev exponent, and $\varepsilon >0$ is a small parameter. First, we show that, unlike the case of weak convergence to zero, interior bubbling solutions with a nonzero weak limit cannot occur in low dimensions. We then treat the general setting by removing the restriction that blow-up points are confined to the interior. Using delicate asymptotic expansions of the gradient of the associated functional, we prove that in dimensions $n=4$ and $n=5$, single blow-up point cannot coexist with residual mass.\ We further elucidate the role of the sign of the normal derivative of the potential $V$ on the boundary: if it is positive, any single blow-up solution with residual mass must occur in the interior; if it is negative at some boundary point, boundary blow-up solutions with residual mass can be constructed. Finally, we construct both simple and non-simple interior blow-up solutions exhibiting residual mass, without any assumption on the sign of the normal derivative of $V$. These results provide new insights into the interaction between the potential, the geometry of the domain, and the critical nonlinearity.

• The paper proves that interior blow-up solutions with nonzero residual mass do not exist in low dimensions (n=3,4,5).
• The study shows that the sign of the normal derivative of V at the boundary critically determines whether blow-up occurs inside or on the boundary.
• For dimensions n≥7, explicit constructions reveal isolated and clustered bubble configurations with precise convergence rates and solution multiplicity.

Elliptic Dirichlet Problems Near Criticality: Residual Mass and Blow-Up Dynamics

Problem Framework and Motivation

The paper investigates the nonlinear elliptic Dirichlet problem

$$-\Delta u + V u = u^{p-\varepsilon}, \quad u > 0 \text{ in } \Omega, \quad u = 0 \text{ on } \partial \Omega,$$

where $\Omega \subset \mathbb{R}^n$ ($n \geq 3$) is bounded, $V \in C^3(\overline{\Omega})$ is strictly positive, and $p+1 = \frac{2n}{n-2}$ is the critical Sobolev exponent. The parameter $\varepsilon > 0$ is small, with the focus on the regime where $\varepsilon \to 0$ ("slightly subcritical").

This problem is archetypal in nonlinear elliptic PDEs, relating to phenomena in optical physics, population dynamics, and geometric equations such as the Yamabe problem. The criticality of the exponent manifests as a loss of compactness, making the limiting process ($\varepsilon \to 0$) nontrivial, and giving rise to blow-up solutions whose energy concentrates at isolated points.

A central question is the interaction between blow-up (bubbling) and "residual mass," i.e., non-vanishing weak limits, especially as it pertains to the geometry, dimension, and boundary behavior of the domain. Unlike previous studies that often restrict blow-up points to interior domains, this work systematically examines the possibility of boundary concentration and clarifies the influence of the sign of the boundary derivative of $V$.

Main Results and Analytical Structure

Non-existence of Residual Mass with Interior Blow-Up in Low Dimensions

For dimensions $n=3,4,5$, the authors rigorously prove that interior blow-up solutions with nonzero residual mass cannot exist. That is, if the limit of the approximate solutions yields a non-trivial weak limit, there cannot be interior energy concentration. This is accomplished via asymptotic expansions of the associated functional and refined balancing conditions among bubble parameters, carefully tracking error terms and boundary interactions. Notably, for $\Omega \subset \mathbb{R}^n$0, a delicate decomposition of error terms is required since leading and error terms are of comparable magnitude.

Non-coexistence for Boundary Blow-Up: Role of the Normal Derivative of $\Omega \subset \mathbb{R}^n$1

The influence of the sign of the normal derivative $\Omega \subset \mathbb{R}^n$2 at the boundary is shown to be critical:

• If $\Omega \subset \mathbb{R}^n$3 everywhere on $\Omega \subset \mathbb{R}^n$4, any blow-up solution with residual mass must concentrate in the interior (for $\Omega \subset \mathbb{R}^n$5).
• If $\Omega \subset \mathbb{R}^n$6 at a boundary point $\Omega \subset \mathbb{R}^n$7, boundary blow-up solutions with residual mass are possible, and explicit construction is achieved.

This dichotomy makes precise the analogy between boundary curvature prescription problems and the current nonlinear setting.

Construction of Complex Blow-Up Configurations in Higher Dimensions

For $\Omega \subset \mathbb{R}^n$8, the authors provide constructive existence proofs for both simple and clustered interior blow-up solutions with residual mass—without constraints on the boundary sign of $\Omega \subset \mathbb{R}^n$9. These include:

• Isolated bubbles: finite sums with spatial separation, with precise rates of concentration and convergence to the underlying $n \geq 3$0.
• Clustered bubbles: configurations where centers coalesce to a common point (critical point of $n \geq 3$1), with concentration rates governed by non-degenerate critical points of an auxiliary function involving $n \geq 3$2 and mutual interaction terms.

The machinery relies on Lyapunov-Schmidt reduction, asymptotic expansions of the functional gradient, and fixed-point arguments exploiting the local geometry of critical points.

Explicit Parameter Estimates and Solution Multiplicity

Quantitative rates of the concentration parameters ($n \geq 3$3) and spatial convergence ($n \geq 3$4) are given in terms of $n \geq 3$5, $n \geq 3$6, and geometric quantities. The multiplicity results depend on the critical points of $n \geq 3$7—each non-degenerate critical point potentially gives rise to a distinct blow-up profile.

For each collection of $n \geq 3$8 critical points, at least $n \geq 3$9 solutions are constructed, highlighting the combinatorial explosion in higher-dimensional settings.

Analytical Techniques and Innovations

The analysis is anchored in precise asymptotic expansions of variational gradients, yielding balancing conditions among bubble parameters. Dimension-dependent scaling and matching of main/error terms are meticulously handled; e.g., odd/even decomposition for $V \in C^3(\overline{\Omega})$0, as well as delicate estimates for boundary terms.

The paper generalizes previous approaches (notably for Neumann and Robin problems) by introducing projections of the bubble functions suited for Dirichlet boundary conditions, which entail new error terms and require sharper analysis.

Explicit interaction functions $V \in C^3(\overline{\Omega})$1 capture the mutual influence of concentrated bubbles at critical points, generalizing the classical concentration-compactness principle.

Implications and Future Directions

Theoretical Impact

The results elucidate the fine interplay between geometry, potential, and criticality in nonlinear elliptic PDEs with Dirichlet conditions. The demonstrated non-existence of interior residual-mass blow-up in low dimensions confirms and refines earlier conjectures and results for Neumann cases.

The role of boundary derivatives (especially their sign) is clarified and structurally tied to possible blow-up configurations, echoing geometric issues in scalar curvature prescription and boundary Yamabe problems.

Practical Relevance

Understanding concentration phenomena in these models is directly linked to applications in physics (e.g., laser optics) and biology (e.g., spatial models of population dynamics), where critical thresholds and boundary effects are important.

Extensions

Potential avenues include:

• Generalization to Robin boundary conditions.
• Investigation of multi-blow-up with residual mass in dimensions 3–6.
• Extension to slightly supercritical regimes ($V \in C^3(\overline{\Omega})$2).
• Classification of clustered versus isolated blow-up in more general potential/geometry settings.

Conclusion

This work provides a comprehensive technical account of blow-up phenomena for nonlinear Dirichlet elliptic problems near criticality, covering existence, non-existence, and construction of solutions with residual mass, all modulated by domain geometry and the boundary behavior of the potential. The dimension-dependent results and explicit solution configurations lay important groundwork for future exploration in both nonlinear analysis and applied PDEs.

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Reference: "Blowing-up Solutions with Residual Mass in a Slightly Subcritical Dirichlet Problem" (2604.23339)