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Uniform boundedness for finite Morse index solutions to supercritical semilinear elliptic equations

Alessio Figalli, Yi Ru-Ya Zhang

TL;DR

The paper proves a universal interior $L^\infty$ bound for finite Morse index solutions to semilinear elliptic equations with supercritical nonlinearities in bounded convex domains, valid in the sharp dimension range $3 \le n \le 9$. The authors develop a toolbox for finite Morse index solutions, combining a stability/compactness framework, uniform $W^{1,2}$-integrability, and an $\varepsilon$-regularity result, under a growth condition linking $f$ and its primitive $F$. The main theorem yields uniform bounds for solutions and, via the continuity method, uniform bounds for Gelfand-type problems; they also establish a detailed bifurcation/turning-point structure for analytic nonlinearies on convex domains. The analysis relies on a blend of interior regularity, Derrick-Pohozaev arguments, and recent stability results for stable solutions (CFRS2020), with extensions to convex nonlinearities and potential adaptation to star-shaped domains. Overall, the results illuminate how finite Morse index controls prevent blow-up in the supercritical regime and provide tools applicable to related nonlinear elliptic problems and bifurcation theory.

Abstract

We consider finite Morse index solutions to semilinear elliptic questions, and we investigate their smoothness. It is well-known that: - For $n=2$, there exist Morse index $1$ solutions whose $L^\infty$ norm goes to infinity. - For $n \geq 3$, uniform boundedness holds in the subcritical case for power-type nonlinearities, while for critical nonlinearities the boundedness of the Morse index does not prevent blow-up in $L^\infty$. In this paper, we investigate the case of general supercritical nonlinearities inside convex domains, and we prove an interior a priori $L^\infty$ bound for finite Morse index solution in the sharp dimensional range $3\leq n\leq 9$. As a corollary, we obtain uniform bounds for finite Morse index solutions to the Gelfand problem constructed via the continuity method. [In the previous version, the proof of Proposition A.1 was not complete. Indeed, Lemma A.2 was proved only for estimates centered at the origin, and Footnote 9 claimed that the same estimates hold at every point; however, no complete proof was provided for this extension. In the current version, Lemma A.2 is restricted to estimates at the origin, and the boundedness at arbitrary points is obtained by combining the results in [9] with the new version of Lemma A.2. This avoids the need to extend Lemma A.2 to arbitrary centers and yields a complete proof of Proposition A.1.]

Uniform boundedness for finite Morse index solutions to supercritical semilinear elliptic equations

TL;DR

The paper proves a universal interior bound for finite Morse index solutions to semilinear elliptic equations with supercritical nonlinearities in bounded convex domains, valid in the sharp dimension range . The authors develop a toolbox for finite Morse index solutions, combining a stability/compactness framework, uniform -integrability, and an -regularity result, under a growth condition linking and its primitive . The main theorem yields uniform bounds for solutions and, via the continuity method, uniform bounds for Gelfand-type problems; they also establish a detailed bifurcation/turning-point structure for analytic nonlinearies on convex domains. The analysis relies on a blend of interior regularity, Derrick-Pohozaev arguments, and recent stability results for stable solutions (CFRS2020), with extensions to convex nonlinearities and potential adaptation to star-shaped domains. Overall, the results illuminate how finite Morse index controls prevent blow-up in the supercritical regime and provide tools applicable to related nonlinear elliptic problems and bifurcation theory.

Abstract

We consider finite Morse index solutions to semilinear elliptic questions, and we investigate their smoothness. It is well-known that: - For , there exist Morse index solutions whose norm goes to infinity. - For , uniform boundedness holds in the subcritical case for power-type nonlinearities, while for critical nonlinearities the boundedness of the Morse index does not prevent blow-up in . In this paper, we investigate the case of general supercritical nonlinearities inside convex domains, and we prove an interior a priori bound for finite Morse index solution in the sharp dimensional range . As a corollary, we obtain uniform bounds for finite Morse index solutions to the Gelfand problem constructed via the continuity method. [In the previous version, the proof of Proposition A.1 was not complete. Indeed, Lemma A.2 was proved only for estimates centered at the origin, and Footnote 9 claimed that the same estimates hold at every point; however, no complete proof was provided for this extension. In the current version, Lemma A.2 is restricted to estimates at the origin, and the boundedness at arbitrary points is obtained by combining the results in [9] with the new version of Lemma A.2. This avoids the need to extend Lemma A.2 to arbitrary centers and yields a complete proof of Proposition A.1.]

Paper Structure

This paper contains 15 sections, 15 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Finite Morse index vs uniform boundedness
  3. Main result: finite Morse index solutions with supercritical nonlinearities are uniformly bounded
  4. About Theorem \ref{['mainthm 1']}: extensions and tools used in the proof
  5. On the convexity of $\Omega$
  6. A result for convex nonlinearities
  7. Main tools
  8. An application to the Gelfand problem associated to analytic supercritical nonlinearities
  9. Structure of the paper
  10. Technical tools on finite Morse index solutions
  11. Uniform finite Morse index: Proof of Theorem \ref{['mainthm 1']}
  12. The case $\lambda_\infty>0$
  13. The case $\lambda_\infty=0$
  14. Boundedness of stable solutions in $B_2\setminus\{0\}$ for $3 \leq n \le 9$
  15. Uniform boundedness of solutions with spectrum bounded below

Key Result

Theorem 1.1

Let $3\le n\le 9$, $\Omega\subset \mathbb{R}^n$ a bounded convex domain, and $c_0>0$. Consider and assume that $\mathcal{K}$ is compact for the $C^1_{\rm loc}(\mathbb{R})$ topology. Let $\hat{\lambda} >0$, and let $u \in C^2(\Omega)$ solve for some $f \in \mathcal{K}$ and $\lambda \in [0,\hat{\lambda} ]$. Finally, assume that ${\rm ind}(u,\Omega)\leq k$ and that there exist $\epsilon,t_0>0$ such

Theorems & Definitions (31)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • proof
  • Remark 1.5
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • ...and 21 more