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• W2963758355 abstract "We study singular radially symmetric solution of the stationary Keller-Segel equation, that is, an elliptic equation with exponential nonlinearity, which is super-critical in dimension N≥3. The solutions are unbounded at the origin and we show that they describe the asymptotics of bifurcation branches of regular solutions. It is shown that for any ball and any k≥0, there is a singular solution that satisfies Neumann boundary condition and oscillates at least k times around the constant equilibrium. Moreover, we prove that in dimension 3≤N≤9 there are regular solutions satisfying Neumann boundary conditions that are close to singular ones when the value at the origin is close to infinity. Hence, it follows that there exist regular solutions on any ball with arbitrarily fast oscillations. For generic radii, we show that the bifurcation branches of regular solutions oscillate in the bifurcation plane when 4≤N≤9 and approach to a singular solution. In dimension N>10, we show that the Morse index of the singular solution is finite, and therefore the existence of regular solutions with fast oscillations is not expected. Nous étudions les solutions singulières radiales de l'équation de Keller-Segel stationnaire, qui est une équation elliptique avec une non-linéarité exponentielle, surcritique en dimension N≥3. Ces solutions ne sont pas bornées à l'origine et nous montrons qu'elles décrivent les asymptotiques des branches de bifurcation des solutions régulières. Nous prouvons également que pour toutes boules et tout k≥0, il existe une solution singulière qui satisfait une condition de Neumann homogène au bord et qui oscille exactement k fois autours de l'équilibre constant. De plus, nous montrons qu'en dimension 3≤N≤9, il existe des solutions régulières satisfaisant la condition de Neumann homogène au bord qui sont proches de nos solutions singulières. Par conséquent, dans toute boule, il existe des solutions régulières qui oscillent arbitrairement rapidement. Pour des rayons génériques, nous montrons que les branches de bifurcation des solutions régulières oscillent dans le plan de bifurcation quand 4≤N≤9. En dimension N>10, nous prouvons que l'indice de Morse des solutions singulières est fini. Par conséquent, nous ne nous attendons pas à trouver des solutions régulières qui oscillent rapidement dans ce cas." @default.
• W2963758355 created "2019-07-30" @default.
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• W2963758355 date "2020-02-01" @default.
• W2963758355 modified "2023-10-18" @default.
• W2963758355 title "Singular radial solutions for the Keller-Segel equation in high dimension" @default.
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• W2963758355 doi "https://doi.org/10.1016/j.matpur.2019.12.002" @default.
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