Ground states of semilinear elliptic equations: a geometric approach

✍ Scribed by Rodrigo Bamón; Isabel Flores; Manuel del Pino

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 252 KB
• Volume: 17
• Category: Article
• ISSN: 0294-1449
• DOI: 10.1016/s0294-1449(00)00126-8

✦ Synopsis

We consider the problem u + u p + u q = 0, in R N 0 < u(x) → 0 as |x| → +∞, where 1 < p < (N + 2)/(N -2) < q. We prove that if q is fixed and we let p approach (N + 2)/(N -2) from below, then this problem has a large number of radial solutions. A similar fact takes place if we fix p > N/(N -2) and then let q approach (N + 2)/(N -2). If we fix q and then let p be close enough to N/(N -2) then no solutions exist. © 2000 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ. -On considère le problème de trouver des solutions de l'equation elliptique u + u p + u q = 0, dans R N avec 0 < u(x) → 0 lorsque |x| → +∞, où 1 < p < (N + 2)/(N -2) < q. Si l'on fixe q et p augmente et tend vers (N + 2)/(N -2) alors il'y a un grand nombre des solutions radials.