We study two different versions of a supercritical biharmonic equation with a power-type nonlinearity. First, we focus on the equation \Delta^2 u = |u|^{p−1}u over the whole space R^n, where n > 4 and p > (n + 4)/(n − 4). Assuming that p < p_c, where p_c is a further critical exponent, we show that all regular radial solutions oscillate around an explicit singular radial solution. As it was already known, on the other hand, no such oscillations occur in the remaining case p ≥ p_c. We also study the Dirichlet problem for the equation \Delta^2 u = λ(1 + u)^p over the unit ball in R^n, where λ > 0 is an eigenvalue parameter, while n > 4 and p > (n+4)/(n−4) as before. When it comes to the extremal solution associated to this eigenvalue problem, we show that it is regular as long as p < p_c. Finally, we show that a singular solution exists for some appropriate λ > 0.
Supercritical biharmonic equations with power-type nonlinearity
FERRERO, ALBERTO;
2009-01-01
Abstract
We study two different versions of a supercritical biharmonic equation with a power-type nonlinearity. First, we focus on the equation \Delta^2 u = |u|^{p−1}u over the whole space R^n, where n > 4 and p > (n + 4)/(n − 4). Assuming that p < p_c, where p_c is a further critical exponent, we show that all regular radial solutions oscillate around an explicit singular radial solution. As it was already known, on the other hand, no such oscillations occur in the remaining case p ≥ p_c. We also study the Dirichlet problem for the equation \Delta^2 u = λ(1 + u)^p over the unit ball in R^n, where λ > 0 is an eigenvalue parameter, while n > 4 and p > (n+4)/(n−4) as before. When it comes to the extremal solution associated to this eigenvalue problem, we show that it is regular as long as p < p_c. Finally, we show that a singular solution exists for some appropriate λ > 0.File | Dimensione | Formato | |
---|---|---|---|
FerGruKar.pdf
file disponibile solo agli amministratori
Descrizione: Articolo principale
Tipologia:
Documento in Pre-print
Licenza:
DRM non definito
Dimensione
141.45 kB
Formato
Adobe PDF
|
141.45 kB | Adobe PDF | Visualizza/Apri Richiedi una copia |
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.