The paper deals with problems of the type -Delta n + a(x)u = vertical bar u vertical bar(p-2)u, u > 0, with zero Dirichlet boundary condition on unbounded domains in R-N, N >= 2, with a(x) >=, c > 0, p > 2 and p < 2N/(N - 2) if N >= 3. The lack of compactness in the problem, related to the unboundedness of the domain, is analysed. Moreover, if the potential a(x) has k suitable ' bumps ' and the domain has h suitable ' holes ', it is proved that the problem has at least 2 (h + k) positive solutions (h or k can be zero). The multiplicity results are obtained under a geometric assumption on Omega at infinity which ensures the validity of a local Palais-Smale condition.
Molina, J., Molle, R. (2006). On elliptic problems in domains with unbounded boundary. PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY, 49(3), 709-734 [10.1017/S0013091504001592].
On elliptic problems in domains with unbounded boundary
MOLLE, RICCARDO
2006-01-01
Abstract
The paper deals with problems of the type -Delta n + a(x)u = vertical bar u vertical bar(p-2)u, u > 0, with zero Dirichlet boundary condition on unbounded domains in R-N, N >= 2, with a(x) >=, c > 0, p > 2 and p < 2N/(N - 2) if N >= 3. The lack of compactness in the problem, related to the unboundedness of the domain, is analysed. Moreover, if the potential a(x) has k suitable ' bumps ' and the domain has h suitable ' holes ', it is proved that the problem has at least 2 (h + k) positive solutions (h or k can be zero). The multiplicity results are obtained under a geometric assumption on Omega at infinity which ensures the validity of a local Palais-Smale condition.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
