In this paper the question of finding infinitely many solutions to the problem $−\Delta u +a(x)u =|u|^{p−2}u$ , in $R^N$, u ∈H^1(R^N), is considered when N≥2, p∈(2, 2N/(N−2)), and the potential a(x) is a positive function which is not required to enjoy symmetry properties. Assuming that a(x)satisfies a suitable “slow decay at infinity” condition and, moreover, that its graph has some “dips”, we prove that the problem admits either infinitely many nodal solutions orinfinitely many constant sign solutions. The proof method is purely variational and allows to describe the shape of the solutions.
Multiplicity of Positive and Nodal Solutions for Scalar Field Equations / Cerami, Giovanna; Molle, R.; Passaseo, D.. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 257:10(2014), pp. 3554-3606. [10.1016/j.jde.2014.07.002]
Multiplicity of Positive and Nodal Solutions for Scalar Field Equations
CERAMI, Giovanna;
2014-01-01
Abstract
In this paper the question of finding infinitely many solutions to the problem $−\Delta u +a(x)u =|u|^{p−2}u$ , in $R^N$, u ∈H^1(R^N), is considered when N≥2, p∈(2, 2N/(N−2)), and the potential a(x) is a positive function which is not required to enjoy symmetry properties. Assuming that a(x)satisfies a suitable “slow decay at infinity” condition and, moreover, that its graph has some “dips”, we prove that the problem admits either infinitely many nodal solutions orinfinitely many constant sign solutions. The proof method is purely variational and allows to describe the shape of the solutions.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
