Fractional eigenvalue problems on RN
Let N ≥ 2 be an integer. For each real number s ∈ (0, 1) we denote by (−∆) s the corresponding fractional Laplace operator. First, we investigate the eigenvalue problem (−∆) su = λV(x)u on RN, where V : RN → R is a given function. Under suitable conditions imposed on V we show the existence of an un...
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| Dokumentumtípus: | Folyóirat |
| Megjelent: |
2020
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| Sorozat: | Electronic journal of qualitative theory of differential equations
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| Kulcsszavak: | Differenciálegyenlet |
| doi: | 10.14232/ejqtde.2020.1.26 |
| Online Access: | http://acta.bibl.u-szeged.hu/69530 |
| Tartalmi kivonat: | Let N ≥ 2 be an integer. For each real number s ∈ (0, 1) we denote by (−∆) s the corresponding fractional Laplace operator. First, we investigate the eigenvalue problem (−∆) su = λV(x)u on RN, where V : RN → R is a given function. Under suitable conditions imposed on V we show the existence of an unbounded, increasing sequence of positive eigenvalues. Next, we perturb the above eigenvalue problem with a fractional (t, p)-Laplace operator, when t ∈ (0, 1) and p ∈ (1, ∞) are such that t < s and s − N/2 = t − N/p. We show that when the function V is nonnegative on RN, the set of eigenvalues of the perturbed eigenvalue problem is exactly the unbounded interval (λ1, ∞), where λ1 stands for the first eigenvalue of the initial eigenvalue problem. |
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| ISSN: | 1417-3875 |