Uniqueness of entire ground states for the fractional plasma problem
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Calculus of Variations
Uniqueness of entire ground states for the fractional plasma problem Hardy Chan1 · María Del Mar González2 · Yanghong Huang3 · Edoardo Mainini4 · Bruno Volzone5 Received: 15 July 2020 / Accepted: 22 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We establish uniqueness of vanishing radially decreasing entire solutions, which we call ground states, to some semilinear fractional elliptic equations. In particular, we treat the fractional plasma equation and the supercritical power nonlinearity. As an application, we deduce uniqueness of radial steady states for nonlocal aggregation-diffusion equations of Keller-Segel type, even in the regime that is dominated by aggregation. Mathematics Subject Classification 35K55 · 35R11 · 49K20
Communicated by Manuel del Pino.
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Bruno Volzone [email protected] Hardy Chan [email protected] María Del Mar González [email protected] Yanghong Huang [email protected] Edoardo Mainini [email protected]
1
Departement Mathematik., ETH Zürich, Rämistrasse 101, 8092 Zurich, Switzerland
2
Departamento de Matemáticas, and ICMAT, Universidad Autónoma de Madrid, 28049 Madrid, Spain
3
Department of Mathematics, University of Manchester, Oxford Rd, Manchester M13 9PL, UK
4
Dipartimento di Ingegneria meccanica, energetica, gestionale e dei trasporti, Università degli studi di Genova, Via all’Opera Pia 15, 16145 Genova, Italy
5
Dipartimento di Scienze e Tecnologie, Università degli Studi di Napoli “Parthenope”, Centro Direzionale Isola C4, 80143 Naples, Italy 0123456789().: V,-vol
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H. Chan et al.
1 Introduction We study positive entire ground states to the fractional semilinear equation p
(−)s u = a(u − C )+ in R N ,
(1.1)
where the parameters are in the range 0 < s < 1,
p ≥ 1,
C ≥ 0,
a > 0.
Here (−)s is the fractional Laplace operator on R N (s < 1/2 if N = 1). Moreover x+ := 0 ∨ x denotes the maximum of 0 and x. By a ground state we mean a bounded positive solution u to (1.1) which is radially decreasing and decays at infinity, i.e., u(x) → 0 for |x| → ∞. In the subcritical case p < (N + 2s)/(N − 2s) with C > 0, the free boundary problem (1.1) is the so called fractional plasma equation, and it is the object of our first main result. Theorem 1.1 Let 1 ≤ p < (N + 2s)/(N − 2s) and C > 0. There exists a unique ground state for equation (1.1). In our second main theorem, we investigate ground states in the critical and supercritical regime p ≥ (N +2s)/(N −2s) to equation (1.1), with the choice C = 0. A nontrivial solution exists only for this special case, as we will show that there are no ground states if C > 0 and p ≥ (N + 2s)/(N − 2s). Theorem 1.2 Let p ≥ (N + 2s)/(N − 2s). Let C = 0 and b > 0. There exists a unique ground state u for equation (1.1) such that u(0) = b. In the above results, ground state solutions are interpreted in the distributional sense. However, these solutions turn out to be continuous (hence smooth) and the equation is also
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