Abstract
We study the nonlocal equation
where \(N \ge 1\), \(\alpha \in (0, N)\), \(I_\alpha (x) = A_\alpha /|x |^{N - \alpha }\) is the Riesz potential and \(\varepsilon > 0\) is a small parameter. We show that if the external potential \(V \in C ({\mathbb {R}}^N; [0, \infty ))\) has a local minimum and \(p \in [2, (N + \alpha )/(N - 2)_+)\) then for all small \(\varepsilon > 0\) the problem has a family of solutions concentrating to the local minimum of \(V\) provided that: either \(p>1 + \max (\alpha , \frac{\alpha + 2}{2})/(N - 2)_+\), or \(p > 2\) and \(\liminf _{|x | \rightarrow \infty } V (x) |x |^2 > 0\), or \(p = 2\) and \(\inf _{x \in {\mathbb {R}}^N} V (x) (1 + |x |^{N-\alpha })>0\). Our assumptions on the decay of \(V\) and admissible range of \(p\ge 2\) are optimal. The proof uses variational methods and a novel nonlocal penalization technique that we develop in this work.
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Notes
It should be noted that lemma 15 in [50] only holds when there exists \(\gamma < 1\) such that \(\liminf _{|x | \rightarrow \infty } V (x) |x |^\gamma > 0\) (in the first term in (10) therein, one should read \(v(y)\) instead of \(v (x)\)). It is known that if \(\limsup _{|x | \rightarrow \infty } V (x) |x |^\gamma = 0\) for some \(\gamma > 1\), the problem (1) considered in [50] does not have a positive solution [41, theorem 3].
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Communicated by P. Rabinowitz.
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Moroz, V., Van Schaftingen, J. Semi-classical states for the Choquard equation. Calc. Var. 52, 199–235 (2015). https://doi.org/10.1007/s00526-014-0709-x
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DOI: https://doi.org/10.1007/s00526-014-0709-x