Ground states of nonlinear Choquard equations with multi-well potentials

Abstract

In this paper, we study minimizers of the Hartree-type energy functional Ea(u)≔∫RN∇u(x)2+V(x)u(x)2dx−ap∫RNIα∗u(x)pu(x)pdx,a≥0 under the mass constraint ∫RNu2dx=1, where p=N+α+2N with α ∈ (0, N) for N ≥ 2 is the mass critical exponent. Here Iα denotes the Riesz potential and the trapping potential 0≤V(x)∈Lloc∞(RN) satisfies limx→∞V(x)=∞. We prove that minimizers exist if and only if a satisfies a<a∗=Q22(p−1), where Q is a positive radially symmetric ground state of −Δu+u=(Iα∗up)up−2u in ℝN. The uniqueness of positive minimizers holds if a > 0 is small enough. The blow-up behavior of positive minimizers as a↗a∗ is also derived under some general potentials. Especially, we prove that minimizers must blow up at the central point of the biggest inscribed sphere of the set Ω ≔ {x ∈ ℝN, V(x) = 0} if Ω>0.

Existence and limit behavior of prescribed L2‐norm solutions for Schrödinger‐Poisson‐Slater systems in

“…Stimulated by other studies, 11,14,16,17 our final result focus on the limit behavior of nonnegative minimizers of (3) as N ↗ N * when one of the above 2 cases holds. (3) for N < N * .…”

“…Thus the study of existence and other properties of minimizers for (3) can provide us some information on prescribed L 2 -norm solutions of (1). Inspired by previous studies, 8,14,16,17 we firstly concern with the existence and nonexistence of minimizers for problem (3) under the assumption of m(x). Here, we assume that m(x) satisfies We find that there exists a critical value N * > 0 such that (3) has at least one minimizer if N ∈ (0, N * ) and no minimizers if N > N * .…”

“…Similar conclusions to Theorem 1.1 were also given elsewhere for different type of problem, see, for instance, other studies. 11,14,16,17,23 However, the methods used in these papers do not work here anymore, due to the possible loss of compactness, and we overcome this difficulty by applying Lions's concentration-compactness method (cf Lions 24,25 ). As an extension of Theorem 1.1, we next concern with the existence and nonexistence of minimizers at the critical value N = N * .…”

“…Especially, (1) shows that the existence of minimizers may occur at the critical value N = N * , which is quit different from that given in previous studies. 11,14,16,17 Based on Theorems 1.1 and 1.2, we start to study the properties of the L 2 -norm solutions for (1) by investigating minimizers of (3).…”

“…Indeed, the inequality (16) implies that E (u) is bounded from below for any ∈ (0, N), where N ∈ (0, N * ) is fixed. Let {v n } ⊂ H 1 (R 3 ) be a minimizing sequence of e( ) so that ||v n || 2 2 = and lim n→∞ E (v n ) = e( ).…”