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Multiple normalized solutions for Choquard equation involving the biharmonic operator and competing potentials in ℝN
Preview AbstractThis paper is concerned with the existence of multiple normalized solutions for a class of Choquard equation involving the biharmonic operator and competing potentials in ℝN:
{Δ2u+V(𝜀x)u=λu+G(𝜀x)(Iμ∗F(u))f(u)in ℝN,∫ℝN|u|2dx=c2,where Δ2 is the biharmonic operator, c,𝜀>0, μ∈(0,N), λ∈ℝ is an unknown parameter that appears as the Lagrange multiplier, the absorption potential V, reaction potential G and nonlinear term f are continuous functions and satisfy some assumptions. With the help of the minimization techniques and Lusternik–Schnirelmann category, we show that the number of normalized solutions is not less than the number of global minimum points of V when the parameter 𝜀 is sufficiently small. Furthermore, and more interestingly, we can prove the existence of multiple solutions for this problem by using the Morse theory. As far as we know, this study seems to be the first contribution regarding the concentration behavior for Choquard equation involving the biharmonic operator.ON THE MAGNETIC PEKAR FUNCTIONAL AND THE EXISTENCE OF BIPOLARONS
Preview AbstractFirst, this paper proves the existence of a minimizer for the Pekar functional including a constant magnetic field and possibly some additional local fields that are energy reducing. Second, the existence of the aforementioned minimizer is used to establish the binding of polarons in the model of Pekar–Tomasevich including external fields.
On fractional Choquard equations
Preview AbstractWe investigate a class of nonlinear Schrödinger equations with a generalized Choquard nonlinearity and fractional diffusion. We obtain regularity, existence, nonexistence, symmetry as well as decays properties.
Groundstates of nonlinear Choquard equations: Hardy–Littlewood–Sobolev critical exponent
Preview AbstractWe consider nonlinear Choquard equation
where N ≥ 3, V ∈ L∞(ℝN) is an external potential and Iα(x) is the Riesz potential of order α ∈ (0, N). The power
in the nonlocal part of the equation is critical with respect to the Hardy–Littlewood–Sobolev inequality. As a consequence, in the associated minimization problem a loss of compactness may occur. We prove that if 
then the equation has a nontrivial solution. We also discuss some necessary conditions for the existence of a solution. Our considerations are based on a concentration compactness argument and a nonlocal version of Brezis–Lieb lemma.A strongly indefinite Choquard equation with critical exponent due to the Hardy–Littlewood–Sobolev inequality
Preview AbstractIn this paper, we are concerned with the following nonlinear Choquard equation
−Δu+V(x)u=(∫ℝNG(y,u)|x−y|μdy)g(x,u)in ℝN,where N≥4, 0<μ<N and G(x,u)=∫u0g(x,s)ds. If 0 lies in a gap of the spectrum of −Δ+V and g(x,u) is of critical growth due to the Hardy–Littlewood–Sobolev inequality, we obtain the existence of nontrivial solutions by variational methods. The main result here extends and complements the earlier theorems obtained in [N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z.248 (2004) 423–443; B. Buffoni, L. Jeanjean and C. A. Stuart, Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc.119 (1993) 179–186; V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc.367 (2015) 6557–6579].Isolated singularities of positive solutions for Choquard equations in sublinear case
Preview AbstractOur purpose of this paper is to study the isolated singularities of positive solutions to Choquard equation in the sublinear case q∈(0,1)
−Δu+u=Iα[up]uqinℝN∖{0},lim|x|→+∞u(x)=0,where p>0,N≥3,α∈(0,N) and Iα[up](x)=∫ℝNup(y)|x−y|N−αdy is the Riesz potential, which appears as a nonlocal term in the equation. We investigate the nonexistence and existence of isolated singular solutions of Choquard equation under different range of the pair of exponent (p,q). Furthermore, we obtain qualitative properties for the minimal singular solutions of the equation.Bifurcation into spectral gaps for strongly indefinite Choquard equations
Preview AbstractIn this paper, we consider the semilinear elliptic equations
{−Δu+V(x)u=(Iα∗|u|p)|u|p−2u+λuforx∈ℝN,u(x)→0as |x|→∞,where Iα is a Riesz potential, p∈(N+αN,N+αN−2), N≥3 and V is continuous periodic. We assume that 0 lies in the spectral gap (a,b) of −Δ+V. We prove the existence of infinitely many geometrically distinct solutions in H1(ℝN) for each λ∈(a,b), which bifurcate from b if N+αN<p<1+2+αN. Moreover, b is the unique gap-bifurcation point (from zero) in [a,b]. When λ=a, we find infinitely many geometrically distinct solutions in H2loc(ℝN). Final remarks are given about the eventual occurrence of a bifurcation from infinity in λ=a.Concentration phenomena for the fractional relativistic Schrödinger–Choquard equation
Preview AbstractWe consider the fractional relativistic Schrödinger–Choquard equation
{(−Δ+m2)su+V(ε x)u=(1|x|μ∗F(u))f(u)in ℝN,u∈Hs(ℝN),u>0 in ℝN,where ε>0 is a small parameter, s∈(0,1), m>0, N>2s, μ∈(0,2s), (−Δ+m2)s is the fractional relativistic Schrödinger operator, V:ℝN→ℝ is a continuous potential having a local minimum, f:ℝ→ℝ is a continuous nonlinearity with subcritical growth at infinity and F(t)=∫t0f(τ)dτ. Exploiting appropriate variational arguments, we construct a family of solutions concentrating around the local minimum of V as ε→0.