Volume 24, Issue 2 - Special Issue: Geometric, topological and variational methods in PDEs

In this paper, we consider the general dual fractional parabolic problem ∂ t α u ( x , t ) + L u ( x , t ) = f ( t , u ( x , t ) ) in R n × R . ${\partial }_{t}^{\alpha }u\left(x,t\right)+\mathcal{L}u\left(x,t\right)=f\left(t,u\left(x,t\right)\right) \text{in} {\mathbb{R}}^{n}{\times}\mathbb{R}.$ We show that the bounded entire solution u satisfying certain one-direction asymptotic assumptions must be monotone increasing and one-dimensional symmetric along that direction under an appropriate decreasing condition on f . Our result here actually solves a well-known problem known as Gibbons’ conjecture in the setting of the dual fractional parabolic equations. To overcome the difficulties caused by the nonlocal divergence type operator L $\mathcal{L}$ and the Marchaud time derivative ∂ t α ${\partial }_{t}^{\alpha }$ , we introduce several new ideas. First, we derive a general weighted average inequality corresponding to the nonlocal operator L $\mathcal{L}$ , which plays a fundamental bridging role in proving maximum principles in unbounded domains. Then we combine these two essential ingredients to carry out the sliding method to establish the Gibbons’ conjecture. It is worth noting that our results are novel even for a special case of L $\mathcal{L}$ , the fractional Laplacian (−Δ) s , and the approach developed in this paper will be adapted to a broad range of nonlocal parabolic equations involving more general Marchaud time derivatives and more general non-local elliptic operators.

In this paper we study the following nonlinear fractional Hartree (or Choquard-Pekar) equation ( − Δ ) s u + μ u = ( I α * F ( u ) ) F ′ ( u )   in R N , ${\left(-{\Delta}\right)}^{s}u+\mu u=\left({I}_{\alpha }{\ast}F\left(u\right)\right){F}^{\prime }\left(u\right)\quad \text{in} {\mathbb{R}}^{N},$ (*) where μ > 0, s ∈ (0, 1), N ≥ 2, α ∈ (0, N ), I α ∼ 1 | x | N − α ${I}_{\alpha }\sim \frac{1}{\vert x{\vert }^{N-\alpha }}$ is the Riesz potential, and F is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions u ∈ H s ( R N ) $u\in {H}^{s}\left({\mathbb{R}}^{N}\right)$ , by assuming F odd or even: we consider both the case μ > 0 fixed and the case ∫ R N u 2 = m > 0 ${\int }_{{\mathbb{R}}^{N}}{u}^{2}=m{ >}0$ prescribed. Here we also simplify some arguments developed for s = 1 (S. Cingolani, M. Gallo, and K. Tanaka, “Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities,” Calc. Var. Partial Differ. Equ. , vol. 61, no. 68, p. 34, 2022). A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions (H. Berestycki and P.-L. Lions, “Nonlinear scalar field equations II: existence of infinitely many solutions,” Arch. Ration. Mech. Anal. , vol. 82, no. 4, pp. 347–375, 1983); for (*) the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as μ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any m > 0. The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a C 1 -regularity.

In this paper, we are concerned with the Hénon-Hardy type systems with exponential nonlinearity on a half space R + 2 ${\mathbb{R}}_{+}^{2}$ : ( − Δ ) α 2 u ( x ) = | x | a u p 1 ( x ) e q 1 v ( x ) , x ∈ R + 2 , ( − Δ ) v ( x ) = | x | b u p 2 ( x ) e q 2 v ( x ) , x ∈ R + 2 , $\begin{cases}{\left(-{\Delta}\right)}^{\frac{\alpha }{2}}u\left(x\right)=\vert x{\vert }^{a}{u}^{{p}_{1}}\left(x\right){e}^{{q}_{1}v\left(x\right)}, x\in {\mathbb{R}}_{+}^{2},\quad \hfill \\ \left(-{\Delta}\right)v\left(x\right)=\vert x{\vert }^{b}{u}^{{p}_{2}}\left(x\right){e}^{{q}_{2}v\left(x\right)}, x\in {\mathbb{R}}_{+}^{2},\quad \hfill \end{cases}$ with Dirichlet boundary conditions, where 0 <  α  < 2 and p 1 , p 2 , q 1 , q 2  > 0. First, we derived the integral representation formula corresponding to the above system under the assumption p 1 ≥ − 2 a α − 1 ${p}_{1}\ge -\frac{2a}{\alpha }-1$ . Then, we prove Liouville theorem for solutions to the above system via the method of scaling spheres.

In this paper, we summarize some of the recent developments in the area of fractional elliptic and parabolic equations with focus on how to apply the sliding method and the method of moving planes to obtain qualitative properties of solutions. We will compare the two methods and point out the pros and cons of each. We will demonstrate how to modify the ideas and techniques in studying fractional elliptic equations and then to employ them to investigate fractional parabolic problems. Besides deriving monotonicity of solutions, some other applications of the sliding method will be illustrated. These results have more or less appeared in a series of previous literatures, in which the ideas were usually submerged in detailed calculations. What we are trying to do here is to single out these ideas and illuminate the inner connections among them by using figures and intuitive languages, so that the readers can see the whole picture and quickly grasp the essence of these useful methods and will be able to apply them to solve a variety of other fractional elliptic and parabolic problems.

In this paper, let α be any real number between 0 and 2, we study the following semi-linear elliptic system involving the fractional Laplacian: ( − Δ ) α / 2 u ( x ) = f ( u ( x ) , v ( x ) ) , x ∈ R n , ( − Δ ) α / 2 v ( x ) = g ( u ( x ) , v ( x ) ) , x ∈ R n . $\begin{cases}{\left(-{\Delta}\right)}^{\alpha /2}u\left(x\right)=f\left(u\left(x\right),v\left(x\right)\right), x\in {\mathbb{R}}^{n},\quad \hfill \\ {\left(-{\Delta}\right)}^{\alpha /2}v\left(x\right)=g\left(u\left(x\right),v\left(x\right)\right), x\in {\mathbb{R}}^{n}.\quad \hfill \end{cases}$ Under nature structure conditions on f and g , we classify the positive solutions for the semi-linear elliptic system involving the fractional Laplacian by using the direct method of the moving spheres introducing by W. Chen, Y. Li, and R. Zhang (“A direct method of moving spheres on fractional order equations,” J. Funct. Anal. , vol. 272, pp. 4131–4157, 2017). In the half space, we establish a Liouville type theorem without any assumption of integrability by combining the direct method of moving planes and moving spheres, which improves the result proved by W. Dai, Z. Liu, and G. Lu (“Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space,” Potential Anal. , vol. 46, pp. 569–588, 2017).

We establish some sharp affine weighted L 2 Sobolev inequalities on the upper half space, which involves a divergent operator with degeneracy on the boundary. Moreover, for some certain exponents cases, we also characterize the extremal functions and best constants. Our approach only relies on the L 2 structure of gradient norm, affine invariance and a class of weighted L 2 Sobolev inequality on the upper half space. This is a simple approach which does not depend on the geometric structure of Euclidean space such as Brunn–Minkowski theory on convex geometry.

In this paper we are concerned with the existence of segregated non-radial solutions for nonlinear Schrödinger systems with a large number of components in a weak fully attractive or repulsive regime in presence of a suitable external radial potential.

We investigate the Dirichelt problem involving the fractional Laplacian in the upper half-space R + n = x ∈ R n ∣ x 1 > 0 ${\mathbb{R}}_{+}^{n}=\left\{x\in {\mathbb{R}}^{n}\mid {x}_{1}{ >}0\right\}$ :   ( − Δ ) s u ( x ) = f ( u ( x ) ) , x ∈ R + n ,   u ( x ) > 0 , x ∈ R + n ,   u ( x ) = 0 , x ∉ R + n . \begin{cases}\quad \hfill & {\left(-{\Delta}\right)}^{s}u\left(x\right)=f\left(u\left(x\right)\right),\qquad x\in {\mathbb{R}}_{+}^{n},\hfill \\ \quad \hfill & \qquad u\left(x\right){ >}0,\qquad x\in {\mathbb{R}}_{+}^{n},\hfill \\ \quad \hfill & \qquad u\left(x\right)=0,\qquad x\notin {\mathbb{R}}_{+}^{n}.\hfill \end{cases}. . We prove the positive solutions are monotonic increasing in the x 1 -direction assuming u ( x ) grows no faster than | x | γ with γ ∈ (0, 2 s ) for | x | large. To start with, we develop a maximum principle on the narrow region. Then we apply a direct method of the moving planes for the fractional Laplacian to derive the monotonicity. As an application of the monotonicity result, we use it to prove nonexistence of bounded positive solutions in R + n ${\mathbb{R}}_{+}^{n}$ for f ( u ) = u p , p ∈ 1 , n − 1 + 2 s n − 1 − 2 s $p\in \left(1,\frac{n-1+2s}{n-1-2s}\right)$ .

We give an upper bound for the least possible energy of a sign-changing solution to the nonlinear scalar field equation − Δ u = f ( u ) , u ∈ D 1,2 ( R N ) , $-{\Delta}u=f\left(u\right), u\in {D}^{1,2}\left({\mathrm{R}}^{N}\right),$ where N ≥ 5 and the nonlinearity f is subcritical at infinity and supercritical near the origin. More precisely, we establish the existence of a nonradial sign-changing solution whose energy is smaller that 12 c 0 if N = 5, 6 and smaller than 10 c 0 if N ≥ 7, where c 0 is the ground state energy.

The existence of L 2 –normalized solutions is studied for the equation − Δ u + μ u = f ( x , u )     in R N , ∫ R N u 2 d x = m . $-{\Delta}u+\mu u=f\left(x,u\right)\quad \quad \text{in} {\mathbf{R}}^{N},\quad {\int }_{{\mathbf{R}}^{N}}{u}^{2} \mathrm{d}x=m.$ Here m > 0 and f ( x , s ) are given, f ( x , s ) has the L 2 -subcritical growth and ( μ , u ) ∈ R × H 1 ( R N ) are unknown. In this paper, we employ the argument in Hirata and Tanaka (“Nonlinear scalar field equations with L 2 constraint: mountain pass and symmetric mountain pass approaches,” Adv. Nonlinear Stud. , vol. 19, no. 2, pp. 263–290, 2019) and find critical points of the Lagrangian function. To obtain critical points of the Lagrangian function, we use the Palais–Smale–Cerami condition instead of Condition (PSP) in Hirata and Tanaka (“Nonlinear scalar field equations with L 2 constraint: mountain pass and symmetric mountain pass approaches,” Adv. Nonlinear Stud. , vol. 19, no. 2, pp. 263–290, 2019). We also prove the multiplicity result under the radial symmetry.

We focus on the following fractional ( p , q )-Choquard problem: ( − Δ ) p s u + ( − Δ ) q s u + V ( ε x ) ( | u | p − 2 u + | u | q − 2 u ) = 1 | x | μ * F ( u ) f ( u )  in  R N , u ∈ W s , p ( R N ) ∩ W s , q ( R N ) , u > 0  in  R N , $\begin{cases}{\left(-{\Delta}\right)}_{p}^{s}u+{\left(-{\Delta}\right)}_{q}^{s}u+V\left(\varepsilon x\right)\left(\vert u{\vert }^{p-2}u+\vert u{\vert }^{q-2}u\right)=\left(\frac{1}{\vert x{\vert }^{\mu }}{\ast}F\left(u\right)\right)f\left(u\right) \,\text{in}\,{\mathbb{R}}^{N},\quad \hfill \\ u\in {W}^{s,p}\left({\mathbb{R}}^{N}\right)\cap {W}^{s,q}\left({\mathbb{R}}^{N}\right), u{ >}0\,\text{in}\,{\mathbb{R}}^{N},\quad \hfill \end{cases}$ where ɛ > 0 is a small parameter, 0 < s < 1, 1 < p < q < N s $1{< }p{< }q{< }\frac{N}{s}$ , 0 < μ < sp , ( − Δ ) r s ${\left(-{\Delta}\right)}_{r}^{s}$ , with r ∈ { p , q }, is the fractional r -Laplacian operator, V : R N → R $V:{\mathbb{R}}^{N}\to \mathbb{R}$ is a positive continuous potential satisfying a local condition, f : R → R $f:\mathbb{R}\to \mathbb{R}$ is a continuous nonlinearity with subcritical growth at infinity and F ( t ) = ∫ 0 t f ( τ ) d τ $F\left(t\right)={\int }_{0}^{t}f\left(\tau \right) \mathrm{d}\tau $ . Applying suitable variational and topological methods, we relate the number of solutions with the topology of the set where the potential V attains its minimum value.