Volume 25, Issue 3

The Willmore flow is the negative gradient flow of the Willmore energy. In this paper, we consider a special kind of solutions to Willmore flow, which we call solitons, and investigate their geometric properties.

In the present paper, we consider the coupled Schrödinger systems with critical exponent: − Δ u i + λ V i ( x ) + a i u i = ∑ j = 1 d β i j u j 3 u i u i    in  R 3 , u i ∈ H 1 ( R N ) , i = 1,2 , … , d , $$\begin{cases}-{\Delta}{u}_{i}+\left(\lambda {V}_{i}\left(x\right)+{a}_{i}\right){u}_{i}=\sum _{j=1}^{d}{\beta }_{ij}{\left\vert {u}_{j}\right\vert }^{3}\left\vert {u}_{i}\right\vert {u}_{i}\quad \,\text{in}\,{\mathbb{R}}^{3},\quad \hfill \\ {u}_{i}\in {H}^{1}\left({\mathbb{R}}^{N}\right),\quad i=1,2,\dots ,d,\quad \hfill \end{cases}$$ where d ≥ 2, β ii > 0 for every i , β ij = β ji when i ≠ j , λ > 0 is a parameter and 0 ≤ V i ∈ L loc  ∞ R N $0\le {V}_{i}\in {L}_{\text{loc\,}}^{\infty }\left({\mathbb{R}}^{N}\right)$ have a common bottom int  V i − 1 ( 0 ) ${V}_{i}^{-1}\left(0\right)$ composed of ℓ 0 ℓ 0 ≥ 1 ${\ell }_{0}\left({\ell }_{0}\ge 1\right)$ connected components Ω k k = 1 ℓ 0 ${\left\{{{\Omega}}_{k}\right\}}_{k=1}^{{\ell }_{0}}$ , where int  V i − 1 ( 0 ) ${V}_{i}^{-1}\left(0\right)$ is the interior of the zero set V i − 1 ( 0 ) = x ∈ R N ∣ V i ( x ) = 0 ${V}_{i}^{-1}\left(0\right)=\left\{x\in {\mathbb{R}}^{N}\mid {V}_{i}\left(x\right)=0\right\}$ of V i . We study the existence of least energy positive solutions to this system which are trapped near the zero sets int  V i − 1 ( 0 ) ${V}_{i}^{-1}\left(0\right)$ for λ > 0 large for weakly cooperative case β i j > 0 s m a l l $\left({\beta }_{ij}{ >}0 \mathrm{s}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{l}\right)$ and for purely competitive case β i j ≤ 0 $\left({\beta }_{ij}\le 0\right)$ . Besides, when d = 2, we construct a one-bump fully nontrivial solution which is localised at one prescribed components Ω k k = 1 ℓ ${\left\{{{\Omega}}_{k}\right\}}_{k=1}^{\ell }$ for large λ .

In this paper, we consider the following nonlocal Schrödinger system − a + b ∫ R 3 | ∇ u 1 | 2 d x Δ u 1 = λ 1 u 1 + μ 1 | u 1 | p 1 − 2 u 1 + β r 1 | u 1 | r 1 − 2 u 1 | u 2 | r 2 , − a + b ∫ R 3 | ∇ u 2 | 2 d x Δ u 2 = λ 2 u 2 + μ 2 | u 2 | p 2 − 2 u 2 + β r 2 | u 1 | r 1 | u 2 | r 2 − 2 u 2 , ∫ R 3 | u 1 | 2 d x = c 1 , ∫ R 3 | u 2 | 2 d x = c 2 . $$\begin{cases}-\left(a+b{\int }_{{\mathbb{R}}^{3}}\vert \nabla {u}_{1}{\vert }^{2}\mathrm{d}x\right){\Delta}{u}_{1}={\lambda }_{1}{u}_{1}+{\mu }_{1}\vert {u}_{1}{\vert }^{{p}_{1}-2}{u}_{1}+\beta {r}_{1}\vert {u}_{1}{\vert }^{{r}_{1}-2}{u}_{1}\vert {u}_{2}{\vert }^{{r}_{2}},\quad \hfill \\ -\left(a+b{\int }_{{\mathbb{R}}^{3}}\vert \nabla {u}_{2}{\vert }^{2}\mathrm{d}x\right){\Delta}{u}_{2}={\lambda }_{2}{u}_{2}+{\mu }_{2}\vert {u}_{2}{\vert }^{{p}_{2}-2}{u}_{2}+\beta {r}_{2}\vert {u}_{1}{\vert }^{{r}_{1}}\vert {u}_{2}{\vert }^{{r}_{2}-2}{u}_{2},\quad \hfill \\ {\int }_{{\mathbb{R}}^{3}}\vert {u}_{1}{\vert }^{2}\mathrm{d}x={c}_{1}, {\int }_{{\mathbb{R}}^{3}}\vert {u}_{2}{\vert }^{2}\mathrm{d}x={c}_{2}.\quad \hfill \end{cases}$$ In the case of b = 0, 2 < p 1 , p 2 < 2*, the existence of a Mountain Pass solution and a global minimizer to the above problem was obtained (see T. Bartsch, L. Jeanjean [Proc. R. Soc. Edinburgh, Sect. A, 148 (2018), pp. 225–242.]). However, in the case of b > 0, p 1 = p 2 = 2*, 2 < r 1 + r 2 < p ̂ $2{< }{r}_{1}+{r}_{2}{< }\hat{p}$ or p ̄ ≤ r 1 + r 2 < 2 * $\bar{p}\le {r}_{1}+{r}_{2}{< }{2}^{{\ast}}$ , similar existence results to the above problem is still unknown, here 2* is Sobolev critical exponent and p ̂ = 10 3 $\hat{p}=\frac{10}{3}$ , p ̄ = 14 3 $\bar{p}=\frac{14}{3}$ are the L 2 -critical exponent for the system when b = 0 and b > 0 respectively. Here we focus on these unknown case. We investigate the existence of positive normalized solutions under different assumptions on β > 0. Figure 1, Tables 1 and 2 will illustrate our main results and the relationship between our work and some related works in the literature.

In this paper, we consider the following nonlinear Choquard equation with prescribed L 2 -norm: − Δ u + λ u = I α ∗ F ( u ) f ( u )  in  R N , ∫ R N | u | 2 d x = a > 0 , u ∈ H 1 ( R N ) , $\begin{cases}-{\Delta}u+\lambda u=\left({I}_{\alpha }\ast F\left(u\right)\right)f\left(u\right) \,\text{in}\, {\mathbb{R}}^{N},\quad \hfill \\ {\int }_{{\mathbb{R}}^{N}}\vert u{\vert }^{2}\mathrm{d}x=a{ >}0, u\in {H}^{1}\left({\mathbb{R}}^{N}\right),\quad \hfill \end{cases}$ where N ≥ 3 , α ∈ ( 0 , N ) , I α ( x ) = A α | x | N − α $N\ge 3,\alpha \in \left(0,N\right),{I}_{\alpha }\left(x\right)=\frac{{A}_{\alpha }}{\vert x{\vert }^{N-\alpha }}$ is the Riesz potential, f ∈ C ( R , R ) $f\in C\left(\mathbb{R},\mathbb{R}\right)$ , F ( s ) = ∫ 0 s f ( t ) d t $F\left(s\right)={\int }_{0}^{s}f\left(t\right)\mathrm{d}t$ and λ is an unknown Lagrange multiplier. Under the general assumption of F and within an appropriate mass range, we prove the existence and multiplicity of solutions to this problem, which may manifest as global minimizer, local minimizer, or mountain pass-type solutions.

In the present paper, we study the well-posedness of the solution to the initial boundary value problem for the damped Kirchhoff-type wave equation with fractional Laplacian. First, the existence and uniqueness of the local solution are established by the Banach fixed point theorem. Then, the global existence and finite time blowup of the solution are derived at the subcritical and critical initial energy levels. Finally, the finite time blowup of the solution and upper bound and lower bound estimate of blowup time are given at the arbitrarily positive initial energy level.

Moving from the seminal papers by Bobkov and Tanaka [“On positive solutions for ( p , q )-Laplace equations with two parameters,” Calc. Var. Partial Differ. Equ. , vol. 54, pp. 3277–3301, 2015, “Remarks on minimizers for ( p , q )-Laplace equations with two parameters,” Commun. Pure Appl. Anal. , vol. 17, pp. 1219–1253, 2018, “Multiplicity of positive solutions for ( p , q )-Laplace equations with two parameters,” Commun. Contemp. Math. , vol. 24, 2022, Art. no. 2150008] on the spectrum of the ( p , q )-Laplacian, we analyze the case of the double-phase operator. We discuss the region of parameters in which existence and non-existence of positive solutions occur. The proofs are based on normalization procedures, the Nehari manifold, and truncation techniques, exploiting Picone-type inequalities and an ad-hoc strong maximum principle.

In this paper, we study a nonlinear system involving a generalized tempered fractional p -Laplacian in B 1 (0): ∂ t u ( x , t ) + ( − Δ − λ f ) p s u ( x , t ) = g ( t , u ( x , t ) ) , ( x , t ) ∈ B 1 ( 0 ) × [ 0 , + ∞ ) , u ( x ) = 0 , ( x , t ) ∈ B 1 c ( 0 ) × [ 0 , + ∞ ) , $$\begin{cases}_{t}u\left(x,t\right)+{\left(-{\Delta}-{\lambda }_{f}\right)}_{p}^{s}u\left(x,t\right)=g\left(t,u\left(x,t\right)\right),\quad \hfill & \left(x,t\right)\in {B}_{1}\left(0\right){\times}\left[0,+\infty \right),\hfill \\ u\left(x\right)=0,\quad \hfill & \left(x,t\right)\in {B}_{1}^{c}\left(0\right){\times}\left[0,+\infty \right),\hfill \end{cases}$$ where 0 < s < 1, p > 2, n ≥ 2. We establish Hopf’s lemma for parabolic equations involving a generalized tempered fractional p -Laplacian. Hopf’s lemma will become powerful tools in obtaining qualitative properties of solutions for nonlocal parabolic equations.

We construct free boundary minimal disc stackings, with any number of strata, in the three-dimensional Euclidean unit ball, and prove uniform, linear lower and upper bounds on the Morse index of all such surfaces. Among other things, our work implies for any positive integer k the existence of k -tuples of distinct, pairwise non-congruent, embedded free boundary minimal surfaces all having the same topological type. In addition, since we prove that the equivariant Morse index of any such free boundary minimal stacking, with respect to its maximal symmetry group, is bounded from below by (the integer part of) half the number of layers, it follows that any possible realization of such surfaces via an equivariant min-max method would need to employ sweepouts with an arbitrarily large number of parameters. This also shows that it is only for N = 2 and N = 3 layers that free boundary minimal disc stackings can be obtained by means of one-dimensional mountain pass schemes.

The first purpose of this paper is to establish the singular Trudinger–Moser inequality in R 2 ${\mathbb{R}}^{2}$ for the Aharonov–Bohm magnetic fields. The second purpose is to derive the singular Hardy–Trudinger–Moser inequality in the unit ball B 2 ${\mathbb{B}}^{2}$ with Aharonov–Bohm magnetic potential. Moreover, we will show the constant 4 π ( 1 − β 2 ) $4\pi \left(1-\frac{\beta }{2}\right)$ is sharp in these two inequalities. The main skills include providing the asymptotic estimates of the related heat kernel and adapting the level set to derive a global Trudinger–Moser inequality from a local one. These results extend the inequalities established by Lu and Yang [Calc. Var. Partial Differ. Equ., 63 (2024)] into the weighted version.

We consider a class of one-dimensional elliptic equations possessing a p ( x )-harmonic functional as a nonlocal coefficient. As part of our results we treat the model case − M ∫ 0 1 u ′ ( x ) p ( x ) d x u ′ ′ ( t ) = λ f t , u ( t ) ,  0 < t < 1 $${-}M\left(\underset{0{}}{\overset{1}{\int }}{\left\vert {u}^{\prime }(x)\right\vert }^{p(x)}\ \mathrm{d}x\right){u}^{\prime \prime }(t)=\lambda f\left(t,u(t)\right)\text{,\hspace{0.17em}}0< t< 1$$ subject to the boundary data u ( 0 ) = 0 = u ( 1 ) . $$u(0)=0=u(1).$$ In addition, we consider a broader class of problems, of which the model case in a special case, by writing the argument of M as a finite convolution. As part of the analysis, a simple but fundamental lemma in introduced that allows the estimation of u ′ ( x ) p ( x ) ${\left\vert {u}^{\prime }(x)\right\vert }^{p(x)}$ in terms of constant exponents; this is the key to circumventing the variable exponent. An unusual array of analytical tools is used, including Sobolev’s inequality. Our results address both existence and nonexistence of solution.

In this paper, we investigate two anisotropic Gaussian curvature flows. Through establishing the long-time existence and congergence for these two flows, we derive the existence results for the Orlicz-Gaussian Minkowski problem in both origin-symmetric and general convex body settings.

A solution of the Allen–Cahn equation in R 3 ${\mathbb{R}}^{3}$ is called a two-end solution if its nodal set is asymptotic to ( x ′ , z ) ∈ R 3 : z = k i ⁡ ln | x ′ | + c i , 1 ≤ i ≤ 2 $\left\{\left({x}^{\prime },z\right)\in {\mathbb{R}}^{3}:z={k}_{i}\mathrm{ln}\vert {x}^{\prime }\vert +{c}_{i},1\le i\le 2\right\}$ at infinity. In this paper, we show that two-end solutions are axially symmetric and monotonic if k 1 , k 2 satisfy k 1 − k 2 > 2 2 ${k}_{1}-{k}_{2}{ >}2\sqrt{2}$ . We also establish the nonexistence of two-end solution with k 1 , k 2 satisfying − 2 2 < k 2 < k 1 < 2 2 $-\frac{\sqrt{2}}{2}{< }{k}_{2}{< }{k}_{1}{< }\frac{\sqrt{2}}{2}$ or k 1 = − k 2 = 2 2 ${k}_{1}=-{k}_{2}=\frac{\sqrt{2}}{2}$ .

In this article, we focus on studying space-time fractional parabolic equations with the nonlocal Bellman operator and the Marchaud fractional derivative. To address the difficulty caused by the space-time non-locality of operator ∂ t α − F s ${\partial }_{t}^{\alpha }-{F}_{s}$ , we first establish the narrow region principle and the maximum principle. Next, we establish a direct method of moving planes applicable to the operator ∂ t α − F s ${\partial }_{t}^{\alpha }-{F}_{s}$ . Finally, combining perturbation techniques and limit arguments, we apply this direct method of moving planes to prove the radial symmetry and monotonicity of solutions for the space-time fractional parabolic equations involving the nonlocal Bellman operator and the Marchaud fractional derivative. Additionally, the Liouville theorem of homogeneous space-time fractional parabolic equation is proved.

Using the normal form theory and the existence of an algebraic inverse integrating factor we characterize the local analytic integrability of the systems whose quasi-homogeneous leading term is a 1 y 3 + a 2 x 3 y , b 1 x 5 + b 2 x 2 y 2 $\left({a}_{1}{y}^{3}+{a}_{2}{x}^{3}y,{b}_{1}{x}^{5}+{b}_{2}{x}^{2}{y}^{2}\right)$ . More specifically we prove that the analytic integrable vector fields inside such family are orbitally equivalent to a semi-quasi-homogeneus system, that is, are not orbitally equivalent to its lowest-degree quasi-homogeneous term.

In this paper, we obtain a general supercritical Sobolev inequality in W 0 , r a d 1 , p ( B ) ${W}_{0,rad}^{1, p}\left(B\right)$ , where B is the unit ball in R N ${\mathbb{R}}^{N}$ . In order to discuss the sharp growth, we consider p -Laplacian elliptic problem − Δ p u = u q ( r )     in   B , u > 0     in   B , u = 0     on   ∂ B , $$\begin{cases}-{{\Delta}}_{p}u={u}^{q\left(r\right)}\quad \hfill & \quad \text{in}\quad B,\hfill \\ u{ >}0\quad \hfill & \quad \text{in}\quad B,\hfill \\ u=0\quad \hfill & \quad \text{on}\quad \partial B,\hfill \end{cases}$$ where Δ p u = div(|∇ u | p −2 ∇ u ), 1 < p < N , p * = N p N − p ${p}^{{\ast}}=\frac{Np}{N-p}$ , r = | x | and the variable exponent q ( r ) ≥ p * − 1 is a differentiable function in [0,1]. We give a Pohozaev identity and prove nonexistence of positive solutions to supercritical p -Laplacian equation in the unit ball B , which is significant for studying the nonlinear problems with variable exponential growth. Moreover, we also give the result of existence for a general nonlinearity case. Finally, we also give some results for a class of concave-convex nonlinear problem.