Advanced Nonlinear Studies

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.

The main result in this paper is that, for singular integral operators associated with standard kernels, local L ¹ -estimates imply global L p -estimates for every p ∈ (1, ∞ ). When combined with the result of Melnikov-Verdera, this yields a complete and self-contained proof of the L p -boundedness of the Cauchy operator on Lipschitz curves and chord-arc curves in the plane.

It is well-understood that partial sums of Fourier series for L ¹ functions don’t necessarily converge to the original function in the pointwise-almost everywhere sense. Estimates for averages of partial sum operators demonstrate that the divergence can be quantified. We establish a bilinear maximal estimate for averages of partial sums projectors.

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.

On general Carnot groups, the definition of a possible hypoelliptic Hodge-Laplacian on forms using the Rumin complex has been considered in (M. Rumin, “Differential geometry on C-C spaces and application to the Novikov-Shubin numbers of nilpotent Lie groups,” C. R. Acad. Sci., Paris Sér. I Math. , vol. 329, no. 11, pp. 985–990, 1999, M. Rumin, “Sub-Riemannian limit of the differential form spectrum of contact manifolds,” Geom. Funct. Anal. , vol. 10, no. 2, pp. 407–452, 2000), where the author introduced a 0-order pseudodifferential operator on forms. However, for questions regarding regularity for example, where one needs sharp estimates, this 0-order operator is not suitable. Up to now, there have only been very few attempts to define hypoelliptic Hodge-Laplacians on forms that would allow for such sharp estimates. Indeed, this question is rather difficult to address in full generality, the main issue being that the Rumin exterior differential d c is not homogeneous on arbitrary Carnot groups. In this note, we consider the specific example of the free Carnot group of step 3 with 2 generators, and we introduce three possible definitions of hypoelliptic Hodge-Laplacians. We compare how these three possible Laplacians can be used to obtain sharp div-curl type inequalities akin to those considered by Bourgain & Brezis and Lanzani & Stein for the de Rham complex, or their subelliptic counterparts obtained by Baldi, Franchi & Pansu for the Rumin complex on Heisenberg groups.

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.

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.

We survey recent results on soliton resolution for the energy critical nonlinear wave equation.

We present global Schauder type estimates in all variables and unique solvability results in kinetic Hölder spaces for kinetic Kolmogorov-Fokker-Planck (KFP) equations. The leading coefficients are Hölder continuous in the x , v variables and are merely measurable in the temporal variable. Our proof is inspired by Campanato’s approach to Schauder estimates and does not rely on the estimates of the fundamental solution of the KFP operator.

We construct the Neumann function in a 1-sided chord-arc domain (i.e., a uniform domain with an Ahlfors regular boundary), and establish size and Hölder continuity estimates up to the boundary. We then obtain a Kenig-Pipher type theorem, in which L p solvability of the Neumann problem is shown to yield solvability in L q for 1 < q < p , and in the Hardy space H ¹ , in 2-sided chord-arc domains, under suitable background hypotheses.

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.

We study the asymptotic behavior of positive least energy solutions to the weakly coupled nonlinear Schrödinger systems with nearly critical exponents. There are at most three kinds of asymptotic behaviors for positive solutions to two-coupled systems: (i) both two components converge; (ii) one component blows up while the other one converges; (iii) both two components blow up. Here we prove that any type of these three asymptotic behaviors will happen under different circumstances.

This paper is concerned with the Dirichlet eigenvalue problem for Laplace operator in a bounded domain with periodic perforation in the case of small volume. We obtain the optimal quantitative error estimates independent of the spectral gaps for an asymptotic expansion, with two leading terms, of Dirichlet eigenvalues. We also establish the convergence rates for the corresponding eigenfunctions. Our approach uses a known reduction to a degenerate elliptic eigenvalue problem for which a quantitative analysis is carried out.

In this paper, we are concerned with n-component Ginzburg-Landau equations on ${\mathbb{R}}^{2}$ . By introducing a diffusion constant for each component, we discuss that the n-component equations are different from n-copies of the single Ginzburg-Landau equations. Then, the results of Brezis-Merle-Riviere for the single Ginzburg-Landau equation can be nontrivially extended to the multi-component case. First, we show that if the solutions have their gradients in L ² space, they are trivial solutions. Second, we prove that if the potential is square summable, then it has quantized integrals, i.e., there exists one-to-one correspondence between the possible values of the potential energy and ${\mathbb{N}}^{n}$ . Third, we show that different diffusion coefficients in the system are important to obtain nontrivial solutions of n-component equations.

This paper is concerned with the existence of normalized solutions to a class of (2, q)-Laplacian equations in all the possible cases with respect to the mass critical exponents 2(1 + 2/N), q(1 + 2/N). In the mass subcritical cases, we study a global minimization problem and obtain a ground state solution. While in the mass critical cases, we prove several nonexistence results. At last, we derive a ground state and infinitely many radial solutions in the mass supercritical case. Compared with the classical Schrödinger equation, the (2, q)-Laplacian equation possesses a quasi-linear term, which brings in some new difficulties and requires a more subtle analysis technique. Moreover, the vector field $\overrightarrow {a}\left(\xi \right)=\vert \xi {\vert }^{q-2}\xi$ corresponding to the q-Laplacian is not strictly monotone when q < 2, so we shall consider separately the case q < 2 and the case q > 2.

This article is intended as an introduction to the distorted Fourier transform associated with a Schrödinger operator on the line or the half-line. This versatile tool has seen numerous applications in nonlinear PDE in recent years. It typically arises in the asymptotic stability analysis of topological solitons in classical field theory, such as kinks in the sine-Gordon or ϕ ⁴ models. The distorted Fourier transform is also a natural technique in the analysis of dispersive equations on manifolds with symmetries. Such models appear in general relativity, for example in the study of waves on a black-hole background. While microlocal methods have proven to be powerful in such applications, the more classical Weyl–Titchmarsh spectral theory and with it, the distorted Fourier transform, continue to play an essential role in the analysis of evolution PDEs. This article explain how it can be derived from Stone’s formula, which also establishes the Plancherel and inversion theorems.

The purpose of this paper is two-fold. First, we derive sharp Trudinger–Moser inequalities with logarithmic weights in fractional dimensions: $\,\underset{{\left(\underset{0}{\overset{1}{\int }}w\left(r\right){\left\vert {u}^{\prime }\left(r\right)\right\vert }^{\beta +2}\mathrm{d}{\lambda }_{\alpha }\right)}^{1/\left(\beta +2\right)}\le 1}{\mathrm{sup}}\underset{0}{\overset{1}{\int }}{\text{e}}^{{\mu }_{\alpha ,\theta ,\gamma }{\left\vert u\right\vert }^{\frac{\beta +2}{\left(\beta +1\right)\left(1-\gamma \right)}}}\mathrm{d}{\lambda }_{\theta }{< }+\infty ,$ where 0 ≤ γ < 1, α = β + 1, ${\mu }_{\alpha ,\theta ,\gamma }{:=}\left(\theta +1\right){\left[{\omega }_{\alpha }^{1/\alpha }\left(1-\gamma \right)\right]}^{\frac{1}{1-\gamma }}$ , $w\left(r\right)={w}_{1}\left(r\right)={\left(\mathrm{log}\frac{1}{r}\right)}^{\gamma \left(\beta +1\right)}$ or $w\left(r\right)={w}_{2}\left(r\right)={\left(\mathrm{log}\frac{e}{r}\right)}^{\gamma \left(\beta +1\right)}$ and λ θ (E) = ω θ ∫ E r θ dr for all $E\subset \mathbb{R}$ . The case γ > 1 and γ = 1 are also be considered in this part to improve our paper. Indeed, we have a continuous embedding X(w 2) ↪ L ∞(0, 1) for γ > 1 and a critical growth of double exponential type for γ = 1. Second, we apply the Lions type Concentration-Compactness principle for Trudinger–Moser inequalities and the precise estimate of normalized concentration limit for normalized concentrating sequence at origin to establish the existence of extremals for Trudinger–Moser inequalities when $w\left(r\right)={w}_{1}\left(r\right)={\left(\mathrm{log}\frac{1}{r}\right)}^{\gamma \left(\beta +1\right)}$ and γ > 0 is sufficiently small.

We consider incompressible inviscid elastodynamics equations with a free surface and establish regularity of solutions for these equations. Compared with previous result on this free boundary problem [X. Gu and F. Wang, Well-posedness of the free boundary problem in incompressible elastodynamic under the mixed type stability condition, J. Math. Anal. Appl., 482, (2020), 123529] in space H ³, we are able to establish regularity in space H 2.5+δ upon the Arbitrary Lagrangian-Eulerian (ALE) coordinates. It is achieved by reformulating the system into a new formulation with the ALE coordinates, presenting uniform estimates for the pressure, tangential estimates for the system, as well as curl and divergence estimates.

In this paper, we extend our convexity theory for C ² cost functions in optimal transportation to more general generating functions, which were originally introduced by the second author to extend the framework of optimal transportation to embrace near field geometric optics. In particular we provide an alternative geometric treatment to the previous analytic approach using differential inequalities, which also gives a different derivation of the invariance of the fundamental regularity conditions under duality. We also extend our local theory to cover the strict version of these conditions for C ² cost and generating functions.

For elliptic partial differential equations, the pure interior C ² estimates and Pogorelov type estimates are important issues. In this paper, we study the interior estimates of $\tilde {{{\Gamma}}_{k}}$ -admissible solutions for curvature quotient equations ${\left(\frac{{\sigma }_{k}\left(\eta \left(\kappa \right)\right)}{{\sigma }_{l}\left(\eta \left(\kappa \right)\right)}\right)}^{\frac{1}{k-l}}=g\left(x,u\right)$ , and establish the pure interior curvature estimate when l = k − 1, 0 < k ≤ n and Pogorelov type estimate when 0 ≤ l < k ≤ n.

Let V be a finite tree with radially decaying weights. We show that there exists a set $E\subset {\mathbb{R}}^{2}$ for which the following two problems are equivalent: (1) Given a (real-valued) function ϕ on the leaves of V, extend it to a function Φ on all of V so that ${\Vert}{\Phi}{{\Vert}}_{{L}^{1,p}\left(V\right)}$ has optimal order of magnitude. Here, L 1,p (V) is a weighted Sobolev space on V. (2) Given a function $f:E\to \mathbb{R}$ , extend it to a function $F\in {L}^{2,p}\left({\mathbb{R}}^{2}\right)$ so that ${\Vert}F{{\Vert}}_{{L}^{2,p}\left({\mathbb{R}}^{2}\right)}$ has optimal order of magnitude.

The Halo Conjecture presents one of the outstanding open problems in the theory of differentiation of integrals. In this paper we discuss the Halo Conjecture and its connections with topics of current interest in harmonic analysis.