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Communications on Pure & Applied Analysis

January 2021 , Volume 20 , Issue 1

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On principal eigenvalues of biharmonic systems
Lingju Kong and Roger Nichols
2021, 20(1): 1-15 doi: 10.3934/cpaa.2020254 +[Abstract](642) +[HTML](89) +[PDF](347.6KB)
Abstract:

We prove the existence, positivity, simplicity, uniqueness up to nonnegative eigenfunctions, and isolation of the principle eigenvalue of a biharmonic system. We also provide the extension of our results to a related system.

A note on Riemann-Liouville fractional Sobolev spaces
Alessandro Carbotti and Giovanni E. Comi
2021, 20(1): 17-54 doi: 10.3934/cpaa.2020255 +[Abstract](687) +[HTML](106) +[PDF](508.61KB)
Abstract:

Taking inspiration from a recent paper by Bergounioux et al., we study the Riemann-Liouville fractional Sobolev space \begin{document}$ W^{s, p}_{RL, a+}(I) $\end{document}, for \begin{document}$ I = (a, b) $\end{document} for some \begin{document}$ a, b \in \mathbb{R}, a < b $\end{document}, \begin{document}$ s \in (0, 1) $\end{document} and \begin{document}$ p \in [1, \infty] $\end{document}; that is, the space of functions \begin{document}$ u \in L^{p}(I) $\end{document} such that the left Riemann-Liouville \begin{document}$ (1 - s) $\end{document}-fractional integral \begin{document}$ I_{a+}^{1 - s}[u] $\end{document} belongs to \begin{document}$ W^{1, p}(I) $\end{document}. We prove that the space of functions of bounded variation \begin{document}$ BV(I) $\end{document} and the fractional Sobolev space \begin{document}$ W^{s, 1}(I) $\end{document} continuously embed into \begin{document}$ W^{s, 1}_{RL, a+}(I) $\end{document}. In addition, we define the space of functions with left Riemann-Liouville \begin{document}$ s $\end{document}-fractional bounded variation, \begin{document}$ BV^{s}_{RL,a+}(I) $\end{document}, as the set of functions \begin{document}$ u \in L^{1}(I) $\end{document} such that \begin{document}$ I^{1 - s}_{a+}[u] \in BV(I) $\end{document}, and we analyze some fine properties of these functions. Finally, we prove some fractional Sobolev-type embedding results and we analyze the case of higher order Riemann-Liouville fractional derivatives.

Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters
Meilan Cai and Maoan Han
2021, 20(1): 55-75 doi: 10.3934/cpaa.2020257 +[Abstract](585) +[HTML](88) +[PDF](511.47KB)
Abstract:

In this paper, we concern with the problem of limit cycle bifurcation for a class of piecewise smooth cubic systems. Using the first order Melnikov function we prove that at least thirteen limit cycles can be bifurcated from periodic solutions surrounding the center.

Scattering of the focusing energy-critical NLS with inverse square potential in the radial case
Kai Yang
2021, 20(1): 77-99 doi: 10.3934/cpaa.2020258 +[Abstract](575) +[HTML](71) +[PDF](427.96KB)
Abstract:

We consider the Cauchy problem of the focusing energy-critical nonlinear Schrödinger equation with an inverse square potential. We prove that if any radial solution obeys the supreme of the kinetic energy over the maximal lifespan is below the kinetic energy of the ground state solution, then the solution exists globally in time and scatters in both time directions.

Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation
Alex H. Ardila and Mykael Cardoso
2021, 20(1): 101-119 doi: 10.3934/cpaa.2020259 +[Abstract](559) +[HTML](91) +[PDF](379.97KB)
Abstract:

Using variational methods we study the stability and strong instability of ground states for the focusing inhomogeneous nonlinear Schrödinger equation (INLS)

We construct two kinds of invariant sets under the evolution flow of (INLS). Then we show that the solution of (INLS) is global and bounded in \begin{document}$ H^{1}(\mathbb{R^{N}}) $\end{document} in the first kind of the invariant sets, while the solution blow-up in finite time in the other invariant set. Consequently, we prove that if the nonlinearity is \begin{document}$ L^{2} $\end{document}-supercritical, then the ground states are strongly unstable by blow-up.

Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential
Noriyoshi Fukaya
2021, 20(1): 121-143 doi: 10.3934/cpaa.2020260 +[Abstract](528) +[HTML](66) +[PDF](406.52KB)
Abstract:

We study uniqueness and nondegeneracy of ground states for stationary nonlinear Schrödinger equations with a focusing power-type nonlinearity and an attractive inverse-power potential. We refine the results of Shioji and Watanabe (2016) and apply it to prove the uniqueness and nondegeneracy of ground states for our equations. We also discuss the orbital instability of ground state-standing waves.

A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle
Gloria Paoli, Gianpaolo Piscitelli and Rossanno Sannipoli
2021, 20(1): 145-158 doi: 10.3934/cpaa.2020261 +[Abstract](539) +[HTML](75) +[PDF](345.76KB)
Abstract:

In this paper we study the first Steklov-Laplacian eigenvalue with an internal fixed spherical obstacle. We prove that the spherical shell locally maximizes the first eigenvalue among nearly spherical sets when both the internal ball and the volume are fixed.

Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains
Teresa D'Aprile
2021, 20(1): 159-191 doi: 10.3934/cpaa.2020262 +[Abstract](500) +[HTML](69) +[PDF](463.44KB)
Abstract:

We are concerned with the existence of blowing-up solutions to the following boundary value problem

where \begin{document}$ \Omega $\end{document} is a smooth and bounded domain in \begin{document}$ \mathbb{R}^2 $\end{document} such that \begin{document}$ 0\in\Omega $\end{document}, \begin{document}$ V $\end{document} is a positive smooth potential, \begin{document}$ N $\end{document} is a positive integer and \begin{document}$ \lambda>0 $\end{document} is a small parameter. Here \begin{document}$ {\mathit{\boldsymbol{\delta}}}_0 $\end{document} defines the Dirac measure with pole at \begin{document}$ 0 $\end{document}. We assume that \begin{document}$ \Omega $\end{document} is \begin{document}$ (N+1) $\end{document}-symmetric and we find conditions on the potential \begin{document}$ V $\end{document} and the domain \begin{document}$ \Omega $\end{document} under which there exists a solution blowing up at \begin{document}$ N+1 $\end{document} points located at the vertices of a regular polygon with center \begin{document}$ 0 $\end{document}.

Spectrum of the Laplacian on regular polyhedra
Evan Greif, Daniel Kaplan, Robert S. Strichartz and Samuel C. Wiese
2021, 20(1): 193-214 doi: 10.3934/cpaa.2020263 +[Abstract](534) +[HTML](69) +[PDF](6925.09KB)
Abstract:

We study eigenvalues and eigenfunctions of the Laplacian on the surfaces of four of the regular polyhedra: tetrahedron, octahedron, icosahedron and cube. We show two types of eigenfunctions: nonsingular ones that are smooth at vertices, lift to periodic functions on the plane and are expressible in terms of trigonometric polynomials; and singular ones that have none of these properties. We give numerical evidence for conjectured asymptotic estimates of the eigenvalue counting function. We describe an enlargement phenomenon for certain eigenfunctions on the octahedron that scales down eigenvalues by a factor of \begin{document}$ \frac{1}{3} $\end{document}.

Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles
Justin Holmer and Chang Liu
2021, 20(1): 215-242 doi: 10.3934/cpaa.2020264 +[Abstract](472) +[HTML](48) +[PDF](724.33KB)
Abstract:

We consider the 1D nonlinear Schrödinger equation (NLS) with focusing point nonlinearity,

where \begin{document}$ {\delta} = {\delta}(x) $\end{document} is the delta function supported at the origin. In the \begin{document}$ L^2 $\end{document} supercritical setting \begin{document}$ p>3 $\end{document}, we construct self-similar blow-up solutions belonging to the energy space \begin{document}$ L_x^\infty \cap \dot H_x^1 $\end{document}. This is reduced to finding outgoing solutions of a certain stationary profile equation. All outgoing solutions to the profile equation are obtained by using parabolic cylinder functions (Weber functions) and solving the jump condition at \begin{document}$ x = 0 $\end{document} imposed by the \begin{document}$ \delta $\end{document} term in (0.1). This jump condition is an algebraic condition involving gamma functions, and existence and uniqueness of solutions is obtained using the intermediate value theorem and formulae for the digamma function. We also compute the form of these outgoing solutions in the slightly supercritical case \begin{document}$ 0<p-3 \ll 1 $\end{document} using the log Binet formula for the gamma function and steepest descent method in the integral formulae for the parabolic cylinder functions.

High-order Wong-Zakai approximations for non-autonomous stochastic $ p $-Laplacian equations on $ \mathbb{R}^N $
Wenqiang Zhao and Yijin Zhang
2021, 20(1): 243-280 doi: 10.3934/cpaa.2020265 +[Abstract](454) +[HTML](78) +[PDF](480.61KB)
Abstract:

In this paper, we investigate the approximations of stochastic \begin{document}$ p $\end{document}-Laplacian equation with additive white noise by a family of piecewise deterministic partial differential equations driven by a stationary stochastic process. We firstly obtain the tempered pullback attractors for the random \begin{document}$ p $\end{document}-Laplacian equation with a general diffusion. We secondly prove the convergence of solutions and the upper semi-continuity of pullback attractors of the Wong-Zakai approximation equations in a Hilbert space for the additive case. Thirdly, by a truncation technique, the uniform compactness of pullback attractor with respect to the quantity of approximations is derived in the space of \begin{document}$ q $\end{document}-times integrable functions, where the upper semi-continuity of the attractors of the approximation equations is well established.

Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles
Chao Wang, Qihuai Liu and Zhiguo Wang
2021, 20(1): 281-300 doi: 10.3934/cpaa.2020266 +[Abstract](489) +[HTML](59) +[PDF](360.15KB)
Abstract:

In this paper, we prove the existence and multiplicity of subharmonic bouncing motions for a Hill's type sublinear oscillator with an obstacle. Furthermore, we also consider the existence, multiplicity and dense distribution of symmetric periodic bouncing solutions when the weight function is even. Based on an appropriate coordinate transformation and the method of phase-plane analysis, we can study our main results via Poincar\begin{document}$ \acute{e} $\end{document} map by applying some suitable fixed point theorems.

Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions
Mengni Li
2021, 20(1): 301-317 doi: 10.3934/cpaa.2020267 +[Abstract](613) +[HTML](56) +[PDF](372.83KB)
Abstract:

In this paper we study the boundary regularity of solutions to the Dirichlet problem for a class of Monge-Ampère type equations with nonzero boundary conditions. We construct global Hölder estimates for convex solutions to the problem and emphasize that the boundary regularity essentially depends on the convexity of the domain. The proof is based on a careful study of the concept of \begin{document}$ (a,\eta) $\end{document} type convex domain and a family of auxiliary functions.

Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional
Bo Chen and Youde Wang
2021, 20(1): 319-338 doi: 10.3934/cpaa.2020268 +[Abstract](448) +[HTML](45) +[PDF](367.37KB)
Abstract:

We follow the idea of Wang [21] to show the existence of global weak solutions to the Cauchy problems of Landau-Lifshtiz type equations and related heat flows from a \begin{document}$ n $\end{document}-dimensional Euclidean domain \begin{document}$ \Omega $\end{document} or a \begin{document}$ n $\end{document}-dimensional closed Riemannian manifold \begin{document}$ M $\end{document} into a 2-dimensional unit sphere \begin{document}$ \mathbb{S}^{2} $\end{document}. Our conclusions extend a series of related results obtained in the previous literature.

Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound
Christopher S. Goodrich, Benjamin Lyons and Mihaela T. Velcsov
2021, 20(1): 339-358 doi: 10.3934/cpaa.2020269 +[Abstract](508) +[HTML](53) +[PDF](4191.53KB)
Abstract:

We investigate the relationship between the sign of the discrete fractional sequential difference \begin{document}$ \big(\Delta_{1+a-\mu}^{\nu}\Delta_a^{\mu}f\big)(t) $\end{document} and the monotonicity of the function \begin{document}$ t\mapsto f(t) $\end{document}. More precisely, we consider the special case in which this fractional difference can be negative and satisfies the lower bound

for some \begin{document}$ \varepsilon>0 $\end{document}. We prove that even though the fractional difference can be negative, the monotonicity of the function \begin{document}$ f $\end{document}, nonetheless, is still implied by the above inequality. This demonstrates a significant dissimilarity between the fractional and non-fractional cases. Because of the challenges of a purely analytical approach, our analysis includes numerical simulation.

Isomorphism between one-dimensional and multidimensional finite difference operators
Anton A. Kutsenko
2021, 20(1): 359-368 doi: 10.3934/cpaa.2020270 +[Abstract](595) +[HTML](53) +[PDF](501.77KB)
Abstract:

Finite difference operators are widely used for the approximation of continuous ones. It is well known that the analysis of continuous differential operators may strongly depend on their dimensions. We will show that the finite difference operators generate the same algebra, regardless of their dimension.

On optimal autocorrelation inequalities on the real line
José Madrid and João P. G. Ramos
2021, 20(1): 369-388 doi: 10.3934/cpaa.2020271 +[Abstract](373) +[HTML](37) +[PDF](367.11KB)
Abstract:

We study autocorrelation inequalities, in the spirit of Barnard and Steinerberger's work [1]. In particular, we obtain improvements on the sharp constants in some of the inequalities previously considered by these authors, and also prove existence of extremizers to these inequalities in certain specific settings. Our methods consist of relating the inequalities in question to other classical sharp inequalities in Fourier analysis, such as the sharp Hausdorff–Young inequality, and employing functional analysis as well as measure theory tools in connection to a suitable dual version of the problem to identify and impose conditions on extremizers.

New general decay result for a system of viscoelastic wave equations with past history
Adel M. Al-Mahdi, Mohammad M. Al-Gharabli and Salim A. Messaoudi
2021, 20(1): 389-404 doi: 10.3934/cpaa.2020273 +[Abstract](477) +[HTML](52) +[PDF](364.45KB)
Abstract:

This work is concerned with a coupled system of viscoelastic wave equations in the presence of infinite-memory terms. We show that the stability of the system holds for a much larger class of kernels. More precisely, we consider the kernels \begin{document}$ g_i : [0, +\infty) \rightarrow (0, +\infty) $\end{document} satisfying

where \begin{document}$ \xi_i $\end{document} and \begin{document}$ H_i $\end{document} are functions satisfying some specific properties. Under this very general assumption on the behavior of \begin{document}$ g_i $\end{document} at infinity, we establish a relation between the decay rate of the solutions and the growth of \begin{document}$ g_i $\end{document} at infinity. This work generalizes and improves earlier results in the literature. Moreover, we drop the boundedness assumptions on the history data, usually made in the literature.

Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials
Anna Canale, Francesco Pappalardo and Ciro Tarantino
2021, 20(1): 405-425 doi: 10.3934/cpaa.2020274 +[Abstract](494) +[HTML](45) +[PDF](420.82KB)
Abstract:

The main results in the paper are the weighted multipolar Hardy inequalities

in \begin{document}$ \mathbb{R}^N $\end{document} for any \begin{document}$ \varphi $\end{document} in a suitable weighted Sobolev space, with \begin{document}$ 0<c\le c_{o,\mu} $\end{document}, \begin{document}$ a_1,\dots,a_n\in \mathbb{R}^N $\end{document}, \begin{document}$ K $\end{document} constant. The weight functions \begin{document}$ \mu $\end{document} are of a quite general type.

The paper fits in the framework of Kolmogorov operators defined on smooth functions

perturbed by multipolar inverse square potentials, and related evolution problems. Necessary and sufficient conditions for the existence of exponentially bounded in time positive solutions to the associated initial value problem are based on weighted Hardy inequalities. For constants \begin{document}$ c $\end{document} beyond the optimal Hardy constant \begin{document}$ c_{o,\mu} $\end{document} we are able to show nonexistence of positive solutions.

Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food
Chao Xing, Jiaojiao Pan and Hong Luo
2021, 20(1): 427-448 doi: 10.3934/cpaa.2020275 +[Abstract](396) +[HTML](38) +[PDF](1782.17KB)
Abstract:

The article aims to investigate the dynamic transitions of a toxin-producing phytoplankton zooplankton model with additional food in a 2D-rectangular domain. The investigation is based on the dynamic transition theory for dissipative dynamical systems. Firstly, we verify the principle of exchange of stability by analysing the corresponding linear eigenvalue problem. Secondly, by using the technique of center manifold reduction, we determine the types of transitions. Our results imply that the model may bifurcate two new steady state solutions, which are either attractors or saddle points. In addition, the model may also bifurcate a new periodic solution as the control parameter passes critical value. Finally, some numerical results are given to illustrate our conclusions.

The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field
Claudianor O. Alves, Rodrigo C. M. Nemer and Sergio H. Monari Soares
2021, 20(1): 449-465 doi: 10.3934/cpaa.2020276 +[Abstract](540) +[HTML](45) +[PDF](376.43KB)
Abstract:

Nonlinear Schrödinger equations with an external magnetic field and a power nonlinearity with subcritical exponent \begin{document}$ p $\end{document} are considered. It is established a lower bound to the number of nontrivial solutions to these equations in terms of the topology of the domains in which the problem is given if \begin{document}$ p $\end{document} is suitably close to the critical exponent \begin{document}$ 2^* = 2N/(N-2) $\end{document}, \begin{document}$ N \geq 3 $\end{document}. To prove this lower bound, based on a proof of a result of Benci and Cerami, it is provided an abstract result that establishes Morse relations that are used to count solutions.

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