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Communications on Pure & Applied Analysis

March 2017 , Volume 16 , Issue 2

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Singular periodic solutions for the p-laplacian ina punctured domain
Shanming Ji, Yutian Li, Rui Huang and Xuejing Yin
2017, 16(2): 373-392 doi: 10.3934/cpaa.2017019 +[Abstract](197) +[HTML](5) +[PDF](459.2KB)
Abstract:

Abstract. In this paper we are interested in studying singular periodic solutions for the p-Laplacian in a punctured domain. We find an interesting phenomenon that there exists a critical exponent pc = N and a singular exponent qs = p-1. Precisely speaking, only if p > pc can singular periodic solutions exist; while if 1 < ppc then all of the solutions have no singularity. By the singular exponent qs = p-1, we mean that in the case when q = qs, completely different from the remaining case qqs, the problem may or may not have solutions depending on the coefficients of the equation.

Long-term stability for kdv solitons in weighted Hs spaces
Brian Pigott and Sarah Raynor
2017, 16(2): 393-416 doi: 10.3934/cpaa.2017020 +[Abstract](163) +[HTML](1) +[PDF](456.8KB)
Abstract:

In this work, we consider the stability of solitons for the KdV equation below the energy space, using spatially-exponentially-weighted norms. Using a combination of the I-method and spectral analysis following Pego and Weinstein, we are able to show that, in the exponentially weighted space, the perturbation of a soliton decays exponentially for arbitrarily long times. The finite time restriction is due to a lack of global control of the unweighted perturbation.

Center conditions for generalized polynomial kukles systems
Jaume Giné
2017, 16(2): 417-426 doi: 10.3934/cpaa.2017021 +[Abstract](144) +[HTML](4) +[PDF](377.9KB)
Abstract:

Abstract. In this paper we study the center problem for certain generalized Kukles systems $\dot{x}= y, \qquad \dot{y}= P_0(x)+ P_1(x)y+P_2(x) y^2+ P_3(x) y^3, $\end{document} where Pi(x) are polynomials of degree n, P0(0) = 0 and P0′(0) < 0. Computing the focal values and using modular arithmetics and Gröbner bases we find the center conditions for such systems when P0 is of degree 2 and Pi for i = 1; 2; 3 are of degree 3 without constant terms. We also establish a conjecture about the center conditions for such systems.

Diffusive predator-prey models with stage structure on prey and beddington-deangelis functional responses
Seong Lee and Inkyung Ahn
2017, 16(2): 427-442 doi: 10.3934/cpaa.2017022 +[Abstract](206) +[HTML](1) +[PDF](395.9KB)
Abstract:

In this paper, we examine a diffusive predator-prey model with Beddington-DeAngelis functional response and stage structure on prey under homogeneous Neumann boundary conditions, where the discrete time delay covers the period from the birth of immature prey to their maturity. We investigate the dynamics of their permanence and the extinction of the predator, and provide sufficient conditions for the global attractiveness and the locally asymptotical stability of the semi-trivial and coexistence equilibria.

Existence and upper semicontinuity of (L2, Lq) pullback attractors for a stochastic p-laplacian equation
Linfang Liu and Xianlong Fu
2017, 6(2): 443-474 doi: 10.3934/cpaa.2017023 +[Abstract](226) +[HTML](7) +[PDF](532.2KB)
Abstract:

In this paper we study the dynamical behavior of solutions for a non-autonomous \begin{document}$p$\end{document}-Laplacian equation driven by a white noise term. We first establish the abstract results on existence and continuity of bi-spatial pullback random attractors for a cocycle. Then by conducting some tail estimates and applying the obtained abstract results we show the existence and upper semi-continuity of \begin{document}$(L^{2}(\mathbb{R}^{n}), L^{q}(\mathbb{R}^{n}))$\end{document}-pullback attractors for this \begin{document}$p$\end{document}-Laplacian equation.

Estimates for eigenvalues of a system of elliptic equations with drift and of bi-drifting laplacian
Feng Du and Adriano Cavalcante Bezerra
2017, 6(2): 475-491 doi: 10.3934/cpaa.2017024 +[Abstract](316) +[HTML](2) +[PDF](388.4KB)
Abstract:

In this paper, we firstly study the eigenvalue problem of a systemof elliptic equations with drift and get some universal inequalities of PayneP′olya-Weinberger-Yang type on a bounded domain in Euclidean spaces and inGaussian shrinking solitons. Furthermore, we study two kinds of the clampedplate problems and the buckling problems for the bi-drifting Laplacian and getsome sharp lower bounds for the first eigenvalue for these eigenvalue problemon compact manifolds with boundary and positive m-weighted Ricci curvatureor on compact manifolds with boundary under some condition on the weightedRicci curvature.

Multi-peak solutions for nonlinear Choquard equation with a general nonlinearity
Minbo Yang, Jianjun Zhang and Yimin Zhang
2017, 16(2): 493-512 doi: 10.3934/cpaa.2017025 +[Abstract](209) +[HTML](3) +[PDF](479.5KB)
Abstract:

In this paper, we study a class of nonlinear Choquard type equations involving a general nonlinearity. By using the method of penalization argument, we show that there exists a family of solutions having multiple concentration regions which concentrate at the minimum points of the potential V. Moreover, the monotonicity of f(s)=s and the so-called Ambrosetti-Rabinowitz condition are not required.

Liouville theorems for elliptic problems in variable exponent spaces
SYLWIA DUDEK and IWONA SKRZYPCZAK
2017, 16(2): 513-532 doi: 10.3934/cpaa.2017026 +[Abstract](189) +[HTML](5) +[PDF](486.3KB)
Abstract:

We investigate nonexistence of nonnegative solutions to a partial differential inequality involving the p(x){Laplacian of the form

in ${\mathbb{R}^n}$, as well as in outer domain $\Omega \subseteq {\mathbb{R}^n}$, where Φ(x; u; ∇u) is a locally integrable Carathéodory’s function. We assume that Φ(x; u; ∇u) ≥ 0 or compatible with p and u. Growth conditions on u and p lead to Liouville-type results for u.

Asymptotic behavior of solutions to a nonlinear plate equation with memory
Yongqin Liu
2017, 16(2): 533-556 doi: 10.3934/cpaa.2017027 +[Abstract](182) +[HTML](1) +[PDF](460.5KB)
Abstract:

In this paper we consider the initial value problem of a nonlinear plate equation with memory-type dissipation in multi-dimensional space. Due to the memory effect, more technique is needed to deal with the global existence and decay property of solutions compared with the frictional dissipation. The model we study is an inertial one and the rotational inertia plays an important role in the part of energy estimates. By exploiting the time-weighted energy method we prove the global existence and asymptotic decay of solutions under smallness and suitable regularity assumptions on the initial data.

Global dynamics of solutions with group invariance for the nonlinear schrödinger equation
Takahisa Inui
2017, 16(2): 557-590 doi: 10.3934/cpaa.2017028 +[Abstract](176) +[HTML](2) +[PDF](585.3KB)
Abstract:

We consider the focusing mass-supercritical and energy-subcritical nonlinear Schrödinger equation (NLS). We are interested in the global behavior of the solutions to (NLS) with group invariance. By the group invariance, we can determine the global behavior of the solutions above the ground state standing waves.

Existence and stability of periodic solutions for relativistic singular equations
Jifeng Chu, Zaitao Liang, Fangfang Liao and Shiping Lu
2017, 16(2): 591-609 doi: 10.3934/cpaa.2017029 +[Abstract](163) +[HTML](2) +[PDF](454.8KB)
Abstract:

In this paper, we study the existence, multiplicity and stability of positive periodic solutions of relativistic singular differential equations. The proof of the existence and multiplicity is based on the continuation theorem of coincidence degree theory, and the proof of stability is based on a known connection between the index of a periodic solution and its stability.

The existence and nonexistence results of ground state nodal solutions for a Kirchhoff type problem
Xiao-Jing Zhong and Chun-Lei Tang
2017, 16(2): 611-628 doi: 10.3934/cpaa.2017030 +[Abstract](151) +[HTML](1) +[PDF](446.3KB)
Abstract:

In this paper, we investigate the existence and nonexistence of ground state nodal solutions to a class of Kirchhoff type problems

where $a, b>0$, $\lambda < a\lambda_1$, $\lambda_1$ is the principal eigenvalue of $(-\triangle, H_0.{1}(\Omega))$. With the help of the Nehari manifold, we obtain that there is $\Lambda>0$ such that the Kirchhoff type problem possesses at least one ground state nodal solution $u_b$ for all $0 < b < \Lambda$ and $\lambda < a\lambda_1$ and prove that its energy is strictly larger than twice that of ground state solutions. Moreover, we give a convergence property of $u_b$ as $b\searrow 0$. Besides, we firstly establish the nonexistence result of nodal solutions for all $b\geq\Lambda$. This paper can be regarded as the extension and complementary work of W. Shuai (2015)[21], X.H. Tang and B.T. Cheng (2016)[22].

Regularity estimates for continuous solutions of α-convex balance laws
Laura Caravenna
2017, 16(2): 629-644 doi: 10.3934/cpaa.2017031 +[Abstract](192) +[HTML](3) +[PDF](451.2KB)
Abstract:

This paper proves new regularity estimates for continuous solutions to the balance equation

when the flux $f$ satisfies a convexity assumption that we denote as 2n-convexity. The results are known in the case of the quadratic flux by very different arguments in [14,10,8]. We prove that the continuity of $u$ must be in fact $1/2n$-Hölder continuity and that the distributional source term $g$ is determined by the classical derivative of $u$ along any characteristics; part of the proof consists in showing that this classical derivative is well defined at any `Lebesgue point' of $g$ for suitable coverings. These two regularity statements fail in general for $C^{\infty}(\mathbb{R})$, strictly convex fluxes, see [3].

S-shaped and broken s-shaped bifurcation curves for a multiparameter diffusive logistic problem with holling type-Ⅲ functional response
Tzung-shin Yeh
2017, 16(2): 645-670 doi: 10.3934/cpaa.2017032 +[Abstract](153) +[HTML](1) +[PDF](548.7KB)
Abstract:

We study exact multiplicity and bifurcation curves of positive solutions for a multiparameter diffusive logistic problem with Holling type-Ⅲ functional response

where u is the population density of the species, p > 1, q, r are two positive dimensionless parameters, and λ > 0 is a bifurcation parameter. For fixed p > 1, assume that q, r satisfy one of the following conditions: (ⅰ) rη1, p* q and (q, r) lies above the curve

(ⅱ) rη2, p* q and (q, r) lies on or below the curve Γ1, where η1, p* and η2, p* are two positive constants, and $C_{p}^{*}={{\left(\frac{{{p}^{2}}+3p-4+p\sqrt{%{{p}^{2}}+6p-7}}{4} \right)}^{1/p}}$. Then on the (λ, ||u||)-plane, we give a classification of three qualitatively different bifurcation curves: an S-shaped curve, a broken S-shaped curve, and a monotone increasing curve. Hence we are able to determine the exact multiplicity of positive solutions by the values of q, r and λ.

A concentration phenomenon of the least energy solution to non-autonomous elliptic problems with a totally degenerate potential
Shun Kodama
2017, 16(2): 671-698 doi: 10.3934/cpaa.2017033 +[Abstract](131) +[HTML](1) +[PDF](504.8KB)
Abstract:

In this paper we study the following non-autonomous singularly perturbed Dirichlet problem:

for a totally degenerate potential K. Here ε > 0 is a small parameter, $\Omega \subset \mathbb{R}^N$ is a bounded domain with a smooth boundary, and f is an appropriate superlinear subcritical function. In particular, f satisfies $0 < \liminf_{ t \to 0+} f(t)/t^q \leq \limsup_{ t \to 0+} f(t)/t^q < + \infty$ for some $1 < q < + \infty$. We show that the least energy solutions concentrate at the maximal point of the modified distance function $D(x) = \min \{ (q+1) d(x, \partial A), 2 d(x, \partial \Omega) \}$, where $A = \{ x \in \bar{ \Omega } \mid K(x) = \max_{ y \in \bar{ \Omega } } K(y) \}$ is assumed to be a totally degenerate set satisfying ${{A}^{{}^\circ }}\ne \emptyset $.

A sustainability condition for stochastic forest model
TÔn Vı$\underset{.}{\overset{\hat{\ }}{\mathop{\text{E}}}}\, $T T$\mathop {\text{A}}\limits_. $, Linhthi hoai Nguyen and Atsushi Yagi
2017, 16(2): 699-718 doi: 10.3934/cpaa.2017034 +[Abstract](194) +[HTML](9) +[PDF](487.0KB)
Abstract:

A stochastic forest model of young and old age class trees is studied. First, we prove existence, uniqueness and boundedness of global nonnegative solutions. Second, we investigate asymptotic behavior of solutions by giving a sufficient condition for sustainability of the forest. Under this condition, we show existence of a Borel invariant measure. Third, we present several sufficient conditions for decline of the forest. Finally, we give some numerical examples.

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