2��-periodic self-similar solutions for the anisotropic affine curve shortening problem

We study the existence of 2??-periodic positive solutions of the equation $$u_{\theta\theta}+u=\displaystyle{\frac{a(\theta)}{u^3}},$$ where a(??) is a positive smooth 2??-periodic function. A priori estimates and sufficient conditions for the existence of solutions of the equation are established.     

Anisotropic elliptic problems with natural growth terms

In this paper we prove existence and regularity of solutions for nonlinear anisotropic elliptic equations of the type $$-\sum_{i=1}^N\frac{\partial}{\partial x_i}\left[\left|\frac{\partial u}{\partial {x}_i}\right|^{p_i-2}\frac{\partial u}{\partial x_i}\right]+g(x,u,\nabla u)=f$$ in a bounded, smooth, domain ??, in ${\mathbb{R}^N}$ , with homogeneous Dirichlet boundary conditions. The right hand side f is assumed to belong to some Lebesgue space and the function g is a nonlinear lower order term.     

Changing-sign bubble solutions for an anisotropic sinh-Poisson equation

We consider the following anisotropic sinh-Poisson equation $${\rm div} (a(x) \nabla u)+ 2\varepsilon^2 a(x) {\rm sinh}\,u=0\ \ {\rm in}\ \Omega, \quad u=0 \ \ {\rm on}\ \partial \Omega,$$ where ${\Omega \subset \mathbb{R}^2}$ is a bounded smooth domain and a(x) is a positive smooth function. We investigate the effect of anisotropic coefficient ${a(x)}$ on the existence of bubbling solutions. We show that there exists a family of solutions u _?? concentrating positively and negatively at ${\bar{x}}$ , a given local critical point of a(x), for ?? sufficiently small, for which with the property $$2\varepsilon^2a(x){\rm sinh} u_\varepsilon \rightharpoonup 8\pi\sum\limits_{j=1}^{m}b_j\delta_{\bar{x}},$$ where ${b_j=\pm 1}$ . This result shows a striking difference with the isotropic case (a(x) ?? Constant) in Bartolucci and Pistoia (IMA J Appl Math 72(6):706?C729, 2007), Jost et?al. (Calc Var Partial Differ Equ 31:263?C276, 2008) and Esposito and Wei (Calc Var Partial Differ Equ 34:341?C375, 2009).     

Rate of P-convergence over equivalence classes of double sequence spaces

This paper presents multidimensional characterizations of equivalence classes of double sequences. These characterizations will be analyzed by presenting theorems of the following type: If A is convergent preserving over ${\{x \in S^{\prime\prime}_{0} : x \;{\rm equivalent \; to}\; b\}}$ then there exists a ${b^{\prime} \in S^{\prime\prime}_{0}}$ such that b converges faster than b?? and A is convergent preserving over ${\{x \in S^{\prime\prime}_{0} : x \; {\rm equivalent \; to}\; b^{\prime}\}.}$      

A Landesman-Lazer Type Theorem for Periodic Solutions of the Resonant Asymmetric ρ-Laplacian Equation

In this paper, we give a Landesman-Lazer type theorem for periodic solutions of the asymmetric 1-dimensional p-Laplacian equation -（｜x＇｜^p-2x＇）＇=λ｜x｜^p-2x＋＋μ｜x｜^p-2x-＋f（t,x）with periodic boundary value.     

Positive solutions for a boundary-value problem with Riemann–Liouville fractional derivative*

In this work, we are mainly concerned with the existence of positive solutions for the fractional boundary-value problem $$ \left\{ {\begin{array}{*{20}{c}} {D_{0+}^{\alpha }D_{0+}^{\alpha }u=f\left( {t,u,{u}^{\prime},-D_{0+}^{\alpha }u} \right),\quad t\in \left[ {0,1} \right],} \hfill \\ {u(0)={u}^{\prime}(0)={u}^{\prime}(1)=D_{0+}^{\alpha }u(0)=D_{0+}^{{\alpha +1}}u(0)=D_{0+}^{{\alpha +1}}u(1)=0.} \hfill \\ \end{array}} \right. $$ Here ?? ?? (2, 3] is a real number, $ D_{0+}^{\alpha } $ is the standard Riemann?CLiouville fractional derivative of order ??. By virtue of some inequalities associated with the fractional Green function for the above problem, without the assumption of the nonnegativity of f, we utilize the Krasnoselskii?CZabreiko fixed-point theorem to establish our main results. The interesting point lies in the fact that the nonlinear term is allowed to depend on u, u??, and $ -D_{0+}^{\alpha } $ .     

Decreasing Regularly Varying Solutions of Sublinearly Perturbed Superlinear Thomas–Fermi Equation

We consider the perturbed Thomas–Fermi equation $$\begin{array}{ll} x^{\prime \prime}\, =\, p(t)|x|^{\gamma-1}x\, +\, q(t)|x|^{\delta-1}x, \qquad \qquad \qquad (A) \end{array}$$ where δ and γ are positive constants with \({\delta < 1 < \gamma}\) and p(t) and q(t) are positive continuous functions on \({[a,\infty), a > 0}\) . Our aim here is to establish criteria for the existence of positive solutions of (A) decreasing to zero as \({t \to \infty}\) in the case where p(t) and q(t) are regularly varying functions (in the sense of Karamata). Generalization of the obtained results to equations of the form $$\begin{array}{ll} \left(r(t)x^{\prime}\right)^{\prime}\, =\, p(t)|x|^{\gamma-1}x \,+ \,q(t)|x|^{\delta-1}x, \qquad \qquad \qquad (B) \end{array}$$ is also given.     

Nonlinear perturbations of a periodic elliptic problem with discontinuous nonlinearity in {\mathbb{R}^{N}}

Using variational methods, we establish existence of positive solutions for a class of elliptic problems like $$-\Delta{u}+V(x)u=H(u-\beta)f(u)\,\,\,\, {\rm in}\,\,\,\mathbb{R}^{N},$$ where ?? >?0, V is a positive, continuous perturbations of a periodic function, H is the Heaviside function and f is a continuous function with subcritical growth.     

Layered solutions with multiple asymptotes for non autonomous Allen–Cahn equations in {\mathbb {R}^{3}}

We consider a class of semilinear elliptic equations of the form $$ \label{eq:abs}-\Delta u(x,y,z)+a(x)W'(u(x,y,z))=0,\quad (x,y,z)\in\mathbb {R}^{3},$$ where ${a:\mathbb {R} \to \mathbb {R}}$ is a periodic, positive, even function and, in the simplest case, ${W : \mathbb {R} \to \mathbb {R}}$ is a double well even potential. Under non degeneracy conditions on the set of minimal solutions to the one dimensional heteroclinic problem $$-\ddot q(x)+a(x)W^{\prime}(q(x))=0,\ x\in\mathbb {R},\quad q(x)\to\pm1\,{\rm as}\, x\to \pm\infty,$$ we show, via variational methods the existence of infinitely many geometrically distinct solutions u of (0.1) verifying u(x, y, z) → ± 1 as x → ± ∞ uniformly with respect to ${(y, z) \in \mathbb {R}^{2}}$ and such that ${\partial_{y}u \not \equiv0, \partial_{z}u \not\equiv 0}$ in ${\mathbb {R}^{3}}$ .     

Existence and regularity of positive solutions of elliptic equations of Schr?dinger type

We prove the existence of positive solutions with optimal local regularity of the homogeneous equation of Schr?dinger type $$ - {\rm{div}}(A\nabla u) - \sigma u = 0{\rm{ in }}\Omega $$ for an arbitrary open ?? ? ? ⁿ under only a form-boundedness assumption on ?? ?? D??(??) and ellipticity assumption on A ?? L ^??(??) ^n×n . We demonstrate that there is a two-way correspondence between form boundedness and existence of positive solutions of this equation as well as weak solutions of the equation with quadratic nonlinearity in the gradient $$ - {\rm{div}}(A\nabla u) = (A\nabla v) \cdot \nabla v + \sigma {\rm{ in }}\Omega $$ As a consequence, we obtain necessary and sufficient conditions for both formboundedness (with a sharp upper form bound) and positivity of the quadratic form of the Schr?dinger type operator H = ?div(A?·)-?? with arbitrary distributional potential ?? ?? D??(??), and give examples clarifying the relationship between these two properties.     

Multiplicity of periodic solutions to symmetric delay differential equations

By applying the method based on the usage of the equivariant gradient degree introduced by G?ba (1997) and the cohomological equivariant Conley index introduced by Izydorek (2001), we establish an abstract result for G-invariant strongly indefinite asymptotically linear functionals showing that the equivariant invariant ${\omega(\nabla \Phi)}$ , expressed as that difference of the G-gradient degrees at infinity and zero, contains rich numerical information indicating the existence of multiple critical points of ${\Phi}$ exhibiting various symmetric properties. The obtained results are applied to investigate an asymptotically linear delay differential equation $$x\prime = - \nabla f \big ({x \big (t - \frac{\pi}{2} \big )} \big ), \quad x \in V \qquad \quad (*)$$ (here ${f : V \rightarrow \mathbb{R}}$ is a continuously differentiable function satisfying additional assumptions) with Γ-symmetries (where Γ is a finite group) using a variational method introduced by Guo and Yu (2005). The equivariant invariant ${\omega(\nabla \Phi) = n_{1}({\bf H}_{1}) + n_{2}({\bf H}_{2}) + \cdots + n_{m}({\bf H}_{m})}$ in the case n _k ≠ 0 (for maximal twisted orbit types (H _k )) guarantees the existence of at least |n _k | different G-orbits of periodic solutions with symmetries at least (H _k). This result generalizes the result by Guo and Yu (2005) obtained in the case without symmetries. The existence of large number of nonconstant periodic solutions for (*) (classified according to their symmetric properties) is established for several groups Γ, with the exact value of ${\omega(\,\nabla \Phi)}$ evaluated.     

Uniform Estimates of Solutions of a Nonlinear Third-Order Differential Equation

One considers the differential equation $$y{\prime\prime\prime}(x) + p\bigl(x, y(x), y{\prime}(x), y{\prime\prime}(x)\bigr) |y(x)|^{k-1} y(x) = 0,$$ where k?>?1, the function p(x, y ₀ , y ₁ , y ₂) is continuous and satisfies the inequalities $$ 0 < p_* \le p(x, y_0, y_1, y_2) \le p^* < \infty,$$ as well as the Lipschitz condition with respect to the last three arguments. Uniform estimates are obtained for the moduli of the solutions with a common domain.     

A local mountain pass type result for a system of nonlinear Schr?dinger equations

We consider a singular perturbation problem for a system of nonlinear Schr?dinger equations: $$ \begin{array}{l} -\varepsilon^2\Delta v_1 +V_1(x)v_1 = \mu_1 v_1^3 + \beta v_1v_2^2 \quad {\rm in}\,\,{\bf R}^N, \\ -\varepsilon^2\Delta v_2 +V_2(x)v_2 = \mu_2 v_2^3 + \beta v_1^2v_2 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x) >0 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x)\in H^1({\bf R}^N), \end{array} \quad\quad\quad\quad\quad (*) $$ where N?=?2, 3, ?? ₁, ?? ₂, ?? > 0 and V ₁(x), V ₂(x): R ^N ?? (0, ??) are positive continuous functions. We consider the case where the interaction ?? > 0 is relatively small and we define for ${P\in{\bf R}^N}$ the least energy level m(P) for non-trivial vector solutions of the rescaled ??limit?? problem: $$ \begin{array}{l} -\Delta v_1 +V_1(P)v_1 = \mu_1 v_1^3 + \beta v_1v_2^2 \quad {\rm in}\,\,{\bf R}^N, \\ -\Delta v_2 +V_2(P)v_2 = \mu_2 v_2^3 + \beta v_1^2v_2 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x) >0 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x)\in H^1({\bf R}^N). \end{array} \quad\quad\quad\quad\quad\quad (**) $$ We assume that there exists an open bounded set ${\Lambda\subset{\bf R}^N}$ satisfying $$ {\mathop {\rm inf} _{P\in\Lambda} m(P)} < {\mathop {\rm inf}_{P\in\partial\Lambda} m(P)}. $$ We show that (*) possesses a family of non-trivial vector positive solutions ${\{(v_{1\varepsilon}(x), v_{2\varepsilon} (x))\}_{\varepsilon\in (0,\varepsilon_0]}}$ which concentrates??after extracting a subsequence ?? _n ?? 0??to a point ${P_0\in\Lambda}$ with ${m(P_0)={\rm inf}_{P\in\Lambda}m(P)}$ . Moreover (v _1?? (x), v _2?? (x)) converges to a least energy non-trivial vector solution of (**) after a suitable rescaling.     

Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps

We consider the Markov chain ${\{X_n^x\}_{n=0}^\infty}$ on ${\mathbb{R}^d}$ defined by the stochastic recursion ${X_{n}^{x}= \psi_{\theta_{n}} (X_{n-1}^{x})}$ , starting at ${x\in\mathbb{R}^d}$ , where ?? ₁, ?? ₂, . . . are i.i.d. random variables taking their values in a metric space ${(\Theta, \mathfrak{r})}$ , and ${\psi_{\theta_{n}} :\mathbb{R}^d\mapsto\mathbb{R}^d}$ are Lipschitz maps. Assume that the Markov chain has a unique stationary measure ??. Under appropriate assumptions on ${\psi_{\theta_n}}$ , we will show that the measure ?? has a heavy tail with the exponent ???>?0 i.e. ${\nu(\{x\in\mathbb{R}^d: |x| > t\})\asymp t^{-\alpha}}$ . Using this result we show that properly normalized Birkhoff sums ${S_n^x=\sum_{k=1}^n X_k^x}$ , converge in law to an ??-stable law for ${\alpha\in(0, 2]}$ .     

Existence, nonexistence and multiplicity of positive solutions for nonlinear fractional differential equations

In this paper, we consider the existence, nonexistence and multiplicity of positive solutions for nonlinear fractional differential equation boundary-value problem $$\left\{ \begin{array}{@{}l}-D^{\alpha}_{0+}u(t)=f(t,u(t)), \quad t\in[0,1]\\[3pt]u(0)=u(1)=u''(0)=0\end{array} \right.$$ where 2<????3 is a real number, and $D^{\alpha}_{0+}$ is the Caputo??s fractional derivative, and f:[0,1]×[0,+??)??[0,+??) is continuous. By means of a fixed-point theorem on cones, some existence, nonexistence and multiplicity of positive solutions are obtained.     

Non-resonant boundary value problems with singular {\phi} -Laplacian operators

In this paper, using Leray–Schauder degree arguments, critical point theory for lower semicontinuous functionals and the method of lower and upper solutions, we give existence results for periodic problems involving the relativistic operator ${u \mapsto \left(\frac{u^\prime}{\sqrt{1-u^\prime 2}}\right)^\prime+r(t)u}$ with ${\int_0^Tr dt\neq 0}$ . In particular we show that in this case we have non-resonance, that is periodic problem $$\left(\frac{u^\prime}{\sqrt{1-u^\prime 2}}\right)^\prime+r(t)u=e(t),\quad u(0)-u(T)=0=u^\prime(0)-u^\prime(T),$$ has at least one solution for any continuous function ${e : [0, T] \to \mathbb {R}}$ . Then, we consider Brillouin and Mathieu-Duffing type equations for which ${r(t) \equiv b_1 + b_2 {\rm cos} t {\rm and} b_1, b_2 \in \mathbb{R}}$ .     

Asymptotic analysis of Emden-Fowler differential equations in the framework of regular variation

Sufficient conditions are established for the existence of slowly varying solution and regularly varying solution of index 1 of the second-order nonlinear differential equation $$x^{\prime\prime}(t)+q(t)|x(t)|^{\gamma}\,{\rm sgn}\, x(t)=0, \quad \quad (A)$$ where γ is a positive constant different from 1 and q : [a, ∞) → (0, ∞) is a continuous integrable function. We show how an application of the theory of regular variation gives the possibility of determining the precise asymptotic behavior of solutions of both superlinear and sublinear equation (A).     

On the Li-Stevi? Integral Type Operators from Weighted Bergman Spaces into β-Zygmund Spaces

For an analytic self-map ?? of the unit disk ${\mathbb{D}}$ and an analytic function g on ${\mathbb{D}}$ , we define the following integral type operators: $$T_{\varphi}^{g}f(z) := \int_{0}^{z} f(\varphi(\zeta))g(\zeta) d\zeta\quad {\rm and}\quad C_{\varphi}^{g}f(z) := \int_{0}^{z}f^{\prime}(\varphi(\zeta))g(\zeta) d\zeta$$ . We give a characterization for the boundedness and compactness of these operators from the weighted Bergman space ${L_{a}^p(dA_{\alpha})}$ into the ??-Zygmund space ${\mathcal{Z}_{\beta}}$ . We will also estimate the essential norm of these type of operators. As an application of results, we characterize the above operator-theoretic properties of Volterra type integral operators and composition operators.     

A nonlinear third order multi-point boundary value problem in the resonance case

In this paper, we discuss the following third order ordinary differential equation $$x^{\prime\prime\prime}(t)=f(t,x(t),x^{\prime}(t),x^{\prime\prime}(t))+e(t),\quad t\in (0,1)$$ with the multi-point boundary conditions $$x^{\prime}(0)=\alpha x^{\prime}(\xi),\qquad x^{\prime\prime}(0)=0,\qquad x(1)=\sum^{m-2}_{j=1}\beta_{j}x(\eta_{j})$$ where β _j (1≤j≤m?2), α∈R, 0<η ₁<η ₂<???<η _m?2<1, 0<ξ<1. When the β _j ’s have no same sign, some existence results are given for the nonlinear problems at resonance case. An example is provided in this paper.