Band 16, Heft 4

In this paper, we establish a unilateral global bifurcation result from intervals for a class of semilinear problems on the unit ball. By applying the above result, we obtain the spectrum of a class of half-linear problems on the unit ball. Our results partially give a positive answer to the open problem proposed by Berestycki.

In this paper, Morse theory is used to study the existence and multiplicity of nontrivial solutions for the following class of quasilinear elliptic equations: { - Δ ⁢ u - Δ ⁢ ( u 2 ) ⁢ u = g ⁢ ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ ⁡ Ω , $\left\{\begin{aligned} &\displaystyle{-}\Delta u-\Delta(u^{2})u=g(x,u),&&% \displaystyle x\in\Omega,\\ &\displaystyle u=0,&&\displaystyle x\in\partial\Omega,\end{aligned}\right.$ where Ω⊂ℝN${\Omega\subset\mathbb{R}^{N}}$ is a bounded open domain with smooth boundary ∂⁡Ω${\partial\Omega}$, N⩾3${N\geqslant 3}$ and g is a Carathéodory function with some additional growth conditions.

We use variational techniques to prove existence and nonexistence results for the following singular elliptic system: { - div ⁡ ( | ∇ ⁡ u | p - 2 ⁢ ∇ ⁡ u ) = θ ⁢ z q u 1 - θ , u > 0 ⁢ in ⁢ Ω , u ∈ W 0 1 , p ⁢ ( Ω ) , - div ⁡ ( | ∇ ⁡ z | p - 2 ⁢ ∇ ⁡ z ) = q ⁢ z q - 1 ⁢ u θ , z > 0 ⁢ in ⁢ Ω , z ∈ W 0 1 , p ⁢ ( Ω ) , $\left\{\begin{aligned} \displaystyle{-}\operatorname{div}(|\nabla u|^{p-2}% \nabla u)&\displaystyle=\theta\frac{z^{q}}{u^{1-\theta}},&&\displaystyle u>0% \text{ in }\Omega,&&\hskip 10.0ptu\in W^{1,p}_{0}(\Omega),\\ \displaystyle-\operatorname{div}(|\nabla z|^{p-2}\nabla z)&\displaystyle=qz^{q% -1}u^{\theta},&&\displaystyle z>0\text{ in }\Omega,&&\hskip 10.0ptz\in W^{1,p}% _{0}(\Omega),\end{aligned}\right.$ where Ω is a bounded open set in ℝN${\operatorname{\mathbb{R}}^{\mathrm{N}}}$ (N≥2${\mathrm{N}\geq 2}$), p>1${p>1}$, q>0${q>0}$ and 0<θ<1${0<\theta<1}$.

Let Ω be a Lipschitz bounded domain of ℝN${\mathbb{R}^{N}}$, N≥2${N\geq 2}$. The fractional Cheeger constant hs⁢(Ω)${h_{s}(\Omega)}$, 0<s<1${0<s<1}$, is defined by h s ⁢ ( Ω ) = inf E ⊂ Ω ⁡ P s ⁢ ( E ) | E | , where   P s ⁢ ( E ) = ∫ ℝ N ∫ ℝ N | χ E ⁢ ( x ) - χ E ⁢ ( y ) | | x - y | N + s ⁢ d x ⁢ d y , $h_{s}(\Omega)=\inf_{E\subset{\Omega}}\frac{P_{s}(E)}{|E|},\quad\text{where}% \quad P_{s}(E)=\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|\chi_{E}(x)-% \chi_{E}(y)|}{|x-y|^{N+s}}\,\mathrm{d}x\,\mathrm{d}y,$ with χE${\chi_{E}}$ denoting the characteristic function of the smooth subdomain E . The main purpose of this paper is to show that lim p → 1 + | ϕ p s | L ∞ ⁢ ( Ω ) 1 - p = h s ( Ω ) = lim p → 1 + | ϕ p s | L 1 ⁢ ( Ω ) 1 - p , $\lim_{p\rightarrow 1^{+}}\lvert\phi_{p}^{s}|_{L^{\infty}(\Omega)}^{1-p}=h_{s}(% \Omega)=\lim_{p\rightarrow 1^{+}}\lvert\phi_{p}^{s}|_{L^{1}(\Omega)}^{1-p},$ where ϕps${\phi_{p}^{s}}$ is the fractional (s,p)${(s,p)}$-torsion function of Ω, that is, the solution of the Dirichlet problem for the fractional p -Laplacian: -(Δ)ps⁢u=1${-(\Delta)_{p}^{s}\,u=1}$ in Ω, u=0${u=0}$ in ℝN∖Ω${\mathbb{R}^{N}\setminus\Omega}$. For this, we derive suitable bounds for the first eigenvalue λ1,ps⁢(Ω)${\lambda_{1,p}^{s}(\Omega)}$ of the fractional p -Laplacian operator in terms of ϕps${\phi_{p}^{s}}$. We also show that ϕps${\phi_{p}^{s}}$ minimizes the (s,p)${(s,p)}$-Gagliardo seminorm in ℝN${\mathbb{R}^{N}}$, among the functions normalized by the L1${L^{1}}$-norm.

In this paper, we study the following biharmonic equations: { Δ 2 ⁢ u - a 0 ⁢ Δ ⁢ u + ( λ ⁢ b ⁢ ( x ) + b 0 ) ⁢ u = f ⁢ ( u ) in ⁢ ℝ N , u ∈ H 2 ⁢ ( ℝ N ) , $\left\{\begin{aligned} &\displaystyle\Delta^{2}u-a_{0}\Delta u+(\lambda b(x)+b% _{0})u=f(u)&\hskip 10.0pt\text{in }\mathbb{R}^{N},\\ &\displaystyle u\in H^{2}(\mathbb{R}^{N}),\end{aligned}\right.$ where N≥3${N\geq 3}$, a0,b0∈ℝ${a_{0},b_{0}\in\mathbb{R}}$ are two constants, λ>0${\lambda>0}$ is a parameter, b⁢(x)≥0${b(x)\geq 0}$ is a potential well and f⁢(t)∈C⁢(ℝ)${f(t)\in C(\mathbb{R})}$ is subcritical and superlinear or asymptotically linear at infinity. By the Gagliardo–Nirenberg inequality, we make some observations on the operator Δ2-a0⁢Δ+λ⁢b⁢(x)+b0${\Delta^{2}-a_{0}\Delta+\lambda b(x)+b_{0}}$ in H2⁢(ℝN)${H^{2}(\mathbb{R}^{N})}$. Based on these observations, we give a new variational setting to the above problem for a0<0${a_{0}<0}$. With this new variational setting in hands, we establish some new existence results of the nontrivial solutions for all a0<0${a_{0}<0}$ with λ sufficiently large by the variational method. The concentration behavior of the nontrivial solution for λ→+∞${\lambda\to+\infty}$ is also obtained. It is worth pointing out that it seems to be the first time that the nontrivial solution of the above problem is obtained for all a0<0${a_{0}<0}$.

In this paper we are mainly concerned with the persistence of invariant tori with prescribed frequency for analytic nearly integrable Hamiltonian systems under the Brjuno–Rüssmann non-resonant condition, when the Kolmogorov non-degeneracy condition is violated. As it is well known, the frequency of the persisting invariant tori may undergo some drifts, when the Kolmogorov non-degeneracy condition is violated. By the method of introducing external parameters and rational approximations, we prove that if the Brouwer topological degree of the frequency mapping is nonzero at some Brjuno–Rüssmann frequency, then the invariant torus with this frequency persists under small perturbation.

We consider a nonlinear Robin problem driven by a nonlinear, nonhomogeneous differential operator, and with a Carathéodory reaction term which is (p-1)${{(p-1)}}$-superlinear near ±∞${\pm\infty}$ without satisfying the Ambrosetti–Rabinowitz condition and which does not have a standard subcritical polynomial growth. Using a combination of variational methods and Morse theoretic techniques, we prove a multiplicity theorem producing three nontrivial solutions (two of which have constant sign). In the process we establish some useful facts about the boundedness of the weak solutions of critical equations and the relation of Sobolev and Hölder local minimizers for functionals with a critical perturbation term.

In this paper, we consider the periodic solutions of a variable coefficient wave equation which models the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. Under Dirichlet–Neumann boundary conditions, we find some important properties for the variable coefficient wave operator. Then, based on these properties, we obtain the existence and multiplicity of periodic solutions by using the Leray–Schauder degree theory.

Goodwin’s celebrated growth cycle model has been widely studied since its introduction in 1967. In recent years several contributions have appeared with the aim of amending the original model so as to improve its economic coherence and enrich its structure. In this article we propose a new and generalized approach, within the theory of planar Hamiltonian systems, for the modeling of Goodwin-type cycles. This new approach, which includes and improves various attempts by the recent literature, is very general and fulfills the essential requirement that the orbits lie entirely in the economically feasible interval. We provide a necessary and sufficient condition for all solutions to be cycles lying entirely in the unit box. In addition, we study the period length of the cycles near the equilibrium and close to the boundary of the domain. Finally, we discuss an example of how small perturbations of the model may affect the qualitative behavior of the solutions.

In this paper we present, in the case of Sobolev spaces, some concentration-compactness theorems, nowadays known as profile decomposition theorems, which imply the most known results in literature, clarifying the connections between the different versions.

We study the Dirichlet problem of the constant mean curvature equation in the steady state space over a bounded domain of a slice. Under suitable conditions on the convexity of the domain, we prove the existence of a spacelike graph with constant mean curvature H , with -1≤H<0${-1\leq H<0}$, and constant boundary values.

By using the fibering method jointly with Nehari manifold techniques, we obtain the existence of multiple solutions to a fractional p -Laplacian system involving critical concave-convex nonlinearities, provided that a suitable smallness condition on the parameters involved is assumed. The result is obtained although there is no general classification for the optimizers of the critical fractional Sobolev embedding.

In this paper, we deal with the singular perturbed fractional elliptic problem ε⁢(-Δ)1/2⁢u+V⁢(z)⁢u=f⁢(u)$\varepsilon(-\Delta)^{1/2}{u}+V(z)u=f(u)$ in ℝ$\mathbb{R}$, where (-Δ)1/2⁢u${(-\Delta)^{1/2}u}$ is the square root of the Laplacian and f⁢(s)${f(s)}$ has exponential critical growth. Under suitable conditions on f⁢(s)${f(s)}$, we construct a localized bound state solution concentrating at an isolated component of the positive local minimum points of the potential of V as ε goes to 0.

This paper is devoted to the family of optimal functional inequalities on the n -dimensional sphere 𝕊n${{\mathbb{S}}^{n}}$, namely ∥ F ∥ L q ⁢ ( 𝕊 n ) 2 - ∥ F ∥ L 2 ⁢ ( 𝕊 n ) 2 q - 2 ≤ 𝖢 q , s ⁢ ∫ 𝕊 n F ⁢ ℒ s ⁢ F ⁢ 𝑑 μ   for all ⁢ F ∈ H s / 2 ⁢ ( 𝕊 n ) , $\frac{\lVert F\rVert_{\mathrm{L}^{q}({\mathbb{S}}^{n})}^{2}-\lVert F\rVert_{% \mathrm{L}^{2}({\mathbb{S}}^{n})}^{2}}{q-2}\leq\mathsf{C}_{q,s}\int_{{\mathbb{% S}}^{n}}{F\mathcal{L}_{s}F}\,d\mu\quad\text{for all }F\in\mathrm{H}^{s/2}({% \mathbb{S}}^{n}),$ where ℒs${\mathcal{L}_{s}}$ denotes a fractional Laplace operator of order s∈(0,n)${s\in(0,n)}$, q∈[1,2)∪(2,q⋆]${q\in[1,2)\cup(2,q_{\star}]}$, q⋆=2⁢nn-s${q_{\star}=\frac{2n}{n-s}}$ is a critical exponent, and d⁢μ${d\mu}$ is the uniform probability measure on 𝕊n${{\mathbb{S}}^{n}}$. These inequalities are established with optimal constants using spectral properties of fractional operators. Their consequences for fractional heat flows are considered. If q>2${q>2}$, these inequalities interpolate between fractional Sobolev and subcritical fractional logarithmic Sobolev inequalities, which correspond to the limit case as q→2${q\to 2}$. For q<2${q<2}$, the inequalities interpolate between fractional logarithmic Sobolev and fractional Poincaré inequalities. In the subcritical range q<q⋆${q<q_{\star}}$, the method also provides us with remainder terms which can be considered as an improved version of the optimal inequalities. The case s∈(-n,0)${s\in(-n,0)}$ is also considered. Finally, weighted inequalities involving the fractional Laplacian are obtained in the Euclidean space, by using the stereographic projection.