Volume 16, Issue 1

Hardy–Littlewood–Sobolev inequalities and the Hardy–Sobolev type system play an important role in analysis and PDEs. In this paper, we consider the very general weighted Hardy–Sobolev type system u ⁢ ( x ) = ∫ ℝ n 1 | x | τ ⁢ | x - y | n - α ⁢ | y | t ⁢ f 1 ⁢ ( u ⁢ ( y ) , v ⁢ ( y ) ) ⁢ 𝑑 y , v ⁢ ( x ) = ∫ ℝ n 1 | x | t ⁢ | x - y | n - α ⁢ | y | τ ⁢ f 2 ⁢ ( u ⁢ ( y ) , v ⁢ ( y ) ) ⁢ 𝑑 y , $u(x)=\int_{\mathbb{R}^{n}}\frac{1}{|x|^{\tau}|x-y|^{n-\alpha}|y|^{t}}f_{1}(u(y% ),v(y))dy,\quad v(x)=\int_{\mathbb{R}^{n}}\frac{1}{|x|^{t}|x-y|^{n-\alpha}|y|^% {\tau}}f_{2}(u(y),v(y))dy,$ where f 1 ⁢ ( u ⁢ ( y ) , v ⁢ ( y ) ) = λ 1 ⁢ u p 1 ⁢ ( y ) + μ 1 ⁢ v q 1 ⁢ ( y ) + γ 1 ⁢ u α 1 ⁢ ( y ) ⁢ v β 1 ⁢ ( y ) , $\displaystyle f_{1}(u(y),v(y))=\lambda_{1}u^{p_{1}}(y)+\mu_{1}v^{q_{1}}(y)+% \gamma_{1}u^{\alpha_{1}}(y)v^{\beta_{1}}(y),$ f 2 ⁢ ( u ⁢ ( y ) , v ⁢ ( y ) ) = λ 2 ⁢ u p 2 ⁢ ( y ) + μ 2 ⁢ v q 2 ⁢ ( y ) + γ 2 ⁢ u α 2 ⁢ ( y ) ⁢ v β 2 ⁢ ( y ) . $\displaystyle f_{2}(u(y),v(y))=\lambda_{2}u^{p_{2}}(y)+\mu_{2}v^{q_{2}}(y)+% \gamma_{2}u^{\alpha_{2}}(y)v^{\beta_{2}}(y).$ Only the special cases when γ1=γ2=0${\gamma_{1}=\gamma_{2}=0}$ and one of λi${\lambda_{i}}$ and μi${\mu_{i}}$ is zero (for both i=1${i=1}$ and i=2${i=2}$) have been considered in the literature. We establish the integrability of the solutions to the above Hardy–Sobolev type system and the C∞${C^{\infty}}$ regularity of solutions to this system away from the origin, which improves significantly the Lipschitz continuity in most works in the literature. Moreover, we also use the moving plane method of [8] in integral forms developed in [6] to prove that each pair (u,v)${(u,v)}$ of positive solutions of the above integral system is radially symmetric and strictly decreasing about the origin.

We consider a fractional Schrödinger–Poisson system with a general nonlinearity in the subcritical and critical case. The Ambrosetti–Rabinowitz condition is not required. By using a perturbation approach, we prove the existence of positive solutions. Moreover, we study the asymptotics of solutions for a vanishing parameter.

In this paper, we study the eigenvalues of the poly-Laplacian with arbitrary order on a bounded domain in an n -dimensional Euclidean space and we obtain a lower bound which generalizes the results due to Cheng and Wei [5] and gives an improvement of the results due to Cheng, Qi and Wei [3].

In [5], Ortega has analyzed “generalized” collision solutions of the periodically forced rectilinear Kepler problem. In this note, we explain a different approach to study these solutions by embedding the non-autonomous Hamiltonian system into the zero-energy level of an autonomous Hamiltonian system and by employing the Levi-Civita regularization to regularize the double collisions. In addition to this, under a certain smoothness hypothesis of the periodic force term, we show that there exists a set of positive measure of generalized quasi-periodic solutions in the extended phase space, each of them accumulated by generalized periodic solutions of the system. The energy of these quasi-periodic solutions can have an arbitrary large absolute value.

The existence of multiple solutions to a Dirichlet problem involving the (p,q)${(p,q)}$-Laplacian is investigated via variational methods, truncation-comparison techniques, and Morse theory. The involved reaction term is resonant at infinity with respect to the first eigenvalue of -Δp${-\Delta_{p}}$ in W01,p⁢(Ω)${W^{1,p}_{0}(\Omega)}$ and exhibits a concave behavior near zero.

We show that the spatially inhomogeneous Allen–Cahn equation -ε2⁢Δ⁢u=u⁢(u-a⁢(x))⁢(1-u)${-\varepsilon^{2}\Delta u=u(u-a(x))(1-u)}$ in a smooth bounded domain Ω⊂ℝN${\Omega\subset\mathbb{R}^{N}}$, u=0${u=0}$ on ∂⁡Ω$\partial\Omega$, with 0<a⁢(⋅)<1${0<a(\,\cdot\,)<1}$ continuous and ε>0${\varepsilon>0}$ a small parameter, cannot have globally minimizing solutions with transition layers in a smooth subdomain of Ω whereon a-12${a-\frac{1}{2}}$ does not change sign and a-12≠0${a-\frac{1}{2}\neq 0}$ on that subdomain’s boundary. Under the assumption of radial symmetry, this property was shown by Dancer and Yan in [5]. Our approach may also be used to simplify some parts of the latter and related references. In particular, for this model, we can give a streamlined new proof of the existence of locally minimizing transition layered solutions with nonsmooth interfaces, considered originally by del Pino in [6] using different techniques. Besides of its simplicity, the main advantage of our proof is that it allows one to deal with more degenerate situations. We also establish analogous results for a class of problems that includes the spatially inhomogeneous Fisher–KPP equation -ε2⁢Δ⁢u=ρ⁢(x)⁢u⁢(1-u)${-\varepsilon^{2}\Delta u=\rho(x)u(1-u)}$ with ρ sign-changing.

We estimate some complex structures related to perturbed Liouville equations defined on a compact Riemannian 2-manifold. As a byproduct, we obtain a quick proof of the mass quantization and we locate the blow-up points.

In this paper, we derive generalized versions of the results on the existence, uniqueness and continuous differentiability of a global implicit function obtained in [5] and we give some examples.

We prove a conjecture recently formulated by Maia, Montefusco and Pellacci saying that minimal energy solutions of the saturated nonlinear Schrödinger system { - Δ ⁢ u + λ 1 ⁢ u = α ⁢ u ⁢ ( α ⁢ u 2 + β ⁢ v 2 ) 1 + s ⁢ ( α ⁢ u 2 + β ⁢ v 2 ) in ⁢ ℝ n , - Δ ⁢ v + λ 2 ⁢ v = β ⁢ v ⁢ ( α ⁢ u 2 + β ⁢ v 2 ) 1 + s ⁢ ( α ⁢ u 2 + β ⁢ v 2 ) in ⁢ ℝ n $\left\{\begin{aligned} \displaystyle-\Delta u+\lambda_{1}u&\displaystyle=\frac% {\alpha u(\alpha u^{2}+\beta v^{2})}{1+s(\alpha u^{2}+\beta v^{2})}&&% \displaystyle\text{in }\mathbb{R}^{n},\\ \displaystyle-\Delta v+\lambda_{2}v&\displaystyle=\frac{\beta v(\alpha u^{2}+% \beta v^{2})}{1+s(\alpha u^{2}+\beta v^{2})}&&\displaystyle\text{in }\mathbb{R% }^{n}\end{aligned}\right.$ are necessarily semitrivial whenever α,β,λ1,λ2>0${\alpha,\hskip 0.5pt\beta,\hskip 0.5pt\lambda_{1},\hskip 0.5pt\lambda_{2}>0}$ and 0<s<max⁡{α/λ1,β/λ2}${0<s<\max\{\alpha/\lambda_{1},\hskip 0.5pt\beta/\lambda_{2}\}}$ except for the symmetric case λ1=λ2${\lambda_{1}=\lambda_{2}}$, α=β${\alpha=\beta}$. Moreover, it is shown that for most parameter samples α,β,λ1,λ2${\alpha,\beta,\lambda_{1},\lambda_{2}}$, there are infinitely many branches containing seminodal solutions which bifurcate from a semitrivial solution curve parametrized by s .

We deal with existence, uniqueness and regularity for solutions of the boundary value problem { ℒ s ⁢ u = μ in ⁢ Ω , u ⁢ ( x ) = 0 on ⁢ ℝ n \ Ω , $\left\{\begin{aligned} \displaystyle\mathcal{L}^{s}u&\displaystyle=\mu&&% \displaystyle\text{in }\Omega,\\ \displaystyle u(x)&\displaystyle=0&&\displaystyle\text{on }\mathbb{R}^{n}% \backslash\Omega,\end{aligned}\right.$ where Ω is a bounded domain of ℝn${\mathbb{R}^{n}}$, μ is a bounded Radon measure on Ω, and ℒs${\mathcal{L}^{s}}$ is a non-local operator of fractional order s whose kernel K is comparable with the one of the fractional Laplacian.

New weighted Gagliardo–Nirenberg inequalities are introduced together with applications to the local/global existence of solutions to nonlinear strongly coupled and uniform parabolic systems. Much weaker sufficient conditions than those existing in literature for solvability of these systems will be established.

The Burgers-αβ equation, which was first introduced by Holm and Staley [4], is considered in the special case where ν=0${\nu=0}$ and b=3${b=3}$. Traveling wave solutions are classified to the Burgers-αβ equation containing four parameters b,α,ν${b,\alpha,\nu}$, and β, which is a nonintegrable nonlinear partial differential equation that coincides with the usual Burgers equation and viscous b -family of peakon equation, respectively, for two specific choices of the parameter β=0${\beta=0}$ and β=1${\beta=1}$. Under the decay condition, it is shown that there are smooth, peaked and cusped traveling wave solutions of the Burgers-αβ equation with ν=0${\nu=0}$ and b=3${b=3}$ depending on the parameter β. Moreover, all traveling wave solutions without the decay condition are parametrized by the integration constant k1∈ℝ${k_{1}\in\mathbb{R}}$. In an appropriate limit β=1${\beta=1}$, the previously known traveling wave solutions of the Degasperis–Procesi equation are recovered.

In [33], Wang proved that for every Finsler three-dimensional sphere (S3,F)${(S^{3},F)}$ with reversibility λ and flag curvature K satisfying (λ/(1+λ))2<K≤1${(\lambda/({1+\lambda}))^{2}<K\leq 1}$, there exist at least three distinct closed geodesics. In this paper, we prove that for every Finsler three-dimensional sphere (S3,F)$(S^{3},F)$ with reversibility λ and flag curvature K satisfying (9/4)⁢(λ/(1+λ))2<K≤1${(9/4)(\lambda/(1+\lambda))^{2}<K\leq 1}$ with λ<2${\lambda<2}$, if there exist exactly three prime closed geodesics, then two of them are irrationally elliptic and the third one is infinitely degenerate.