Volume 21, Issue 1

In this paper we consider the fractional nonlinear Schrödinger equation ε 2 ⁢ s ⁢ ( - Δ ) s ⁢ v + V ⁢ ( x ) ⁢ v = f ⁢ ( v ) , x ∈ ℝ N , \varepsilon^{2s}(-\Delta)^{s}v+V(x)v=f(v),\quad x\in\mathbb{R}^{N}, where s∈(0,1){s\in(0,1)}, N≥2{N\geq 2}, f is a nonlinearity satisfying Berestycki–Lions type conditions and V∈C⁢(ℝN,ℝ){V\in C(\mathbb{R}^{N},\mathbb{R})} is a positive potential. For ε>0{\varepsilon>0} small, we prove the existence of at least cupl⁢(K)+1{{\rm cupl}(K)+1} positive solutions, where K is a set of local minima in a bounded potential well and cupl⁢(K){{\rm cupl}(K)} denotes the cup-length of K . By means of a variational approach, we analyze the topological difference between two levels of an indefinite functional in a neighborhood of expected solutions. Since the nonlocality comes in the decomposition of the space directly, we introduce a new fractional center of mass, via a suitable seminorm. Some other delicate aspects arise strictly related to the presence of the nonlocal operator. By using regularity results based on fractional De Giorgi classes, we show that the found solutions decay polynomially and concentrate around some point of K for ε small.

In this paper, we study the existence of multiple periodic solutions for the following fractional equation: ( - Δ ) s ⁢ u + F ′ ⁢ ( u ) = 0 , u ⁢ ( x ) = u ⁢ ( x + T )   x ∈ ℝ . (-\Delta)^{s}u+F^{\prime}(u)=0,\qquad u(x)=u(x+T)\quad x\in\mathbb{R}. For an even double-well potential, we establish more and more periodic solutions for a large period T . Without the evenness of F we give the existence of two periodic solutions of the problem. We make use of variational arguments, in particular Clark’s theorem and Morse theory.

We consider the elliptic quasilinear equation -Δm⁢u=up⁢|∇⁡u|q{-\Delta_{m}u=u^{p}\lvert\nabla u\rvert^{q}} in ℝN{\mathbb{R}^{N}}, q≥m{q\geq m} and p>0{p>0}, 1<m<N{1<m<N}. Our main result is a Liouville-type property, namely, all the positive C1{C^{1}} solutions in ℝN{\mathbb{R}^{N}} are constant. We also give their asymptotic behaviour; all the solutions in an exterior domain ℝN∖Br0{\mathbb{R}^{N}\setminus B_{r_{0}}} are bounded. The solutions in Br0∖{0}{B_{r_{0}}\setminus\{0\}} can be extended as continuous functions in Br0{B_{r_{0}}}. The solutions in ℝN∖{0}{\mathbb{R}^{N}\setminus\{0\}} has a finite limit l≥0{l\geq 0} as |x|→∞{\lvert x\rvert\to\infty}. Our main argument is a Bernstein estimate of the gradient of a power of the solution, combined with a precise Osserman-type estimate for the equation satisfied by the gradient.

In this article, we first study the existence of nontrivial solutions to the nonlocal elliptic problems in ℝN{\mathbb{R}^{N}} involving fractional Laplacians and the Hardy–Sobolev–Maz’ya potential. Using variational methods, we investigate the attainability of the corresponding minimization problem, and then obtain the existence of solutions. We also consider another Choquard type equation involving the p -Laplacian and critical nonlinearities in ℝN{\mathbb{R}^{N}}.

Consider the equation div⁡(φ2⁢∇⁡σ)=0{\operatorname{div}(\varphi^{2}\nabla\sigma)=0} in ℝN{\mathbb{R}^{N}}, where φ>0{\varphi>0}. Berestycki, Caffarelli and Nirenberg proved in [H. Berestycki, L. Caffarelli and L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 25 1997, 69–94] that if there exists C>0{C>0} such that ∫BR(φ⁢σ)2≤C⁢R2\int_{B_{R}}(\varphi\sigma)^{2}\leq CR^{2} for every R≥1{R\geq 1}, then σ is necessarily constant. In this paper, we provide necessary and sufficient conditions on 0<Ψ∈C⁢([1,∞)){0<\Psi\in C([1,\infty))} for which this result remains true if we replace C⁢R2{CR^{2}} by Ψ⁢(R){\Psi(R)} in any dimension N . In the case of the convexity of Ψ for large R>1{R>1} and Ψ′>0{\Psi^{\prime}>0}, this condition is equivalent to ∫1∞1Ψ′=∞.\int_{1}^{\infty}\frac{1}{\Psi^{\prime}}=\infty.

We consider the coupled system given by the first variation of the conformal Dirac–Einstein functional. We will show existence of solutions by means of perturbation methods.

In this paper, global-in-time existence and blow-up results are shown for a reaction-diffusion equation appearing in the theory of aggregation phenomena (including chemotaxis). Properties of the corresponding steady-state problem are also presented. Moreover, the stability around constant equilibria and the non-existence of nonconstant solutions are studied in certain cases.

In this paper, we investigate the non-autonomous Choquard equation - Δ ⁢ u + λ ⁢ V ⁢ ( x ) ⁢ u = ( I α ∗ F ⁢ ( u ) ) ⁢ F ′ ⁢ ( u )   in ⁢ R N , -\Delta u+\lambda V(x)u=(I_{\alpha}\ast F(u))F^{\prime}(u)\quad\text{in}\ \mathbb{R}^{N}, where N≥4N\geq 4, λ>0\lambda>0, V∈C⁢(RN,R)V\in C(\mathbb{R}^{N},\mathbb{R}) is bounded from below and has a potential well, IαI_{\alpha} is the Riesz potential of order α∈(0,N)\alpha\in(0,N) and F⁢(u)=12α*⁢|u|2α*+12*α⁢|u|2*αF(u)=\frac{1}{2_{\alpha}^{*}}\lvert u\rvert^{2_{\alpha}^{*}}+\frac{1}{2_{*}^{\alpha}}\lvert u\rvert^{2_{*}^{\alpha}}, in which 2α*=N+αN-22_{\alpha}^{*}=\frac{N+\alpha}{N-2} and 2*α=N+αN2_{*}^{\alpha}=\frac{N+\alpha}{N} are upper and lower critical exponents due to the Hardy–Littlewood–Sobolev inequality, respectively. Based on the variational methods, by combining the mountain pass theorem and Nehari manifold, we obtain the existence and concentration of positive ground state solutions for 𝜆 large enough if 𝑉 is nonnegative in RN\mathbb{R}^{N}; further, by the linking theorem, we prove the existence of nontrivial solutions for 𝜆 large enough if 𝑉 changes sign in RN\mathbb{R}^{N}.

In this paper, we consider a fully nonlinear curvature flow of a convex hypersurface in the Euclidean 𝑛-space. This flow involves 𝑘-th elementary symmetric function for principal curvature radii and a function of support function. Under some appropriate assumptions, we prove the long-time existence and convergence of this flow. As an application, we give the existence of smooth solutions to the Orlicz–Christoffel–Minkowski problem.

Through conformal map, isoperimetric inequalities are equivalent to the Hardy–Littlewood–Sobolev (HLS) inequalities involved with the Poisson-type kernel on the upper half space. From the analytical point of view, we want to know whether there exists a reverse analogue for the Poisson-type kernel. In this work, we give an affirmative answer to this question. We first establish the reverse Stein–Weiss inequality with the Poisson-type kernel, finding that the range of index 𝑝,q′q^{\prime} appearing in the reverse inequality lies in the interval (0,1)(0,1), which is perfectly consistent with the feature of the index for the classical reverse HLS and Stein–Weiss inequalities. Then we give the existence and asymptotic behaviors of the extremal functions of this inequality. Furthermore, for the reverse HLS inequalities involving the Poisson-type kernel, we establish the regularity for the positive solutions to the corresponding Euler–Lagrange system and give the sufficient and necessary conditions of the existence of their solutions. Finally, in the conformal invariant index, we classify the extremal functions of the latter reverse inequality and compute the sharp constant. Our methods are based on the reversed version of the Hardy inequality in high dimension, Riesz rearrangement inequality and moving spheres.

We deal with Dirichlet problems of the form { Δ ⁢ u + f ⁢ ( u ) = 0 in  ⁢ Ω , u = 0 on  ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}\Delta u+f(u)&\displaystyle=0&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. where Ω is a bounded domain of ℝn{\mathbb{R}^{n}}, n≥3{n\geq 3}, and f has supercritical growth from the viewpoint of Sobolev embedding. In particular, we consider the case where Ω is a tubular domain Tε⁢(Γk){T_{\varepsilon}(\Gamma_{k})} with thickness ε>0{{\varepsilon}>0} and center Γk{\Gamma_{k}}, a k -dimensional, smooth, compact submanifold of ℝn{\mathbb{R}^{n}}. Our main result concerns the case where k=1{k=1} and Γk{\Gamma_{k}} is contractible in itself. In this case we prove that the problem does not have nontrivial solutions for ε>0{{\varepsilon}>0} small enough. When k≥2{k\geq 2} or Γk{\Gamma_{k}} is noncontractible in itself we obtain weaker nonexistence results. Some examples show that all these results are sharp for what concerns the assumptions on k and f .

The purpose of the article is to study the existence, regularity, stabilization and blow-up results of weak solution to the following parabolic (p,q){(p,q)}-singular equation: ($\mathrm{P}_{t}$) { u t - Δ p ⁢ u - Δ q ⁢ u = ϑ ⁢ u - δ + f ⁢ ( x , u ) , u > 0 in  ⁢ Ω × ( 0 , T ) , u = 0 on  ⁢ ∂ ⁡ Ω × ( 0 , T ) , u ⁢ ( x , 0 ) = u 0 ⁢ ( x ) in  ⁢ Ω , \displaystyle{}\left\{\begin{aligned} \displaystyle{}u_{t}-\Delta_{p}u-\Delta_% {q}u&\displaystyle=\vartheta u^{-\delta}+f(x,u),\quad u>0&&\displaystyle% \phantom{}\text{in }\Omega\times(0,T),\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega\times(0,T),\\ \displaystyle u(x,0)&\displaystyle=u_{0}(x)&&\displaystyle\phantom{}\text{in }% \Omega,\end{aligned}\right. where Ω is a bounded domain in ℝN{\mathbb{R}^{N}} with C2{C^{2}} boundary ∂⁡Ω{\partial\Omega}, 1<q<p<∞{1<q<p<\infty}, 0<δ,T>0{0<\delta,T>0}, N≥2{N\geq 2} and ϑ>0{\vartheta>0} is a parameter. Moreover, we assume that f:Ω×[0,∞)→ℝ{f:\Omega\times[0,\infty)\to\mathbb{R}} is a bounded below Carathéodory function, locally Lipschitz with respect to the second variable uniformly in x∈Ω{x\in\Omega} and u0∈L∞⁢(Ω)∩W01,p⁢(Ω){u_{0}\in L^{\infty}(\Omega)\cap W^{1,p}_{0}(\Omega)}. We distinguish the cases as q -subhomogeneous and q -superhomogeneous depending on the growth of f (hereafter we will drop the term q ). In the subhomogeneous case, we prove the existence and uniqueness of the weak solution to problem (Pt{\mathrm{P}_{t}}) for δ<2+1p-1{\delta<2+\frac{1}{p-1}}. For this, we first study the stationary problems corresponding to (Pt{\mathrm{P}_{t}}) by using the method of sub- and supersolutions and subsequently employing implicit Euler method, we obtain the existence of a solution to (Pt{\mathrm{P}_{t}}). Furthermore, in this case, we prove the stabilization result, that is, the solution u⁢(t){u(t)} of (Pt{\mathrm{P}_{t}}) converges to u∞{u_{\infty}}, the unique solution to the stationary problem, in L∞⁢(Ω){L^{\infty}(\Omega)} as t→∞{t\rightarrow\infty}. For the superhomogeneous case, we prove the local existence theorem by taking help of nonlinear semigroup theory. Subsequently, we prove finite time blow-up of solution to problem (Pt{\mathrm{P}_{t}}) for small parameter ϑ>0{\vartheta>0} in the case δ≤1{\delta\leq 1} and for all ϑ>0{\vartheta>0} in the case δ>1{\delta>1}. Moreover, we prove higher Sobolev integrability of the solution to purely singular problem corresponding to the steady state of (Pt{\mathrm{P}_{t}}), which is of independent interest. As a consequence of this, we improve the Sobolev regularity of solution to (Pt{\mathrm{P}_{t}}) for the case δ<2+1p-1{\delta<2+\frac{1}{p-1}}.