Volume 17, Issue 3

This paper ascertains the exact boundary blow-up rates of the large positive solutions of a class of cooperative logistic systems involving n species in a general domain of ℝN{\mathbb{R}^{N}} of class 𝒞2+ν{\mathcal{C}^{2+\nu}}, 0<ν<1{0<\nu<1}. The problem models a population divided in groups whose individuals compete with those of the same group, while simultaneously they cooperate with the members of the remaining groups. Our main result provides with the exact blow-up rates along the edges of Ω from the values of the blow-up rates of the underlying uncoupled system. Rather astonishingly, these blow-up rates are independent of the strength of the cooperative effects, which play a secondary role in the analysis carried out in this paper. No previous result of this nature is available in the specialized literature for more than n=2{n=2} species.

This paper deals with the existence and the asymptotic behavior of nontrivial solutions for some classes of stationary Kirchhoff problems driven by a fractional integro-differential operator and involving a Hardy potential and different critical nonlinearities. In particular, we cover the delicate degenerate case, that is, when the Kirchhoff function M is zero at zero. To overcome the difficulties due to the lack of compactness as well as the degeneracy of the models, we have to make use of different approaches.

In this paper, we use a suitable transform of quasi-conformal mapping type to investigate the sharp constants and optimizers for the following Caffarelli–Kohn–Nirenberg inequalities for a large class of parameters (r,p,q,s,μ,σ){(r,p,q,s,\mu,\sigma)} and 0≤a≤1{0\leq a\leq 1}: ( ∫ | u | r d ⁢ x | x | s ) 1 r ≤ C ( ∫ | ∇ u | p d ⁢ x | x | μ ) a p ( ∫ | u | q d ⁢ x | x | σ ) 1 - a q . \bigg{(}\int\lvert u|^{r}\frac{dx}{|x|^{s}}\bigg{)}^{\frac{1}{r}}\leq C\bigg{(% }\int\lvert\nabla u|^{p}\frac{dx}{\lvert x|^{\mu}}\bigg{)}^{\frac{a}{p}}\bigg{% (}\int\lvert u|^{q}\frac{dx}{\lvert x|^{\sigma}}\bigg{)}^{\frac{1-a}{q}}. We compute the best constants and the explicit forms of the extremal functions in numerous cases. When 0<a<1{0<a<1}, we can deduce the existence and symmetry of optimizers for a wide range of parameters. Moreover, in the particular cases r=p⁢q-1p-1{r=p\frac{q-1}{p-1}} and q=p⁢r-1p-1{q=p\frac{r-1}{p-1}}, the forms of maximizers will also be provided in the spirit of Del Pino and Dolbeault [14, 15]. In the case a=1{a=1}, that is, the Caffarelli–Kohn–Nirenberg inequality without the interpolation term, we will provide the exact maximizers for all the range of μ≥0{\mu\geq 0}. The Caffarelli–Kohn–Nirenberg inequalities with arbitrary norms on Euclidean spaces will also be considered in the spirit of Cordero-Erausquin, Nazaret and Villani [13]. Due to the absence of the classical Polyá–Szegö inequality in the weighted case, we establish a symmetrization inequality with power weights which is of independent interest.

We investigate the existence of periodic trajectories of a particle, subject to a central force, which can hit a sphere or a cylinder. We will also provide a Landesman–Lazer-type condition in the case of a nonlinearity satisfying a double resonance condition. Afterwards, we will show how such a result can be adapted to obtain a new result for the impact oscillator at double resonance.

We construct a second-order equation x¨=h⁢(t)/xp{\ddot{x}=h(t)/x^{p}}, with p>1{p>1} and the sign-changing, periodic weight function h having negative mean, which does not have periodic solutions. This contrasts with earlier results which state that, in many cases, such periodic problems are solvable.

We deal with very weak positive supersolutions to the Hénon–Lane–Emden system on neighborhoods of the origin. In our main theorem we prove a sharp nonexistence result.

The aim of this paper is to establish a connection between the method of period segments and the relative Nielsen fixed point theory. We prove that if W is a periodic segment over [0,T]{[0,T]} for the T -periodic semi-process Φ, then the Poincaré map P has at least N⁢(mW,W0∖W0--¯){N(m_{W},\overline{W_{0}\setminus W_{0}^{--}})} fixed points with trajectories contained in W , where N⁢(mW,W0∖W0--¯){N(m_{W},\overline{W_{0}\setminus W_{0}^{--}})} is the relative Nielsen number defined by Zhao. It is also shown that if the sequence N⁢(m¯n){N(\overline{m}^{n})} is bounded and N∞⁢(m)>1{N^{\infty}(m)>1}, then the Poincaré map has infinitely many periodic points. We prove that there exists a compact set I⊂W0{I\subset W_{0}}, invariant for the Poincaré map, such that the topological entropy h⁢(P|I){h(P|_{I})} is bounded from below by log⁡N∞⁢(m)-h⁢(m¯){\log N^{\infty}(m)-h(\overline{m})}. In particular, if h⁢(m¯)=0{h(\overline{m})=0}, then h⁢(P|I)≥log⁡N∞⁢(m).{h(P|_{I})\geq\log N^{\infty}(m).} We adapt the result obtained by Jiang to get a concrete example of a braid-like periodic segment with N∞⁢(m)>1{N^{\infty}(m)>1}.

In this paper, we consider a class of Schrödinger equations involving fractional Laplacian and indefinite potentials. By modifying the definition of the Nehari–Pankov manifold, we prove the existence and asymptotic behavior of least energy solutions. As the fractional Laplacian is nonlocal, when the bottom of the potentials contains more than one isolated components, the least energy solutions may localize near all the isolated components simultaneously. This phenomenon is different from the Laplacian.

We prove the existence of a nontrivial groundstate solution for the class of nonlinear Choquard equations - Δ ⁢ u + u = ( I α * F ⁢ ( u ) ) ⁢ F ′ ⁢ ( u )   in  ⁢ ℝ 2 , -\Delta u+u=\bigl{(}I_{\alpha}*F(u)\bigr{)}F^{\prime}(u)\quad\text{in }\mathbb% {R}^{2}, where Iα{I_{\alpha}} is the Riesz potential of order α on the plane ℝ2{\mathbb{R}^{2}} under general nontriviality, growth and subcriticality on the nonlinearity F∈C1⁢(ℝ,ℝ){F\in C^{1}(\mathbb{R},\mathbb{R})}.

We consider the problem -Δp⁢u+a⁢(x)⁢|u|p-2⁢u=|u|q-2⁢u{-\Delta_{p}u+a(x)\lvert u\rvert^{p-2}u=\lvert u\rvert^{q-2}u} in ℝN{\mathbb{R}^{N}}, where 1<p<N{1<p<N}, p<q<p*=N⁢pN-p{p<q<p^{*}=\frac{Np}{N-p}}, Δp{\Delta_{p}} is the p -Laplacian operator, and the potential function a is positive, bounded and verifies suitable decay assumptions. The existence of infinitely many solutions of the equation is proved via the truncation method.

This paper concerns itself with the nonexistence and multiplicity of solutions for the following fractional Kirchhoff-type problem involving the critical Sobolev exponent: [ a + b ⁢ ( ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ) θ - 1 ] ⁢ ( - Δ ) p s ⁢ u = | u | p s * - 2 ⁢ u + λ ⁢ f ⁢ ( x )   in  ⁢ ℝ N , \Biggl{[}a+b\biggl{(}\iint_{\mathbb{R}^{2N}}\frac{\lvert u(x)-u(y)\rvert^{p}}{% \lvert x-y\rvert^{N+ps}}\,dx\,dy\biggr{)}^{\theta-1}\Biggr{]}(-\Delta)_{p}^{s}% u=\lvert u\rvert^{p_{s}^{*}-2}u+\lambda f(x)\quad\text{in }\mathbb{R}^{N}, where a≥0{a\kern-1.0pt\geq\kern-1.0pt0}, b>0,θ>1{b\kern-1.0pt>\kern-1.0pt0,\theta\kern-1.0pt>\kern-1.0pt1}, (-Δ)ps{(-\Delta)_{p}^{s}} is the fractional p -Laplacian with 0<s<1{0\kern-1.0pt<\kern-1.0pts\kern-1.0pt<\kern-1.0pt1} and 1<p<N/s{1\kern-1.0pt<\kern-1.0ptp\kern-1.0pt<\kern-1.0ptN/s}, ps*=N⁢p/(N-p⁢s){p_{s}^{*}\kern-1.0pt=\kern-1.0ptNp/(N-ps)} is the critical Sobolev exponent, λ≥0{\lambda\geq 0} is a parameter, and f∈Lps*/(ps*-1)⁢(ℝN)∖{0}{f\in L^{p_{s}^{*}/(p_{s}^{*}-1)}(\mathbb{R}^{N})\setminus\{0\}} is a nonnegative function. When λ=0{\lambda=0}, we show that the multiplicity and nonexistence of solutions for the above problem are related with N , θ, s , p , a , and b . When λ>0{\lambda>0}, by using Ekeland’s variational principle and the mountain pass theorem, we show that there exists λ**>0{\lambda^{**}>0} such that the above problem admits at least two nonnegative solutions for all λ∈(0,λ**){\lambda\in(0,\lambda^{**})}. In the latter case, in order to overcome the loss of compactness, we derive a fractional version of the principle of concentration compactness in the setting of the fractional p -Laplacian.