Volume 18, Issue 3

In this paper, we establish a sharp concentration-compactness principle associated with the singular Adams inequality on the second-order Sobolev spaces in ℝ4{\mathbb{R}^{4}}. We also give a new Sobolev compact embedding which states W2,2⁢(ℝ4){W^{2,2}(\mathbb{R}^{4})} is compactly embedded into Lp⁢(ℝ4,|x|-β⁢d⁢x){L^{p}(\mathbb{R}^{4},|x|^{-\beta}\,dx)} for p≥2{p\geq 2} and 0<β<4{0<\beta<4}. As applications, we establish the existence of ground state solutions to the following bi-Laplacian equation with critical nonlinearity: Δ 2 ⁢ u + V ⁢ ( x ) ⁢ u = f ⁢ ( x , u ) | x | β   in  ⁢ ℝ 4 , \displaystyle\Delta^{2}u+V(x)u=\frac{f(x,u)}{|x|^{\beta}}\quad\mbox{in }% \mathbb{R}^{4}, where V⁢(x){V(x)} has a positive lower bound and f⁢(x,t){f(x,t)} behaves like exp⁡(α⁢|t|2){\exp(\alpha|t|^{2})} as t→+∞{t\to+\infty}. In the case β=0{\beta=0}, because of the loss of Sobolev compact embedding, we use the principle of symmetric criticality to obtain the existence of ground state solutions by assuming f⁢(x,t){f(x,t)} and V⁢(x){V(x)} are radial with respect to x and f⁢(x,t)=o⁢(t){f(x,t)=o(t)} as t→0{t\rightarrow 0}.

In this paper, we consider the problem of prescribing the Gaussian and geodesic curvature on a disk and its boundary, respectively, via a conformal change of the metric. This leads us to a Liouville-type equation with a non-linear Neumann boundary condition. We address the question of existence by setting the problem in a variational framework which seems to be completely new in the literature. We are able to find minimizers under symmetry assumptions.

In this paper, we establish existence and multiplicity results for systems of first-order differential equations. To this end, we introduce the method of solution-regions. It generalizes the method of upper and lower solutions and the method of solution-tubes. Our results can also be seen as viability results since we obtain solutions remaining in suitable regions. We give conditions insuring the existence of at least three viable solutions of a system of first-order differential equations. Many examples are presented to show that a large variety of sets can be solution-regions.

In this paper, we are concerned with the existence and concentration phenomena of solutions for the following singularly perturbed fractional Schrödinger problem: ε 2 ⁢ s ⁢ ( - Δ ) s ⁢ u + V ⁢ ( x ) ⁢ u = f ⁢ ( u )   in  ⁢ ℝ N , \varepsilon^{2s}(-\Delta)^{s}u+V(x)u=f(u)\quad\text{in }\mathbb{R}^{N}, where N>2⁢s{N>2s} and the nonlinearity f has critical growth. By using the variational approach, we construct a localized bound-state solution concentrating around an isolated component of the positive minimum point of V as ε→0{\varepsilon\rightarrow 0}. Our result improves the study made in [X. He and W. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. Partial Differential Equations 55 2016, 4, Article ID 91], in the sense that, in the present paper, the Ambrosetti–Rabinowitz condition and the monotonicity condition on f⁢(t)/t{f(t)/t} are not required.

We prove the finite dimensionality of the global attractor and estimate the numbers of the determining modes for the 2D Boussinesq system in a periodic domain with fractional Laplacian in the subcritical case.

This paper deals with the Cauchy problem to the Keller–Segel model coupled with the incompressible 3-D Navier–Stokes equations. Based on so-called Gevrey regularity estimates, which are motivated by the works of Foias and Temam [20], we prove that the solutions are analytic for a small interval of time with values in a Gevrey class of functions. As a consequence of Gevrey estimates, we particularly imply higher-order derivatives of solutions in Besov and Lebesgue spaces. Moreover, we prove that the existence of a positive constant C~{\tilde{C}} such that the initial data (u0,n0,c0):=(u0h,u03,n0,c0){(u_{0},n_{0},c_{0}):=(u_{0}^{h},u_{0}^{3},n_{0},c_{0})} satisfy C ~ ⁢ ( ∥ ( n 0 , c 0 ) ∥ B ˙ q , 1 - 2 + 3 / q ⁢ ( ℝ 3 ) × B ˙ q , 1 3 / q ⁢ ( ℝ 3 ) + ∥ u 0 h ∥ B ˙ p , 1 - 1 + 3 / p ⁢ ( ℝ 3 ) + ∥ u 0 h ∥ B ˙ p , 1 - 1 + 3 / p ⁢ ( ℝ 3 ) α ⁢ ∥ u 0 3 ∥ B ˙ p , 1 - 1 + 3 / p ⁢ ( ℝ 3 ) 1 - α ) ≤ 1 \tilde{C}\bigl{(}\lVert(n_{0},c_{0})\rVert_{\dot{B}^{-2+3/q}_{q,1}(\mathbb{R}^% {3})\times\dot{B}^{3/q}_{q,1}(\mathbb{R}^{3})}+\lVert u_{0}^{h}\rVert_{\dot{B}% ^{-1+3/p}_{p,1}(\mathbb{R}^{3})}+\lVert u_{0}^{h}\rVert_{\dot{B}^{-1+3/p}_{p,1% }(\mathbb{R}^{3})}^{\alpha}\lVert u_{0}^{3}\rVert_{\dot{B}^{-1+3/p}_{p,1}(% \mathbb{R}^{3})}^{1-\alpha}\bigr{)}\leq 1 for certain conditions on p,q{p,q} and α implies the global existence of solutions with large initial vertical velocity component.

Let a⁢(x){a(x)} be non-constant and let bj⁢(x){b_{j}(x)}, for j=0,1,2,3{j=0,1,2,3}, be real or complex polynomials in the variable x . Then the real or complex equivariant polynomial Abel differential equation a⁢(x)⁢y˙=b1⁢(x)⁢y+b3⁢(x)⁢y3{a(x)\dot{y}=b_{1}(x)y+b_{3}(x)y^{3}}, with b3⁢(x)≠0{b_{3}(x)\neq 0}, and the real or complex polynomial equivariant polynomial Abel differential equation of the second kind a⁢(x)⁢y⁢y˙=b0⁢(x)+b2⁢(x)⁢y2{a(x)y\dot{y}=b_{0}(x)+b_{2}(x)y^{2}}, with b2⁢(x)≠0{b_{2}(x)\neq 0}, have at most 7 polynomial solutions. Moreover, there exist equations of this type having this maximum number of polynomial solutions.

Though there has been an extensive study on first-order Caffarelli–Kohn–Nirenberg inequalities, not much is known for the existence of extremal functions for higher-order ones. The higher-order derivative of the Caffarelli–Kohn–Nirenberg inequality established by Lin [14] states ( ∫ ℝ N | D j u | r d ⁢ x | x | s ) 1 r ≤ C ( ∫ ℝ N | D m u | p d ⁢ x | x | μ ) a p ( ∫ ℝ N | u | q d ⁢ x | x | σ ) 1 - a q , \bigg{(}\int_{\mathbb{R}^{N}}\lvert D^{j}u|^{r}\frac{dx}{|x|^{s}}\bigg{)}^{% \frac{1}{r}}\leq C\bigg{(}\int_{\mathbb{R}^{N}}\lvert D^{m}u|^{p}\frac{dx}{|x|% ^{\mu}}\bigg{)}^{\frac{a}{p}}\bigg{(}\int_{\mathbb{R}^{N}}\lvert u|^{q}\frac{% dx}{|x|^{\sigma}}\bigg{)}^{\frac{1-a}{q}}, where C=C⁢(p,q,r,μ,σ,s,m,j){C=C(p,q,r,\mu,\sigma,s,m,j)} and p,q,r,μ,σ,s,m,j{p,q,r,\mu,\sigma,s,m,j} are parameters satisfying some balanced conditions. The main purpose of this paper is to establish the existence of extremal functions for a family of this higher-order derivatives of Caffarelli–Kohn–Nirenberg inequalities under numerous circumstances of parameters. Moreover, we study the compactness of the weighted Sobolev space for higher-order derivatives and prove that H˙μm,p⁢(Ω)∩Lσq⁢(Ω)↪H˙sj,r⁢(Ω){\dot{H}^{m,p}_{\mu}(\Omega)\cap L^{q}_{\sigma}(\Omega)\hookrightarrow\dot{H}^% {j,r}_{s}(\Omega)} is a compact embedding within some range of the parameters.

This paper deals with the problem { - Δ ⁢ u = λ ⁢ u q ⁢ ( x ) , x ∈ Ω , u = 0 , x ∈ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle-\Delta u&\displaystyle=\lambda u^{q(x)},&% \hskip 10.0ptx&\displaystyle\in\Omega,\\ \displaystyle u&\displaystyle=0,&\hskip 10.0ptx&\displaystyle\in\partial\Omega% ,\end{aligned}\right. where Ω⊂ℝN{\Omega\subset\mathbb{R}^{N}} is a bounded smooth domain, λ>0{\lambda>0} is a parameter and the reaction order q⁢(x){q(x)} is a Hölder continuous positive function satisfying q⁢(x)>1{q(x)>1} for all x∈Ω{x\in\Omega}. The relevant feature here is that q is assumed to achieve the value one on ∂⁡Ω{\partial\Omega}. By assuming that q is subcritical, our main result states the existence of a positive solution for all λ>0{\lambda>0}. We also study its asymptotic behavior as λ→0{\lambda\to 0} and as λ→∞{\lambda\to\infty}. It should be noticed that the fact that q=1{q=1} somewhere in ∂⁡Ω{\partial\Omega} gives rise to serious difficulties when looking for critical points of the functional associated with the problem above. This work is a continuation of [13] where q is assumed to take values both greater and smaller than one in Ω, but is constrained to satisfy q⁢(x)>1{q(x)>1} on ∂⁡Ω{\partial\Omega}.

In this paper, we use the rearrangement-free argument, in the spirit of the work by Li, Lu and Zhu [25], on the concentration-compactness principle on the Heisenberg group to establish a sharpened version of the singular Lions concentration-compactness principle for the Trudinger–Moser inequality in ℝn{\mathbb{R}^{n}}. Then we prove a compact embedding theorem, which states that W1,n⁢(ℝn){W^{1,n}(\mathbb{R}^{n})} is compactly embedded into Lp⁢(ℝn,|x|-β⁢d⁢x){L^{p}(\mathbb{R}^{n},|x|^{-\beta}\,dx)} for p≥n{p\geq n} and 0<β<n{0<\beta<n}. As an application of the above results, we establish sufficient conditions for the existence of ground state solutions to the following n -Laplace equation with critical nonlinearity: ($*$) { - div ⁡ ( | ∇ ⁡ u | n - 2 ⁢ ∇ ⁡ u ) + V ⁢ ( x ) ⁢ | u | n - 2 ⁢ u = f ⁢ ( x , u ) | x | β , u ∈ W 1 , n ⁢ ( ℝ n ) , u ≥ 0 , {}\left\{\begin{aligned} &\displaystyle{-}\operatorname{div}(|\nabla u|^{n-2}% \nabla u)+V(x)|u|^{n-2}u=\frac{f(x,u)}{|x|^{\beta}},\\ &\displaystyle u\in W^{1,n}(\mathbb{R}^{n}),\quad u\geq 0,\end{aligned}\right. where V⁢(x)≥c0{V(x)\geq c_{0}} for some positive constant c0{c_{0}} and f⁢(x,t){f(x,t)} behaves like exp⁡(α⁢|t|nn-1){\exp(\alpha|t|^{\frac{n}{n-1}})} as t→+∞{t\rightarrow+\infty}. This work improves substantially related results found in the literature.

We are concerned with the following singularly perturbed fractional Schrödinger equation: { ε 2 ⁢ s ⁢ ( - Δ ) s ⁢ u + V ⁢ ( x ) ⁢ u = f ⁢ ( u ) in  ⁢ ℝ N , u ∈ H s ⁢ ( ℝ N ) , u > 0 ⁢  on  ⁢ ℝ N , \left\{\begin{aligned} &\displaystyle{\varepsilon^{2s}}{(-\Delta)^{s}}u+V(x)u=% f(u)&&\displaystyle{\text{in }}{\mathbb{R}^{N}},\\ &\displaystyle u\in{H^{s}}({\mathbb{R}^{N}}),&&\displaystyle u>0{\text{ on }}{% \mathbb{R}^{N}},\end{aligned}\right. where ε is a small positive parameter, N>2⁢s{N>2s}, and (-Δ)s{{(-\Delta)^{s}}}, with s∈(0,1){s\in(0,1)}, is the fractional Laplacian. Using variational technique, we construct a family of positive solutions uε∈Hs⁢(ℝN){{u_{\varepsilon}}\in{H^{s}}({\mathbb{R}^{N}})} which concentrates around the local minima of V as ε→0{\varepsilon\to 0} under general conditions on f which we believe to be almost optimal.

In this paper, we consider the following Kirchhoff equation: { - ( a + b ∫ Ω | ∇ u | 2 d x ) Δ u = λ u + | u | p - 2 u in  ⁢ Ω , u = 0 on  ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} &\displaystyle{-}\bigg{(}a+b\int_{\Omega}\lvert\nabla u% |^{2}\,dx\bigg{)}\Delta u=\lambda u+|u|^{p-2}u&&\displaystyle\text{in }\Omega,% \\ &\displaystyle u=0&&\displaystyle\text{on }\partial\Omega,\end{aligned}\right. where Ω⊂ℝN{\Omega\subset\mathbb{R}^{N}} (N≥3{N\geq 3}) is a bounded domain with smooth boundary ∂⁡Ω{\partial\Omega}, 2<p<2*=2⁢NN-2{2<p<2^{*}=\frac{2N}{N-2}} is the Sobolev exponent and a , b , λ are positive parameters. By the variational method, we obtain some existence and multiplicity results of the sign-changing solutions (including the radial sign-changing solution in the case of Ω=𝔹R{\Omega=\mathbb{B}_{R}}) for this problem. Some further properties of these sign-changing solutions, such as the numbers of the nodal domains, the concentration behaviors as b→0+{b\to 0^{+}}, the estimates of the energy values and so on, are also obtained. Our results generalize and improve some known results in the literature. Moreover, we also obtain a uniqueness result of the radial positive solution.