Abstract In this paper, we consider the following parabolic variational inequality containing a multivalued term and a convex functional: Find u ∈ L p ⁢ ( 0 , T ; W 0 1 , p ⁢ ( Ω ) ) {u\in L^{p}(0,T;W^{1,p}_{0}(\Omega))} and f ∈ F ⁢ ( ⋅ , ⋅ , u ) {f\in F(\cdot,\cdot,u)} such that u ⁢ ( ⋅ , 0 ) = u 0 {u(\cdot,0)=u_{0}} and 〈 u t + A ⁢ u , v - u 〉 + Ψ ⁢ ( v ) - Ψ ⁢ ( u ) ≥ ∫ Q f ⁢ ( v - u ) ⁢ 𝑑 x ⁢ 𝑑 t for all ⁢ v ∈ L p ⁢ ( 0 , T ; W 0 1 , p ⁢ ( Ω ) ) , \langle u_{t}+Au,v-u\rangle+\Psi(v)-\Psi(u)\geq\int_{Q}f(v-u)\,dx\,dt\quad% \text{for all }v\in L^{p}(0,T;W^{1,p}_{0}(\Omega)), where A is the principal term; F is a multivalued lower-order term; Ψ ⁢ ( u ) = ∫ 0 T ψ ⁢ ( t , u ) ⁢ 𝑑 t {\Psi(u)=\int_{0}^{T}\psi(t,u)\,dt} is a convex functional. Moreover, we study the existence and other properties of solutions of this inequality assuming certain growth conditions on the lower-order term F.

Abstract 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 ( u 0 , n 0 , c 0 ) := ( u 0 h , u 0 3 , n 0 , c 0 ) {(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.

Abstract In this paper we focus on the periodic boundary value problem associated with the Liénard differential equation x ′′ + f ⁢ ( x ) ⁢ x ′ + g ⁢ ( t , x ) = s {x^{\prime\prime}+f(x)x^{\prime}+g(t,x)=s} , where s is a real parameter, f and g are continuous functions and g is T-periodic in the variable t. The classical framework of Fabry, Mawhin and Nkashama, related to the Ambrosetti–Prodi periodic problem, is modified to include conditions without uniformity, in order to achieve the same multiplicity result under local coercivity conditions on g. Analogous results are also obtained for Neumann boundary conditions.

Abstract 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 ˙ μ m , p ⁢ ( Ω ) ∩ L σ q ⁢ ( Ω ) ↪ H ˙ s j , 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.

Abstract In this paper, we study the quasilinear Schrödinger equation - Δ ⁢ u + V ⁢ ( x ) ⁢ u - γ 2 ⁢ ( Δ ⁢ u 2 ) ⁢ u = | u | p - 2 ⁢ u {-\Delta u+V(x)u-\frac{\gamma}{2}(\Delta u^{2})u=|u|^{p-2}u} , x ∈ ℝ N {x\in\mathbb{R}^{N}} , where V ⁢ ( x ) : ℝ N → ℝ {V(x):\mathbb{R}^{N}\to\mathbb{R}} is a given potential, γ > 0 {\gamma>0} , and either p ∈ ( 2 , 2 * ) {p\in(2,2^{*})} , 2 * = 2 ⁢ N N - 2 {2^{*}=\frac{2N}{N-2}} for N ≥ 4 {N\geq 4} or p ∈ ( 2 , 4 ) {p\in(2,4)} for N = 3 {N=3} . If γ ∈ ( 0 , γ 0 ) {\gamma\in(0,\gamma_{0})} for some γ 0 > 0 {\gamma_{0}>0} , we establish the existence of a positive solution u γ {u_{\gamma}} satisfying max x ∈ ℝ N ⁡ | γ μ ⁢ u γ ⁢ ( x ) | → 0 {\max_{x\in\mathbb{R}^{N}}|\gamma^{\mu}u_{\gamma}(x)|\to 0} as γ → 0 + {\gamma\to 0^{+}} for any μ > 1 2 {\mu>\frac{1}{2}} . Particularly, if V ⁢ ( x ) = λ > 0 {V(x)=\lambda>0} , we prove the existence of a positive classical radial solution u γ {u_{\gamma}} and up to a subsequence, u γ → u 0 {u_{\gamma}\to u_{0}} in H 2 ⁢ ( ℝ N ) ∩ C 2 ⁢ ( ℝ N ) {H^{2}(\mathbb{R}^{N})\cap C^{2}(\mathbb{R}^{N})} as γ → 0 + {\gamma\to 0^{+}} , where u 0 {u_{0}} is the ground state of the problem - Δ ⁢ u + λ ⁢ u = | u | p - 2 ⁢ u {-\Delta u+\lambda u=|u|^{p-2}u} , x ∈ ℝ N {x\in\mathbb{R}^{N}} .

Abstract We consider the Cauchy problem for a parabolic-elliptic system in ℝ 2 {\mathbb{R}^{2}} , called the parabolic-elliptic Keller–Segel equation, which appears in various fields in biology and physics. In the critical mass case where the total mass of the initial data is 8 ⁢ π {8\pi} , the unboundedness of nonnegative solutions to the Cauchy problem was shown by Blanchet, Carrillo and Masmoudi [7] under some conditions on the initial data, on the other hand, conditions for boundedness were given by Blanchet, Carlen and Carrillo [6] and López-Gómez, Nagai and Yamada [23]. In this paper, we investigate further the boundedness of nonnegative solutions.

Abstract This paper discusses a high-order Melnikov method for periodically perturbed equations. We introduce a new method to compute M k ⁢ ( t 0 ) {M_{k}(t_{0})} for all k ≥ 0 {k\geq 0} , among which M 0 ⁢ ( t 0 ) {M_{0}(t_{0})} is the traditional Melnikov function, and M 1 ⁢ ( t 0 ) , M 2 ⁢ ( t 0 ) , … {M_{1}(t_{0}),M_{2}(t_{0}),\ldots\,} are its high-order correspondences. We prove that, for all k ≥ 0 {k\geq 0} , M k ⁢ ( t 0 ) {M_{k}(t_{0})} is a sum of certain multiple integrals, the integrand of which we can explicitly compute. In particular, we obtain explicit integral formulas for M 0 ⁢ ( t 0 ) {M_{0}(t_{0})} and M 1 ⁢ ( t 0 ) {M_{1}(t_{0})} . We also study a concrete equation for which the explicit formula of M 1 ⁢ ( t 0 ) {M_{1}(t_{0})} is used to prove the existence of a transversal homoclinic intersection in the case of M 0 ⁢ ( t 0 ) ≡ 0 {M_{0}(t_{0})\equiv 0} .

Abstract 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 - 1 p - 1 {r=p\frac{q-1}{p-1}} and q = p ⁢ r - 1 p - 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.