Band 16, Heft 2

We prove nonexistence and uniqueness results of solutions for biharmonic equations under the Dirichlet boundary conditions on a smooth bounded domain. We carry on the work in [10] where the Navier boundary conditions were considered, we define the h -starlikeness of Ω with respect to the Dirichlet boundary conditions and a classifying number M*⁢(Ω)${M^{*}(\Omega)}$. This allows us to give a generalized critical exponent for these domains which play the role of the classical critical exponent N+4N-4${\frac{N+4}{N-4}}$. Our approach is based on the Rellich–Pohozaev type identity [20, 23]. We study some examples of Dirichlet h -starlike domains with either rich topology or rich geometry where our results can apply.

In this paper, we prove a new continuation theorem for the solvability of periodic boundary value problems for nonlinear vector equations. By applying the continuation theorem, we prove the existence of a periodic solution for a class of semi-linear weekly-coupled systems with time-dependent potential.

This article is about the weak 16th Hilbert problem, i.e. we analyze how many limit cycles can bifurcate from the periodic orbits of a given polynomial differential center when it is perturbed inside a class of polynomial differential systems. More precisely, we consider the uniform isochronous centers x ˙ = - y + x 2 ⁢ y ⁢ ( x 2 + y 2 ) n , y ˙ = x + x ⁢ y 2 ⁢ ( x 2 + y 2 ) n , $\dot{x}=-y+x^{2}y(x^{2}+y^{2})^{n},\quad\dot{y}=x+xy^{2}(x^{2}+y^{2})^{n},$ of degree 2⁢n+3${2n+3}$ and we perturb them inside the class of all polynomial differential systems of degree 2⁢n+3${2n+3}$. For n=0,1${n=0,1}$ we provide the maximum number of limit cycles, 3 and 8 respectively, that can bifurcate from the periodic orbits of these centers using averaging theory of first order, or equivalently Abelian integrals. For n=2${n=2}$ we show that at least 12 limit cycles can bifurcate from the periodic orbits of the center.

We are concerned with the class of equations with exponential nonlinearities - Δ ⁢ u = ρ 1 ⁢ ( h ⁢ e u ∫ Σ h ⁢ e u ⁢ 𝑑 V g - 1 | Σ | ) - ρ 2 ⁢ ( h ⁢ e - u ∫ Σ h ⁢ e - u ⁢ 𝑑 V g - 1 | Σ | ) $-\Delta u=\rho_{1}\Biggl{(}\frac{he^{u}}{\int_{\Sigma}he^{u}\,dV_{g}}-\frac{1}% {|\Sigma|}\Biggr{)}-\rho_{2}\Biggl{(}\frac{he^{-u}}{\int_{\Sigma}he^{-u}\,dV_{% g}}-\frac{1}{|\Sigma|}\Biggr{)}$ on a compact surface Σ, which describes the mean field equation of equilibrium turbulence with arbitrarily signed vortices. Here, h is a smooth positive function and ρ1,ρ2${\rho_{1},\rho_{2}}$ are two positive parameters. We provide the first multiplicity result for this class of equations by using Morse theory.

In this paper, we study the nonlinear fractional Schrödinger system in ℝN${\mathbb{R}^{N}}$. Applying the finite reduction method, we prove that the nonlinear fractional Schrödinger system has multi-peak positive solutions under some suitable conditions.

In this paper, we consider the dynamics for damped generalized incompressible Navier–Stokes equations defined on ℝ2${\mathbb{R}^{2}}$. The generalized Navier–Stokes equations here refer to the equations obtained by replacing the Laplacian in the classical Navier–Stokes equations by the more general operator (-Δ)α${(-\Delta)^{\alpha}}$ with α∈(12,1)${\alpha\in(\frac{1}{2},1)}$. We prove that the rate of dissipation of enstrophy vanishes as ν→0+${\nu\to 0^{+}}$, where ν is the viscosity parameter. Moreover, we prove the existence and finite dimensionality of a global attractor in (H1⁢(ℝ2))2${(H^{1}(\mathbb{R}^{2}))^{2}}$ as ν>0${\nu>0}$ is kept fixed for the generalized Navier–Stokes equations.

We show the existence and multiplicity of radial solutions for the problems with minimum and maximum involving mean curvature operators in the Minkowski space: { div ⁡ ( ϕ N ⁢ ( ∇ ⁡ v ) ) = F ⁢ ( v ) ⁢ ( | x | )   for a.e. ⁢ R 1 < | x | < R 2 , x ∈ ℝ N , N ≥ 2 , min ⁡ { v ⁢ ( x ) ∣ R 1 ≤ | x | ≤ R 2 } = A , max ⁡ { v ⁢ ( x ) ∣ R 1 ≤ | x | ≤ R 2 } = B , $\left\{\begin{aligned} &\displaystyle\operatorname{div}(\phi_{N}(\nabla v))=F(% v)(\lvert x\rvert)\quad\text{for a.e. }R_{1}<\lvert x\rvert<R_{2},\,x\in% \mathbb{R}^{N},\,N\geq 2,\\ &\displaystyle\min\bigl{\{}v(x)\mid R_{1}\leq\lvert x\rvert\leq R_{2}\bigr{\}}% =A,\quad\max\bigl{\{}v(x)\mid R_{1}\leq\lvert x\rvert\leq R_{2}\bigr{\}}=B,% \end{aligned}\right.$ where ϕN⁢(z)=z/1-|z|2${\phi_{N}(z)=z/\sqrt{1-\lvert z\rvert^{2}}}$, z∈ℝN${z\in\mathbb{R}^{N}}$, R1,R2,A,B∈ℝ${R_{1},R_{2},A,B\in\mathbb{R}}$ are constants satisfying 1<R1<R2-1${1<R_{1}<R_{2}-1}$ and A<B${A<B}$; |⋅|${\lvert\cdot\rvert}$ denotes the Euclidean norm in ℝN${\mathbb{R}^{N}}$, and F:C1⁢[R1,R2]→L1⁢[R1,R2]${F:C^{1}[R_{1},R_{2}]\to L^{1}[R_{1},R_{2}]}$ is an unbounded operator. By using the Leray–Schauder degree theory and the Borsuk theorem, we prove that the problem has at least two different radial solutions.

We consider a nonlinear Robin problem driven by a nonhomogeneous differential operator which incorporates the p -Laplacian as special case. The reaction f⁢(z,x)${f(z,x)}$ is a Carathéodory function which need not satisfy a global growth condition and is only assumed to be odd near zero. Using variational tools, we show that the problem has a whole sequence of distinct nodal (that is, sign-changing) solutions.

A classical model is proposed in which two nonlinear Klein–Gordon fields interact via the electromagnetic field. Scaling is such that solitons in the two fields can be interpreted as electrons and protons, respectively. Even though the masses are very different, the magnitude of the charge of the electron-like soliton is the same as that of the proton-like soliton. Attraction and repulsion occur in the desired way through the interaction with the electromagnetic field.

In this paper we study the dynamics of the third exotic contact form of Gonzalo and Varela [5] on S3${S^{3}}$ in the framework of contact form geometry. This dynamics has an impact on the variational problem associated to the contact form such as the violation of the Fredholm condition and the Morse indices of different periodic orbits.

The present paper studies the nonlinear elliptic equation { Δ ⁢ u + λ | x | s ⁢ u p - 1 + ∑ i = 1 l λ i | x | s i ⁢ u 2 ∗ ⁢ ( s i ) - 1 + u 2 ∗ - 1 = 0 in ⁢ Ω , u > 0 in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , $\left\{\begin{aligned} \displaystyle\Delta u+\frac{\lambda}{|x|^{s}}u^{p-1}+% \sum_{i=1}^{l}\frac{\lambda_{i}}{|x|^{s_{i}}}u^{2^{\ast}(s_{i})-1}+u^{2^{\ast}% -1}&\displaystyle=0&&\displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle>0&&\displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\end{% aligned}\right.$ which involves multiple Hardy–Sobolev critical exponents, where λ≠0${\lambda\neq 0}$, s∈[0,2)${s\in[0,2)}$, p∈(2,2∗⁢(s))${p\in(2,2^{\ast}(s))}$, 0<s1<⋯<sl<2$0<s_{1}<\cdots<s_{l}<2$, λ1,…,λk>0${\lambda_{1},\dots,\lambda_{k}>0}$, λk+1,…,λl<0${\lambda_{k+1},\dots,\lambda_{l}<0}$ for some k∈[1,l]${k\in[1,l]}$ and Ω is a C1$C^{1}$ open bounded domain in ℝN$\mathbb{R}^{N}$, N≥3${N\geq 3}$, containing the origin. The existence of a positive ground state solution is established when λ>0${\lambda>0}$ and p≥2∗⁢(sk)${p\geq 2^{\ast}(s_{k})}$.

We study a class of non-periodic damped systems, where the nonlinearity H is subquadratic at 0 and general at infinity (i.e., it can be subquadratic, asymptotically quadratic or superquadratic at infinity, three examples are constructed to illustrate the three different cases). Besides, the global evenness condition (H⁢(t,-u)=H⁢(t,u)${H(t,-u)=H(t,u)}$) and the global non-negative condition (H⁢(t,u)≥0${H(t,u)\geq 0}$) used before are, respectively, replaced by the local evenness condition (H⁢(t,-u)=H⁢(t,u)${H(t,-u)=H(t,u)}$, |u|≤r${|u|\leq r}$) and the local non-negative condition (H⁢(t,u)≥0${H(t,u)\geq 0}$, |u|≤r${|u|\leq r}$). Under the above weaker and more general conditions, we obtain infinitely many nontrivial small negative energy homoclinic orbits for this class of damped systems by an extension of Clark’s theorem.

Fix two differential operators L1${L_{1}}$ and L2${L_{2}}$, and define a sequence of functions inductively by considering u1${u_{1}}$ as the solution to the Dirichlet problem for an operator L1${L_{1}}$ and then un${u_{n}}$ as the solution to the obstacle problem for an operator Li${L_{i}}$ (i=1,2${i=1,2}$ alternating them) with obstacle given by the previous term un-1${u_{n-1}}$ in a domain Ω and a fixed boundary datum g on ∂⁡Ω${\partial\Omega}$. We show that in this way we obtain an increasing sequence that converges uniformly to a viscosity solution to the minimal operator associated with L1${L_{1}}$ and L2${L_{2}}$, that is, the limit u verifies min⁡{L1⁢u,L2⁢u}=0${\min\{L_{1}u,L_{2}u\}=0}$ in Ω with u=g${u=g}$ on ∂⁡Ω${\partial\Omega}$.

In this paper, motivated by recent works on the study of the equations which model electrostatic MEMS devices, we study the quasilinear elliptic equation (Pλ) { - ( r α ⁢ | u ′ | β ⁢ u ′ ) ′ = λ ⁢ r γ ⁢ f ⁢ ( r ) ( 1 - u ) 2 , r ∈ ( 0 , 1 ) , 0 ≤ u ⁢ ( r ) < 1 , r ∈ ( 0 , 1 ) , u ′ ⁢ ( 0 ) = u ⁢ ( 1 ) = 0 . ${}\begin{cases}-(r^{\alpha}|u^{\prime}|^{\beta}u^{\prime})^{\prime}=\dfrac{% \lambda r^{\gamma}f(r)}{(1-u)^{2}},&r\in(0,1),\\ 0\leq u(r)<1,&r\in(0,1),\\ u^{\prime}(0)=u(1)=0.&\end{cases}$ According to the choice of the parameters α, β, and γ, the differential operator which we are dealing with corresponds to the radial form of the Laplacian, the p -Laplacian, and the k -Hessian. We prove the existence of an extremal parameter λ∗>0${\lambda^{\ast}>0}$ such that, for λ∈(0,λ∗)${\lambda\in(0,\lambda^{\ast})}$, there exists a minimal solution u¯λ${\underline{u}_{\lambda}}$ and, for λ>λ∗${\lambda>\lambda^{\ast}}$, there exists no solution of any kind. We also study the behavior of the minimal branch of solutions and we prove uniqueness of solutions of (Pλ∗$P_{\lambda^{\ast}}$) for β>-1${\beta>-1}$.

We study the existence of ground states for the coupled Schrödinger system { - Δ ⁢ u i + λ i ⁢ u i = μ i ⁢ | u i | 2 ⁢ q - 2 ⁢ u i + ∑ j ≠ i b i ⁢ j ⁢ | u j | q ⁢ | u i | q - 2 ⁢ u i , u i ∈ H 1 ⁢ ( ℝ n ) , i = 1 , … , d , $\left\{\begin{aligned} &\displaystyle-\Delta u_{i}+\lambda_{i}u_{i}=\mu_{i}|u_% {i}|^{2q-2}u_{i}+\sum_{j\neq i}b_{ij}|u_{j}|^{q}|u_{i}|^{q-2}u_{i},\\ &\displaystyle u_{i}\in H^{1}(\mathbb{R}^{n}),\quad i=1,\ldots,d,\end{aligned}\right.$ n≥1${n\geq 1}$, for λi,μi>0${\lambda_{i},\mu_{i}>0}$, bi⁢j=bj⁢i>0${b_{ij}=b_{ji}>0}$ (the so-called “symmetric attractive case”) and 1<q<n/(n-2)+${1<q<n/(n-2)^{+}}$. We prove the existence of a nonnegative ground state (u1*,…,ud*)${(u_{1}^{*},\ldots,u_{d}^{*})}$ with ui*${u_{i}^{*}}$ radially decreasing. Moreover, we show that if in addition q<2${q<2}$, such ground states are positive in all dimensions and for all values of the parameters.