Band 18, Heft 2

We present existence and multiplicity theorems for periodic mild solutions to parabolic evolution equations. Their peculiarity is a link with the spectrum of the generator of the semigroup rather than with the spectrum of the linearized periodic BVP for the evolution equation. They provide a positive solution to the open problem risen by Becker [3], they extend some results of Castro and Lazer [5] from scalar to systems of parabolic equations, and they are new even for finite-dimensional ODEs.

We consider a parameter dependent parabolic logistic population model with diffusion and degenerate logistic term allowing for refuges for the population. The aim of the paper is to remove quite restrictive geometric and smoothness conditions on the refuge used in the existing literature. The key is a new simplified construction of a supersolution that does not make use of any regularity condition of the refuge. At the same time we also simplify other arguments commonly used in the literature.

We prove the existence of a solution of (-Δ)s⁢u+f⁢(u)=0{(-\Delta)^{s}u+f(u)=0} in a smooth bounded domain Ω with a prescribed boundary value μ in the class of Radon measures for a large class of continuous functions f satisfying a weak singularity condition expressed under an integral form. We study the existence of a boundary trace for positive moderate solutions. In the particular case where f⁢(u)=up{f(u)=u^{p}} and μ is a Dirac mass, we show the existence of several critical exponents p . We also demonstrate the existence of several types of separable solutions of the equation (-Δ)s⁢u+up=0{(-\Delta)^{s}u+u^{p}=0} in ℝ+N{\mathbb{R}^{N}_{+}}.

In this paper, we consider the following parabolic variational inequality containing a multivalued term and a convex functional: Find u∈Lp⁢(0,T;W01,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)=u0{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)=∫0Tψ⁢(t,u)⁢𝑑t{\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 .

This paper is concerned with the boundary behavior of the unique convex solution to a singular Dirichlet problem for the Monge–Ampère equation det ⁡ D 2 ⁢ u = b ⁢ ( x ) ⁢ g ⁢ ( - u ) , u < 0 , x ∈ Ω , u | ∂ ⁡ Ω = 0 , \operatorname{det}D^{2}u=b(x)g(-u),\quad u<0,\,x\in\Omega,\qquad u|_{\partial% \Omega}=0, where Ω is a strictly convex and bounded smooth domain in ℝN{\mathbb{R}^{N}}, with N≥2{N\geq 2}, g∈C1⁢((0,∞),(0,∞)){g\in C^{1}((0,\infty),(0,\infty))} is decreasing in (0,∞){(0,\infty)} and satisfies lims→0+⁡g⁢(s)=∞{\lim_{s\rightarrow 0^{+}}g(s)=\infty}, and b∈C∞⁢(Ω){b\in C^{\infty}(\Omega)} is positive in Ω, but may vanish or blow up on the boundary. We find a new structure condition on g which plays a crucial role in the boundary behavior of such solution.

In the present paper, we study the p -Laplace equation in a hollow symmetric bounded domain. Let H and G be closed subgroups of the orthogonal group such that H⊊G{H\varsubsetneq G}. Then we prove the existence of a positive solution which is H -invariant and G -non-invariant. Furthermore, we give several examples of H , G and Ω, and find symmetric and asymmetric solutions.

In this paper, we study an optimal shape design problem for the first eigenvalue of the fractional p -Laplacian with mixed boundary conditions. The optimization variable is the set where the Dirichlet condition is imposed (that is restricted to have measure equal to a prescribed quantity α). We show existence of an optimal design and analyze the asymptotic behavior when the fractional parameter s converges to 1, and thus obtain asymptotic bounds that are independent of α.

We consider the Cauchy problem for a parabolic-elliptic system in ℝ2{\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.

We study the existence of measure-valued solutions for a class of degenerate elliptic equations with measure data. The notion of solution is natural, since it is obtained by a regularization procedure which also relies on a standard approximation of the datum μ. We provide partial uniqueness results and qualitative properties of the constructed solutions concerning, in particular, the structure of their diffuse part with respect to the harmonic-capacity.

We study the following fractional Yamabe-type equation: { A s ⁢ u = u n + 2 ⁢ s n - 2 ⁢ s , u > 0 in  ⁢ Ω , u = 0 on  ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle A_{s}u&\displaystyle=u^{\frac{n+2s}{n-2s}% },\\ \displaystyle u&\displaystyle>0&&\displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\end{% aligned}\right. Here Ω is a regular bounded domain of ℝn{\mathbb{R}^{n}}, n≥2{n\geq 2}, and As{A_{s}}, s∈(0,1){s\in(0,1)}, represents the fractional Laplacian operator (-Δ)s{(-\Delta)^{s}} in Ω with zero Dirichlet boundary condition. We investigate the effect of the topology of Ω on the existence of solutions. Our result can be seen as the fractional counterpart of the Bahri–Coron theorem [3].

In this paper, we prove the existence of radial solutions for the nonlinear elliptic problem - div ⁢ ( A ⁢ ( | x | ) ⁢ ∇ ⁡ u ) + V ⁢ ( | x | ) ⁢ u = K ⁢ ( | x | ) ⁢ f ⁢ ( u )   in  ⁢ ℝ N , -\mathrm{div}(A(\lvert x\rvert)\nabla u)+V(\lvert x\rvert)u=K(\lvert x\rvert)f% (u)\quad\text{in }\mathbb{R}^{N}, with suitable hypotheses on the radial potentials A , V , K . We first get compact embeddings of radial weighted Sobolev spaces into sums of weighted Lebesgue spaces, and then we apply standard variational techniques to get existence results.