Band 18, Heft 4

We study an optimization problem with SPDE constraints, which has the peculiarity that the control parameter s is the s -th power of the diffusion operator in the state equation. Well-posedness of the state equation and differentiability properties with respect to the fractional parameter s are established. We show that under certain conditions on the noise, optimality conditions for the control problem can be established.

We consider the Hardy–Schrödinger operator Lγ:=-Δ𝔹n-γ⁢V2{L_{\gamma}:=-\Delta_{\mathbb{B}^{n}}-\gamma{V_{2}}} on the Poincaré ball model of the hyperbolic space 𝔹n{\mathbb{B}^{n}} (n≥3{n\geq 3}). Here V2{V_{2}} is a radially symmetric potential, which behaves like the Hardy potential around its singularity at 0, i.e., V2⁢(r)∼1r2{V_{2}(r)\sim\frac{1}{r^{2}}}. As in the Euclidean setting, Lγ{L_{\gamma}} is positive definite whenever γ<(n-2)24{\gamma<\frac{(n-2)^{2}}{4}}, in which case we exhibit explicit solutions for the critical equation Lγ⁢u=V2*⁢(s)⁢u2*⁢(s)-1{L_{\gamma}u=V_{2^{*}(s)}u^{2^{*}(s)-1}} in 𝔹n,{\mathbb{B}^{n},} where 0≤s<2{0\leq s<2}, 2*⁢(s)=2⁢(n-s)n-2{2^{*}(s)=\frac{2(n-s)}{n-2}}, and V2*⁢(s){V_{2^{*}(s)}} is a weight that behaves like 1rs{\frac{1}{r^{s}}} around 0. In dimensions n≥5{n\geq 5}, the equation Lγ⁢u-λ⁢u=V2*⁢(s)⁢u2*⁢(s)-1{L_{\gamma}u-\lambda u=V_{2^{*}(s)}u^{2^{*}(s)-1}} in a domain Ω of 𝔹n{\mathbb{B}^{n}} away from the boundary but containing 0 has a ground state solution, whenever 0<γ≤n⁢(n-4)4{0<\gamma\leq\frac{n(n-4)}{4}}, and λ>n-2n-4⁢(n⁢(n-4)4-γ){\lambda>\frac{n-2}{n-4}(\frac{n(n-4)}{4}-\gamma)}. On the other hand, in dimensions 3 and 4, the existence of solutions depends on whether the domain has a positive “hyperbolic mass” a notion that we introduce and analyze therein.

In the following, we show the strong comparison principle for the fractional p -Laplacian, i.e. we analyze { ( - Δ ) p s ⁢ v + q ⁢ ( x ) ⁢ | v | p - 2 ⁢ v ≥ 0 in  D , ( - Δ ) p s ⁢ w + q ⁢ ( x ) ⁢ | w | p - 2 ⁢ w ≤ 0 in  D , v ≥ w in  ℝ N , \quad\left\{\begin{aligned} \displaystyle(-\Delta)^{s}_{p}v+q(x)\lvert v\rvert% ^{p-2}v&\displaystyle\geq 0&&\displaystyle\phantom{}\text{in ${D}$},\\ \displaystyle(-\Delta)^{s}_{p}w+q(x)\lvert w\rvert^{p-2}w&\displaystyle\leq 0&% &\displaystyle\phantom{}\text{in ${D}$},\\ \displaystyle v&\displaystyle\geq w&&\displaystyle\phantom{}\text{in ${\mathbb% {R}^{N}}$},\end{aligned}\right. where s∈(0,1){s\in(0,1)}, p>1{p>1}, D⊂ℝN{D\subset\mathbb{R}^{N}} is an open set, and q∈L∞⁢(ℝN){q\in L^{\infty}(\mathbb{R}^{N})} is a nonnegative function. Under suitable conditions on s , p and some regularity assumptions on v , w , we show that either v≡w{v\equiv w} in ℝN{\mathbb{R}^{N}} or v>w{v>w} in D . Moreover, we apply this result to analyze the geometry of nonnegative solutions in starshaped rings and in the half space.

In this paper, we give a unitary approach for the simultaneous study of the convergence of discrete and integral operators described by means of a family of linear continuous functionals acting on functions defined on locally compact Hausdorff topological groups. The general family of operators introduced and studied includes very well-known operators in the literature. We give results of uniform convergence and modular convergence in the general setting of Orlicz spaces. The latter result allows us to cover many other settings as the Lp{L^{p}}-spaces, the interpolation spaces, the exponential spaces and many others.

In this paper, we are concerned with the existence and asymptotic behavior of minimizers of a minimization problem related to some quasilinear elliptic equations. Firstly, we prove that there exist minimizers when the exponent q is the critical one q*=2+4N{q^{*}=2+\frac{4}{N}}. Then, we prove that all minimizers are compact as q tends to the critical case q*{q^{*}} when a<aq*{a<a_{q^{*}}} is fixed. Moreover, we find that all the minimizers must blow up as the exponent q tends to the critical case q*{q^{*}} for any fixed a>aq*{a>a_{q^{*}}}.

We establish the existence of a positive solution to the problem - Δ ⁢ u + V ⁢ ( x ) ⁢ u = f ⁢ ( u ) , u ∈ D 1 , 2 ⁢ ( ℝ N ) , -\Delta u+V(x)u=f(u),\quad u\in D^{1,2}(\mathbb{R}^{N}), for N≥3{N\geq 3}, when the nonlinearity f is subcritical at infinity and supercritical near the origin, and the potential V vanishes at infinity. Our result includes situations in which the problem does not have a ground state. Then, under a suitable decay assumption on the potential, we show that the problem has a positive bound state.

Let n≥2{n\geq 2} be an integer, P=diag⁢(-In-κ,Iκ,-In-κ,Iκ){P=\mathrm{diag}(-I_{n-\kappa},I_{\kappa},-I_{n-\kappa},I_{\kappa})} for some integer κ∈[0,n]{\kappa\in[0,n]}, and let Σ⊂ℝ2⁢n{\Sigma\subset{\mathbb{R}}^{2n}} be a partially symmetric compact convex hypersurface, i.e., x∈Σ{x\in\Sigma} implies P⁢x∈Σ{Px\in\Sigma}, and (r,R){(r,R)}-pinched. In this paper, we prove that when R/r<5/3{{R/r}<\sqrt{5/3}} and 0≤κ≤[n-12]{0\leq\kappa\leq[\frac{n-1}{2}]}, there exist at least E⁢(n-2⁢κ-12)+E⁢(n-2⁢κ-13){E(\frac{n-2\kappa-1}{2})+E(\frac{n-2\kappa-1}{3})} non-hyperbolic P -invariant closed characteristics on Σ. In addition, when R/r<3/2{{R/r}<\sqrt{3/2}}, [n+12]≤κ≤n{[\frac{n+1}{2}]\leq\kappa\leq n} and Σ carries exactly n P -invariant closed characteristics, then there exist at least 2⁢E⁢(2⁢κ-n-14)+E⁢(n-κ-13){2E(\frac{2\kappa-n-1}{4})+E(\frac{n-\kappa-1}{3})} non-hyperbolic P -invariant closed characteristics on Σ, where the function E⁢(a){E(a)} is defined as E⁢(a)=min⁡{k∈ℤ∣k≥a}{E(a)=\min{\{k\in{\mathbb{Z}}\mid k\geq a\}}} for any a∈ℝ{a\in\mathbb{R}}.

In this paper, we study the regularizing effect of lower order terms in elliptic problems involving a Hardy potential. Concretely, our model problem is the differential equation - Δ ⁢ u + h ⁢ ( x ) ⁢ | u | p - 1 ⁢ u = λ ⁢ u | x | 2 + f ⁢ ( x )   in  ⁢ Ω , -\Delta u+h(x)|u|^{p-1}u=\lambda\frac{u}{|x|^{2}}+f(x)\quad\text{in }\Omega, with Dirichlet boundary condition on ∂⁡Ω{\partial\Omega}, where p>1{p>1} and f∈Lhm⁢(Ω){f\in L^{m}_{h}(\Omega)} (i.e. |f|m⁢h∈L1⁢(Ω){|f|^{m}h\in L^{1}(\Omega)}) with m≥p+1p{m\geq\frac{p+1}{p}}. We prove that there is a solution of the above problem even for λ greater than the Hardy constant; i.e., λ≥ℋ=(N-2)24{\lambda\geq\mathcal{H}=\frac{(N-2)^{2}}{4}} and nonnegative functions h∈L1⁢(Ω){h\in L^{1}(\Omega)} which could vanish in a subset of Ω. Moreover, we show that all the solutions are in Lhp⁢m⁢(Ω){L^{pm}_{h}(\Omega)}. These results improve and generalize the case h⁢(x)≡h0{h(x)\equiv h_{0}} treated in [2, 10].

When 2⁢N/(N+1)<p<2{2N/(N+1)<p<2} and 0<q<p/2{0<q<p/2}, non-negative solutions to the singular diffusion equation with gradient absorption ∂ t ⁡ u - Δ p ⁢ u + | ∇ ⁡ u | q = 0   in  ⁢ ( 0 , ∞ ) × ℝ N \partial_{t}u-\Delta_{p}u+|\nabla u|^{q}=0\quad\text{in }(0,\infty)\times% \mathbb{R}^{N} vanish after a finite time. This phenomenon is usually referred to as finite-time extinction and takes place provided the initial condition u0{u_{0}} decays sufficiently rapidly as |x|→∞{|x|\to\infty}. On the one hand, the optimal decay of u0{u_{0}} at infinity guaranteeing the occurrence of finite-time extinction is identified. On the other hand, assuming further that p-1<q<p/2{p-1<q<p/2}, optimal extinction rates near the extinction time are derived.

In this paper, we investigate the existence of multiple solutions for the following two fractional problems: { ( - Δ Ω ) s ⁢ u - λ ⁢ u = f ⁢ ( x , u ) in  ⁢ Ω , u = 0 in  ⁢ ∂ ⁡ Ω    and    { ( - Δ ℝ N ) s ⁢ u - λ ⁢ u = f ⁢ ( x , u ) in  ⁢ Ω , u = 0 in  ⁢ ℝ N ∖ Ω , \left\{\begin{aligned} \displaystyle(-\Delta_{\Omega})^{s}u-\lambda u&% \displaystyle=f(x,u)&&\displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{in }\partial\Omega\end{% aligned}\right.\qquad\text{and}\qquad\left\{\begin{aligned} \displaystyle(-% \Delta_{\mathbb{R}^{N}})^{s}u-\lambda u&\displaystyle=f(x,u)&&\displaystyle% \text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{in }\mathbb{R}^{N}% \setminus\Omega,\end{aligned}\right. where s∈(0,1){s\in(0,1)}, N>2⁢s{N>2s}, Ω is a smooth bounded domain of ℝN{\mathbb{R}^{N}}, and f:Ω¯×ℝ→ℝ{f:\bar{\Omega}\times\mathbb{R}\to\mathbb{R}} is a superlinear continuous function which does not satisfy the well-known Ambrosetti–Rabinowitz condition. Here (-ΔΩ)s{(-\Delta_{\Omega})^{s}} is the spectral Laplacian and (-ΔℝN)s{(-\Delta_{\mathbb{R}^{N}})^{s}} is the fractional Laplacian in ℝN{\mathbb{R}^{N}}. By applying variational theorems of mixed type due to Marino and Saccon and the Linking Theorem, we prove the existence of multiple solutions for the above problems.

By introducing simple topological constraints and applying a binary decomposition method, we show the existence of a set of prograde double-double orbits for any rotation angle θ∈(0,π/7]{\theta\in(0,\pi/7]} in the equal-mass four-body problem. A new geometric argument is introduced to show that for any θ∈(0,π/2){\theta\in(0,\pi/2)}, the action of the minimizer corresponding to the prograde double-double orbit is strictly greater than the action of the minimizer corresponding to the retrograde double-double orbit. This geometric argument can also be applied to study orbits in the planar three-body problem, such as the retrograde orbits, the prograde orbits, the Schubart orbit and the Hénon orbit.

The purpose of this paper is to study properties of continua (closed connected sets) of nontrivial solutions of non-cooperative elliptic systems considered on geodesic balls in Sn{S^{n}}. In particular, we show that if the geodesic ball is a hemisphere, then all these continua are unbounded. It is also shown that the phenomenon of global symmetry-breaking bifurcation of such solutions occurs. Since the problem is variational and SO⁡(n){\operatorname{SO}(n)}-symmetric, we apply the techniques of equivariant bifurcation theory to prove the main results of this article. As the topological tool, we use the degree theory for SO⁡(n){\operatorname{SO}(n)}-invariant strongly indefinite functionals defined in [A. Gołȩbiewska and S. A. Rybicki, Global bifurcations of critical orbits of G -invariant strongly indefinite functionals, Nonlinear Anal. 74 2011, 5, 1823–1834].