Band 19, Heft 4

We prove that any sufficiently differentiable space-like hypersurface of ℝ1+N{{\mathbb{R}}^{1+N}} coincides locally around any of its points with the blow-up surface of a finite-energy solution of the focusing nonlinear wave equation ∂t⁢t⁡u-Δ⁢u=|u|p-1⁢u{\partial_{tt}u-\Delta u=|u|^{p-1}u} on ℝ×ℝN{{\mathbb{R}}\times{\mathbb{R}}^{N}}, for any 1≤N≤4{1\leq N\leq 4} and 1<p≤N+2N-2{1<p\leq\frac{N+2}{N-2}}. We follow the strategy developed in our previous work (2018) on the construction of solutions of the nonlinear wave equation blowing up at any prescribed compact set. Here to prove blow-up on a local space-like hypersurface, we first apply a change of variable to reduce the problem to blowup on a small ball at t=0{t=0} for a transformed equation. The construction of an appropriate approximate solution is then combined with an energy method for the existence of a solution of the transformed problem that blows up at t=0{t=0}. To obtain a finite-energy solution of the original problem from trace arguments, we need to work with H2×H1{H^{2}\times H^{1}} solutions for the transformed problem.

This paper deals with the general Choquard equation - Δ ⁢ u + V ⁢ ( | x | ) ⁢ u = ( I α * | u | p ) ⁢ | u | p - 2 ⁢ u   in  ⁢ ℝ N , -\Delta u+V(|x|)u=(I_{\alpha}*|u|^{p})|u|^{p-2}u\quad\text{in }\mathbb{R}^{N}, where V∈C⁢([0,∞),ℝ+){V\in C([0,\infty),\mathbb{R}^{+})} is bounded below by a positive constant, and Iα{I_{\alpha}} denotes the Riesz potential of order α∈(0,N){\alpha\in(0,N)}. In view of the convolution term, the nonlocal property makes the variational functional completely different from the one for local pure power-type nonlinearity. By combining the Brouwer degree and developing some new techniques, a family of radial solutions with a prescribed number of zeros is constructed for p∈[2,N+αN-2){p\in[2,\frac{N+\alpha}{N-2})}, while the degeneracy happens for p∈(N+αN,2){p\in(\frac{N+\alpha}{N},2)}. This result complements and improves the ones in the literature in the aspect of the range of p .

We study a Schrödinger system of four equations with linear coupling functions and nonlinear couplings, including the case that the corresponding elliptic operators are indefinite. For any given nonlinear coupling β>0{\beta>0}, we first use minimizing sequences on a normalized set to obtain a minimizer, which implies the existence of positive solutions for some linear coupling constants μβ,νβ{\mu_{\beta},\nu_{\beta}} by Lagrange multiplier rules. Then, as β→∞{\beta\to\infty}, we prove that the limit configurations to the competing system are segregated in two groups, develop a variant of Almgren’s monotonicity formula to reveal the Lipschitz continuity of the limit profiles and establish a kind of local Pohozaev identity to obtain the extremality conditions. Finally, we study the relation between the limit profiles and the optimal partition for principal eigenvalue of the elliptic system and obtain an optimal partition for principal eigenvalues of elliptic systems.

The aim of this article is to provide a functional analytical framework for defining the fractional powers As{A^{s}} for -1<s<1{-1<s<1} of maximal monotone (possibly multivalued and nonlinear) operators A in Hilbert spaces. We investigate the semigroup {e-As⁢t}t≥0{\{e^{-A^{s}t}\}_{t\geq 0}} generated by -As{-A^{s}}, prove comparison principles and interpolations properties of {e-As⁢t}t≥0{\{e^{-A^{s}t}\}_{t\geq 0}} in Lebesgue and Orlicz spaces. We give sufficient conditions implying that As{A^{s}} has a sub-differential structure. These results extend earlier ones obtained in the case s=1/2{s=1/2} for maximal monotone operators [H. Brézis, Équations d’évolution du second ordre associées à des opérateurs monotones, Israel J. Math. 12 1972, 51–60], [V. Barbu, A class of boundary problems for second order abstract differential equations, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 19 1972, 295–319], [V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International, Leiden, 1976], [E. I. Poffald and S. Reich, An incomplete Cauchy problem, J. Math. Anal. Appl. 113 1986, 2, 514–543], and the recent advances for linear operators A obtained in [L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 2007, 7–9, 1245–1260], [P. R. Stinga and J. L. Torrea, Extension problem and Harnack’s inequality for some fractional operators, Comm. Partial Differential Equations 35 2010, 11, 2092–2122].

We consider the Hénon problem { - Δ ⁢ u = | x | α ⁢ u N + 2 + 2 ⁢ α N - 2 - ε in  ⁢ B 1 , u > 0 in  ⁢ B 1 , u = 0 on  ⁢ ∂ ⁡ B 1 , \left\{\begin{aligned} &\displaystyle{-}\Delta u=\lvert x\rvert^{\alpha}u^{% \frac{N+2+2\alpha}{N-2}-\varepsilon}&&\displaystyle\phantom{}\text{in }B_{1},% \\ &\displaystyle u>0&&\displaystyle\phantom{}\text{in }B_{1},\\ &\displaystyle u=0&&\displaystyle\phantom{}\text{on }\partial B_{1},\end{% aligned}\right. where B1{B_{1}} is the unit ball in ℝN{\mathbb{R}^{N}} and N⩾3{N\geqslant 3}. For ε>0{\varepsilon>0} small enough, we use α as a parameter and prove the existence of a branch of nonradial solutions that bifurcates from the radial one when α is close to an even positive integer.

In this paper, we establish an ω+{\omega^{+}}-type index theory for paths in the general linear group GL+⁢(2){\mathrm{GL}^{+}(2)}. This is done by the complete homotopy classification for such paths. We also compare this index theory with the ω index theory for paths in the symplectic group Sp⁢(2){\mathrm{Sp}(2)} and obtain a generalization of Bott formula for iterated paths in GL+⁢(2){\mathrm{GL}^{+}(2)}. As applications, the minimal periodic solution problem and the linear stability of general differential systems are studied.

In this paper, we consider the Kirchhoff equation with Hartree-type nonlinearity { - ( ε 2 ⁢ a + ε ⁢ b ⁢ ∫ ℝ 3 | ∇ ⁡ u | 2 ⁢ d ⁢ x ) ⁢ Δ ⁢ u + V ⁢ ( x ) ⁢ u = ε μ - 3 ⁢ ( ∫ ℝ 3 K ⁢ ( y ) ⁢ F ⁢ ( u ⁢ ( y ) ) | x - y | μ ⁢ d ⁢ y ) ⁢ K ⁢ ( x ) ⁢ f ⁢ ( u ) , u ∈ H 1 ⁢ ( ℝ 3 ) , \left\{\begin{aligned} \displaystyle-&\displaystyle\biggl{(}\varepsilon^{2}a+% \varepsilon b\int_{\mathbb{R}^{3}}\lvert\nabla u\rvert^{2}\mathop{}\!dx\biggr{% )}\Delta u+V(x)u=\varepsilon^{\mu-3}\biggl{(}\int_{\mathbb{R}^{3}}\frac{K(y)F(% u(y))}{\lvert x-y\rvert^{\mu}}\mathop{}\!dy\biggr{)}K(x)f(u),\\ &\displaystyle u\in H^{1}(\mathbb{R}^{3}),\end{aligned}\right. where ε>0{\varepsilon>0} is a small parameter, a,b>0{a,b>0}, μ∈(0,3){\mu\in(0,3)}, V,K{V,K} are two positive continuous function and F is the primitive function of f which is superlinear but subcritical at infinity in the sense of the Hardy–Littlewood–Sobolev inequality. We show that the equation admits a positive ground state solution for ε>0{\varepsilon>0} sufficiently small. Furthermore, we prove that these ground state solutions concentrate around such points which are both the minima points of the potential V and the maximum points of the potential K as ε→0{\varepsilon\to 0}.

We study a concept of renormalized solution to the problem { - Δ p ⁢ u = 0 in  ⁢ ℝ + N , | ∇ ⁡ u | p - 2 ⁢ u ν + g ⁢ ( u ) = μ on  ⁢ ∂ ⁡ ℝ + N , \begin{cases}-\Delta_{p}u=0&\mbox{in }{\mathbb{R}}^{N}_{+},\\ \lvert\nabla u\rvert^{p-2}u_{\nu}+g(u)=\mu&\mbox{on }\partial{\mathbb{R}}^{N}_% {+},\end{cases} where 1<p≤N{1<p\leq N}, N≥2{N\geq 2}, ℝ+N={(x′,xN):x′∈ℝN-1,xN>0}{{\mathbb{R}}^{N}_{+}=\{(x^{\prime},x_{N}):x^{\prime}\in{\mathbb{R}}^{N-1},\,x% _{N}>0\}}, uν{u_{\nu}} is the normal derivative of u , μ is a bounded Radon measure, and g:ℝ→ℝ{g:{\mathbb{R}}\rightarrow{\mathbb{R}}} is a continuous function. We prove stability results and, using the symmetry of the domain, apriori estimates on hyperplanes, and potential methods, we obtain several existence results. In particular, we show existence of solutions for problems with nonlinear terms of absorption type in both the subcritical and supercritical case. For the problem with source we study the power nonlinearity g⁢(u)=-uq{g(u)=-u^{q}}, showing existence in the supercritical case, and nonexistence in the subcritical one. We also give a characterization of removable sets when μ≡0{\mu\equiv 0} and g⁢(u)=-uq{g(u)=-u^{q}} in the supercritical case.