Band 17, Heft 4

We consider the critical p -Laplacian system { - Δ p ⁢ u - λ ⁢ a p ⁢ | u | a - 2 ⁢ u ⁢ | v | b = μ 1 ⁢ | u | p ∗ - 2 ⁢ u + α ⁢ γ p ∗ ⁢ | u | α - 2 ⁢ u ⁢ | v | β , x ∈ Ω , - Δ p ⁢ v - λ ⁢ b p ⁢ | u | a ⁢ | v | b - 2 ⁢ v = μ 2 ⁢ | v | p ∗ - 2 ⁢ v + β ⁢ γ p ∗ ⁢ | u | α ⁢ | v | β - 2 ⁢ v , x ∈ Ω , u , v ⁢  in  ⁢ D 0 1 , p ⁢ ( Ω ) , \left\{\begin{aligned} &\displaystyle{-}\Delta_{p}u-\frac{\lambda a}{p}\lvert u% \rvert^{a-2}u\lvert v\rvert^{b}=\mu_{1}\lvert u\rvert^{p^{\ast}-2}u+\frac{% \alpha\gamma}{p^{\ast}}\lvert u\rvert^{\alpha-2}u\lvert v\rvert^{\beta},&&% \displaystyle x\in\Omega,\\ &\displaystyle{-}\Delta_{p}v-\frac{\lambda b}{p}\lvert u\rvert^{a}\lvert v% \rvert^{b-2}v=\mu_{2}\lvert v\rvert^{p^{\ast}-2}v+\frac{\beta\gamma}{p^{\ast}}% \lvert u\rvert^{\alpha}\lvert v\rvert^{\beta-2}v,&&\displaystyle x\in\Omega,\\ &\displaystyle u,v\text{ in }D_{0}^{1,p}(\Omega),\end{aligned}\right. where Δp⁢u:=div⁡(|∇⁡u|p-2⁢∇⁡u){\Delta_{p}u:=\operatorname{div}(\lvert\nabla u\rvert^{p-2}\nabla u)} is the p -Laplacian operator defined on D 1 , p ⁢ ( ℝ N ) := { u ∈ L p ∗ ⁢ ( ℝ N ) : | ∇ ⁡ u | ∈ L p ⁢ ( ℝ N ) } , D^{1,p}(\mathbb{R}^{N}):=\bigl{\{}u\in L^{p^{\ast}}(\mathbb{R}^{N}):\lvert% \nabla u\rvert\in L^{p}(\mathbb{R}^{N})\bigr{\}}, endowed with the norm ∥u∥D1,p:=(∫ℝN|∇⁡u|p⁢𝑑x)1p{{\lVert u\rVert_{D^{1,p}}:=(\int_{\mathbb{R}^{N}}\lvert\nabla u\rvert^{p}\,dx% )^{\frac{1}{p}}}}, N≥3{N\geq 3}, 1<p<N{1<p<N}, λ,μ1,μ2≥0{\lambda,\mu_{1},\mu_{2}\geq 0}, γ≠0{\gamma\neq 0}, a,b,α,β>1{a,b,\alpha,\beta>1} satisfy a+b=p{a+b=p}, α+β=p∗:=N⁢pN-p{\alpha+\beta=p^{\ast}:=\frac{Np}{N-p}}, the critical Sobolev exponent, Ω is ℝN{\mathbb{R}^{N}} or a bounded domain in ℝN{\mathbb{R}^{N}} and D01,p⁢(Ω){D_{0}^{1,p}(\Omega)} is the closure of C0∞⁢(Ω){C_{0}^{\infty}(\Omega)} in D1,p⁢(ℝN){D^{1,p}(\mathbb{R}^{N})}. Under suitable assumptions, we establish the existence and nonexistence of a positive least energy solution of this system. We also consider the existence and multiplicity of the nontrivial nonnegative solutions.

In this paper, we investigate a class of Kirchhoff type problems in ℝ3{\mathbb{R}^{3}} involving a critical nonlinearity, namely, - ( 1 + b ∫ ℝ 3 | ∇ u | 2 d x ) △ u = λ f ( x ) u + | u | 4 u , u ∈ D 1 , 2 ( ℝ 3 ) , -\biggl{(}1+b\int_{\mathbb{R}^{3}}\lvert\nabla u|^{2}\,dx\biggr{)}\triangle u=% \lambda f(x)u+|u|^{4}u,\quad u\in D^{1,2}(\mathbb{R}^{3}), where b>0{b>0}, λ>λ1{\lambda>\lambda_{1}} and λ1{\lambda_{1}} is the principal eigenvalue of -△⁢u=λ⁢f⁢(x)⁢u{-\triangle u=\lambda f(x)u}, u∈D1,2⁢(ℝ3){u\in D^{1,2}(\mathbb{R}^{3})}. We prove that there exists δ>0{\delta>0} such that the above problem has at least two positive solutions for λ1<λ<λ1+δ{\lambda_{1}<\lambda<\lambda_{1}+\delta}. Furthermore, we obtain the existence of ground state solutions. Our tools are the Nehari manifold and the concentration compactness principle. This paper can be regarded as an extension of Naimen’s work [21].

In this paper, we show the existence of nontrivial solutions for doubly critical nonlocal elliptic problems in ℝN{\mathbb{R}^{N}} involving fractional Laplacians. Our approach is elementary without the use of neither the concentration-compactness principle nor the extension argument developed in [4].

Let N≥2{N\geq 2} and 1<p<(N+2)/(N-2)+{1<p<(N+2)/(N-2)_{+}}. Consider the Lane–Emden equation Δ⁢u+up=0{\Delta u+u^{p}=0} in ℝN{\mathbb{R}^{N}} and recall the classical Liouville type theorem: if u is a non-negative classical solution of the Lane–Emden equation, then u≡0{u\equiv 0}. The method of moving planes combined with the Kelvin transform provides an elegant proof of this theorem. A classical approach of Serrin and Zou, originally used for the Lane–Emden system, yields another proof but only in lower dimensions. Motivated by this, we further refine this approach to find an alternative proof of the Liouville type theorem in all dimensions.

This work is concerned with constructing a finite dimensional form (named determining form) by adding a feedback control term through an interpolation operator. The dynamics of the determining form is consistent with those of the original system.

This work is concerned about the existence of solutions to the nonlocal semilinear problem { - ∫ ℝ N J ⁢ ( x - y ) ⁢ ( u ⁢ ( y ) - u ⁢ ( x ) ) ⁢ 𝑑 y + h ⁢ ( u ⁢ ( x ) ) = f ⁢ ( x ) , x ∈ Ω , u = g , x ∈ ℝ N ∖ Ω , \left\{\begin{aligned} &\displaystyle{-}\int_{{\mathbb{R}}^{N}}J(x-y)(u(y)-u(x% ))\,dy+h(u(x))=f(x),&&\displaystyle x\in\Omega,\\ &\displaystyle u=g,&&\displaystyle x\in{\mathbb{R}}^{N}\setminus\Omega,\end{% aligned}\right. verifying that limx→∂⁡Ω,x∈Ω⁡u⁢(x)=+∞{\lim_{x\to\partial\Omega,\,x\in\Omega}u(x)=+\infty}, known in the literature as large solutions. We find out that the relation between the diffusion and the absorption term is not enough to ensure such existence, not even assuming that the boundary datum g blows up close to ∂⁡Ω{\partial\Omega}. On the contrary, the role to obtain large solutions is played only by the interior source f , which gives rise to large solutions even without the presence of the absorption. We determine necessary and sufficient conditions on f providing large solutions and compute the blow-up rates of such solutions in terms of h and f . Finally, we also study the uniqueness of large solutions.

The main aim of the paper is to prove a fountain theorem without assuming the τ-upper semi-continuity condition on the variational functional. Using this improved fountain theorem, we may deal with more general strongly indefinite elliptic problems with various sign-changing nonlinear terms. As an application, we obtain infinitely many solutions for a semilinear Schrödinger equation with strongly indefinite structure and sign-changing nonlinearity.

The generalized Forchheimer flows are studied for slightly compressible fluids in porous media with time-dependent Dirichlet boundary data for the pressure. No restrictions are imposed on the degree of the Forchheimer polynomial. We derive, for all time, the interior L∞{L^{\infty}}-estimates for the pressure, its gradient and time derivative, and the interior L2{L^{2}}-estimates for its Hessian. The De Giorgi and Ladyzhenskaya–Uraltseva iteration techniques are used taking into account the special structures of the equations for both pressure and its gradient. These are combined with the uniform Gronwall-type bounds in establishing the asymptotic estimates when time tends to infinity.

In this paper, we use the critical point theory for convex, lower semicontinuous perturbations of C1{C^{1}}-functionals to obtain the existence of multiple nontrivial solutions for one parameter potential systems involving the operator u↦div⁡(∇⁡u1-|∇⁡u|2){u\mapsto\operatorname{div}(\frac{\nabla u}{\sqrt{1-|\nabla u|^{2}}})}. The solvability of a general non-potential system is also established.

We are concerned with the study of a class of non-autonomous eigenvalue problems driven by two non-homogeneous differential operators with variable (p1,p2){(p_{1},p_{2})}-growth. The main result of this paper establishes the existence of a continuous spectrum consisting in an unbounded interval and the nonexistence of eigenvalues in a neighbourhood of the origin. The abstract results of this paper are described by two Rayleigh-type quotients and the proofs rely on variational arguments.

This paper discusses a high-order Melnikov method for periodically perturbed equations. We introduce a new method to compute Mk⁢(t0){M_{k}(t_{0})} for all k≥0{k\geq 0}, among which M0⁢(t0){M_{0}(t_{0})} is the traditional Melnikov function, and M1⁢(t0),M2⁢(t0),…{M_{1}(t_{0}),M_{2}(t_{0}),\ldots\,} are its high-order correspondences. We prove that, for all k≥0{k\geq 0}, Mk⁢(t0){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 M0⁢(t0){M_{0}(t_{0})} and M1⁢(t0){M_{1}(t_{0})}. We also study a concrete equation for which the explicit formula of M1⁢(t0){M_{1}(t_{0})} is used to prove the existence of a transversal homoclinic intersection in the case of M0⁢(t0)≡0{M_{0}(t_{0})\equiv 0}.

By applying our variational method, we show that there exist 24 local action minimizers connecting two prescribed configurations: a collinear configuration and a double isosceles configuration in H1⁢([0,1],χ){H^{1}([0,1],\chi)} in the planar equal-mass four-body problem. Among the 24 local action minimizers, we prove that the one with the smallest action has no collision singularity and it can be extended to a periodic or quasi-periodic orbit. Furthermore, if all the 24 local action minimizers are free of collision, we show that they can generate sixteen different periodic orbits.

In [1], for 1<p<∞{1<p<\infty}, we proved the Wloc2⁢s,p{W^{2s,p}_{\mathrm{loc}}} local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian (-Δ)s{(-\Delta)^{s}} on an arbitrary bounded open set of ℝN{\mathbb{R}^{N}}. Here we make a more precise and rigorous statement. In fact, for 1<p<2{1<p<2} and s≠12{s\neq\frac{1}{2}}, local regularity does not hold in the Sobolev space Wloc2⁢s,p{W^{2s,p}_{\mathrm{loc}}}, but rather in the larger Besov space (Bp,22⁢s)loc{(B^{2s}_{p,2})_{\mathrm{loc}}}.