Periodic solutions of superlinear autonomous Hamiltonian systems with prescribed period

AbstractIn this paper we prove an existence theorem of nonconstant periodic solution of superlinear autonomous Hamiltonian system x˙(t)=J∇H(x(t)) with prescribed period under an assumption weaker than Ambrosetti–Rabinowitz-type condition:0<μH(x)⩽〈∇H(x),x〉,μ>2,|x|⩾R>0. Our result extends the pioneering work of Rabinowitz of 1978

Existence of solutions for fourth order elliptic equations of Kirchhoff type on RN

In this paper, we study the positive solutions to a class of fourth order elliptic equations of Kirchhoff type on $R^N$ by using variational methods and the truncation method

On Fourth-Order Elliptic Equations of Kirchhoff Type with Dependence on the Gradient and the Laplacian

We consider a nonlocal fourth-order elliptic equation of Kirchhoff type with dependence on the gradient and Laplacian Δ2u-a+b∫Ω∇u2dxΔu=fx,u,∇u,Δu, in Ω, u=0, Δu=0, on ∂Ω, where a, b are positive constants. We will show that there exists b⁎>0 such that the problem has a nontrivial solution for 0<b<b⁎ through an iterative method based on the mountain pass lemma and truncation method developed by De Figueiredo et al., 2004

Nonhomogeneous fractional p-Kirchhoff problems involving a critical nonlinearity

This paper is concerned with the existence of solutions for a kind of nonhomogeneous critical p-Kirchhoff type problem driven by an integro-differential operator L p K . In particular, we investigate the equation: M �ZZ R2n |v(x) − v(y)| p |x − y| n+ps dxdy� L p K v(x) = µg(x)|v| q−2 v + |v| p s −2 v + µ f(x) in R n where g(x) > 0, and f(x) may change sign, µ > 0 is a real parameter, 0 ps, 1 < q < p < p s , p s = np n−ps is the critical exponent of the fractional Sobolev space W s,p K (Rn ). By exploiting Ekeland’s variational principle, we show the existence of non-trivial solutions. The main feature and difficulty of this paper is the fact that M may be zero and lack of compactness at critical level L p s (Rn Our conclusions improve the related results on this topic

Nonhomogeneous fractional p-Kirchhoff problems involving a critical nonlinearity

This paper is concerned with the existence of solutions for a kind of nonhomogeneous critical $p$-Kirchhoff type problem driven by an integro-differential operator $\mathcal{L}^{p}_{K}$. In particular, we investigate the equation: \begin{align*} \mathcal{M}\left(\iint_{\mathbb{R}^{2n}}\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+ps}}dxdy\right) \mathcal{L}^{p}_{K}v(x)=\mu g(x)|v|^{q-2}v+|v|^{p_{s}^{*}-2}v+\mu f(x) \quad\mbox{in}~\mathbb{R}^{n}, \end{align*} where $g(x)>0$, and $f(x)$ may change sign, $\mu>0$ is a real parameter, $0ps$, $1<q<p<p_{s}^{*}$, $p_{s}^{*}=\frac{np}{n-ps}$ is the critical exponent of the fractional Sobolev space $W^{s,p}_{K}(\mathbb{R}^{n}).$ By exploiting Ekeland's variational principle, we show the existence of non-trivial solutions. The main feature and difficulty of this paper is the fact that $\mathcal{M}$ may be zero and lack of compactness at critical level $L^{p_{s}^{*}}(\mathbb{R}^{n})$. Our conclusions improve the related results on this topic

Existence of infinitely many periodic solutions for second-order Hamiltonian systems

By using the variant of the fountain theorem, we study the existence of infinitely many periodic solutions for a class of superquadratic nonautonomous second-order Hamiltonian systems