Nonstationary homoclinic orbit for an infinite-dimensional fractional reaction-diffusion system

<p style='text-indent:20px;'>This paper study nonstationary homoclinic-type solutions for a fractional reaction-diffusion system with asymptotically linear and local super linear nonlinearity. The resulting problem engages two major difficulties: one is that the associated functional is strongly indefinite, the second lies in verifying the link geometry and showing the boundedness of Cerami sequences when the nonlinearity is not super quadratic at infinity globally. These enable us to develop a direct approach and new tricks to overcome the difficulties. We establish the existence of homoclinic orbit under some weak assumptions on nonlinearity.</p>

The cone Moser–Trudinger inequalities and applications

In this paper, we first study the cone Moser–Trudinger inequalities and their best exponents on both bounded and unbounded domains R + 2 . Then, using the cone Moser–Trudinger inequalities, we study the asymptotic behavior of Cerami sequences and the existence of weak solutions to the nonlinear equation − Δ B u = f ( x , u ) , in x ∈ int ( B ) , u = 0 , on ∂ B , where Δ B is an elliptic operator with conical degeneration on the boundary x 1 = 0, and the nonlinear term f has the subcritical exponential growth or the critical exponential growth.

Hamiltonian systems of Schrödinger equations with vanishing potentials

In this paper, by means of minimax techniques involving Cerami sequences, we prove the existence of at least one pair of positive solutions for a Hamiltonian system of Schrödinger equations in [Formula: see text] with potentials vanishing at infinity and subcritical nonlinearities which are superlinear at the origin and at infinity. We establish new estimates to prove the boundedness of a Cerami sequence.

Ground state solutions for nonlinear fractional Kirchhoff–Schrödinger–Poisson systems

In this paper, we study the existence of ground state solutions for the following fractional Kirchhoff–Schrödinger–Poisson systems with general nonlinearities:$$\left\{\begin{array}{ll}\left(a+b{\left[u\right]}_{s}^{2}\right)\,{\left(-{\Delta}\right)}^{s}u+u+\phi \left(x\right)u=\left({\vert x\vert }^{-\mu }\ast F\left(u\right)\right)f\left(u\right)\hfill & \mathrm{in}\text{\ }{\mathrm{&#x211d;}}^{3}\,\text{,}\hfill \\ {\left(-{\Delta}\right)}^{t}\phi \left(x\right)={u}^{2}\hfill & \mathrm{in}\text{\ }{\mathrm{&#x211d;}}^{3}\,\text{,}\hfill \end{array}\right.$$where$${\left[u\right]}_{s}^{2}={\int }_{{\mathrm{&#x211d;}}^{3}}{\vert {\left(-{\Delta}\right)}^{\frac{s}{2}}u\vert }^{2}\,\mathrm{d}x={\iint }_{{\mathrm{&#x211d;}}^{3}{\times}{\mathrm{&#x211d;}}^{3}}\frac{{\vert u\left(x\right)-u\left(y\right)\vert }^{2}}{{\vert x-y\vert }^{3+2s}}\,\mathrm{d}x\mathrm{d}y\,\text{,}$$$s,t\in \left(0,1\right)$ with $2t+4s{ >}3,0{< }\mu {< }3-2t,$$f:{\mathrm{&#x211d;}}^{3}{\times}\mathrm{&#x211d;}\to \mathrm{&#x211d;}$ satisfies a Carathéodory condition and (−Δ)s is the fractional Laplace operator. There are two novelties of the present paper. First, the nonlocal term in the equation sets an obstacle that the bounded Cerami sequences could not converge. Second, the nonlinear term f does not satisfy the Ambrosetti–Rabinowitz growth condition and monotony assumption. Thus, the Nehari manifold method does not work anymore in our setting. In order to overcome these difficulties, we use the Pohozǎev type manifold to obtain the existence of ground state solution of Pohozǎev type for the above system.

An Abstract Linking Theorem Applied to Indefinite Problems Via Spectral Properties

An abstract linking result for Cerami sequences is proved without the Cerami condition. It is applied directly in order to prove the existence of critical points for a class of indefinite problems in infinite-dimensional Hilbert Spaces. The applications are given to Schrödinger equations. Here spectral properties inherited by the potential features are exploited in order to establish a linking structure, and hence hypotheses of monotonicity on the nonlinearities are discarded.

New Super-quadratic Conditions on Ground State Solutions for Superlinear Schrödinger Equation

Consider the semilinear Schrödinger equationwhere f is a superlinear, subcritical nonlinearity. We mainly study the case where both V and f are periodic in x and 0 belongs to a spectral gap of −Δ + V. Based on the work of Szulkin and Weth [J Funct Anal 257: 3802-3822, 2009], we develop a new technique to show the boundedness of Cerami sequences and derive a new super-quadratic condition that there exists a θfor the existence a “ground state solution” which minimizes the corresponding energy among all nontrivial solutions. Our result unifies and improves some known ones and the recent ones of Szulkin and Weth [J Funct Anal 257: 3802-3822, 2009] and Liu [Calc. Var. 45: 1-9, 2012].

The Use of Cerami Sequences in Critical Point Theory

The concept of linking was developed to produce Palais-Smale (PS) sequencesG(uk)→a,G'(uk)→0forC1functionalsGthat separate linking sets. These sequences produce critical points if they have convergent subsequences (i.e., ifGsatisfies the PS condition). In the past, we have shown that PS sequences can be obtained even when linking does not exist. We now show that such situations produce more useful sequences. They not only produce PS sequences, but also Cerami sequences satisfyingG(uk)→a,(1+||uk||)G'(uk)→ 0as well. A Cerami sequence can produce a critical point even when a PS sequence does not. In this situation, it is no longer necessary to show thatGsatisfies the PS condition, but only that it satisfies the easier Cerami condition (i.e., that Cerami sequences have convergent subsequences). We provide examples and applications. We also give generalizations to situations when the separating criterion is violated.