Semiclassical states for coupled nonlinear Schrödinger equations with critical frequency

In this paper, we study the existence and concentration of positive ground state solutions for the semilinear Schrödinger-Poisson system where ε > 0 is a small parameter and λ ≠ 0 is a real parameter, f is a continuous superlinear and subcritical nonlinearity. Suppose that b(x) has a maximum. We prove that the system has a positive ground state solution for all λ ≠ 0 and sufficiently small ε > 0. Moreover, for each λ ≠ 0 we show that uε converges to the positive ground state solution of the associated limit problem and concentrates to a maximum point of b(x) in certain sense as ε → 0. Furthermore, we obtain some sufficient conditions for the nonexistence of positive ground state solutions.

For a nonempty set S, Banach spaces X and Y over a field K = R or C, and a nonempty open set \(U\,\subset\,X\), let G, F, and H be vector spaces of functions acting from S into X,from U into Y, and from S into Y, respectively. Usually, but not always, F, G, and H will be normed spaces. Given a function \((u,s)\longmapsto,\psi(u,s)\) from U × S into Y, and a function g : \(S\,\rightarrow\,U\), the Nemytskii operator \(N\psi\) is defined by \(N\psi(g)(s)\quad:=\quad,N\psi\,g)(s)\quad:=\quad\psi(g(s),s)\qquad s\,\epsilon\,S.\) (6.1) Other authors call such an operator a superposition operator, e.g. Appell and Zabrejko [3].We use the term Nemytskii operator (as many others have) partly to distinguish it from the two-function composition operator \((F,G)\mapsto FoG\) to be treated in Chapter 8. Recall also that for a linear operator A we write Ax := A(x). We will often apply this rule also when A is a Nemytskii operator.KeywordsBanach SpaceBanach AlgebraMinkowski InequalityNonempty Open SubsetPoint PartitionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we consider the existence and concentration behavior of positive ground state solution to the following problem � −h p Δpu+V(x)|u| p−2 u = K(x)|u| q−2 u+ |u| p ∗ −2 u, x ∈ R N , u ∈W 1,p (R N ), u > 0, x ∈ R N , where h is a small positive parameter, 1 0, the equation has a positive ground state solution. Furthermore, we establish the concentration property of such solutions when h tends to zero.

We investigate a class of fractional Schrödinger–Poisson systems with critical exponents and steep potential well. By using the constraint variational method and a quantitative deformation lemma, the existence of positive ground state solutions and sign-changing ground state solutions is obtained. It is shown that the energy of the sign-changing ground state solution is strictly larger than twice the energy of the positive ground state solution. Moreover, we also study the asymptotic behavior of the sign-changing ground state

<p style='text-indent:20px;'>In this paper we prove the existence of at least one positive solution for nonlocal semipositone problem of the type</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ (P_\lambda^\mu)\left\{ \begin{array}{rcl} (-\Delta)^s u& = & \lambda(u^{q}-1)+\mu u^r \text{ in } \Omega\\ u&>&0 \text{ in } \Omega\\ u&\equiv &0 \text{ on }{\mathbb R^N\setminus\Omega}. \end{array}\right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>when the positive parameters <inline-formula><tex-math id="M1">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> belong to certain range. Here <inline-formula><tex-math id="M3">\begin{document}$ \Omega\subset \mathbb{R}^N $\end{document}</tex-math></inline-formula> is assumed to be a bounded open set with smooth boundary, <inline-formula><tex-math id="M4">\begin{document}$ s\in (0, 1), N> 2s $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ 0<q<1<r\leq \frac{N+2s}{N- 2s}. $\end{document}</tex-math></inline-formula> First we consider <inline-formula><tex-math id="M6">\begin{document}$ (P_ \lambda^\mu) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M7">\begin{document}$ \mu = 0 $\end{document}</tex-math></inline-formula> and prove that there exists <inline-formula><tex-math id="M8">\begin{document}$ \lambda_0\in(0, \infty) $\end{document}</tex-math></inline-formula> such that for all <inline-formula><tex-math id="M9">\begin{document}$ \lambda> \lambda_0 $\end{document}</tex-math></inline-formula> the problem <inline-formula><tex-math id="M10">\begin{document}$ (P_ \lambda^0) $\end{document}</tex-math></inline-formula> admits at least one positive solution. In fact we will show the existence of a continuous branch of maximal solutions of <inline-formula><tex-math id="M11">\begin{document}$ (P_\lambda^0) $\end{document}</tex-math></inline-formula> emanating from infinity. Next for each <inline-formula><tex-math id="M12">\begin{document}$ \lambda>\lambda_0 $\end{document}</tex-math></inline-formula> and for all <inline-formula><tex-math id="M13">\begin{document}$ 0<\mu<\mu_{\lambda} $\end{document}</tex-math></inline-formula> we establish the existence of at least one positive solution of <inline-formula><tex-math id="M14">\begin{document}$ (P_\lambda^\mu) $\end{document}</tex-math></inline-formula> using variational method. Also in the sub critical case, i.e., for <inline-formula><tex-math id="M15">\begin{document}$ 1<r<\frac{N+2s}{N-2s} $\end{document}</tex-math></inline-formula>, we show the existence of second positive solution via mountain pass argument.</p>

Bour ded point evaluations for nonminimal measures

In this paper, we first define Teodorescu operator related to the -Dirac operator in Clifford analysis. Then we discuss the existence, q-times integrability, differentiability and mapping properties of Teodorescu operator in a nonempty open set , which is bounded connected.

Definition: A subset Y of a space X is said to be dense in X if every nonempty open set in X contains at least one point in Y.

In this paper we prove a form of the Zilber's trichotomy conjecture for reducts of algebraically closed valued fields of characteristic 0 which are expansions of the valued vector space structure. We prove first that a non-modular reduct of a nontrivially valued algebraically closed field containing the valued vector space structure defines a non-semilinear curve. Then we show that the expansion of such a reduct by a non-semilinear curve defines multiplication on a nonempty open set.

Transitive and series transitive maps on [formula omitted

Densities and liftings for derived algebras

Throughout the present paper, {Xn} denotes a sequence of random quantities which are regarded as exchangeable, and which are assessed with a probability measure P(•) which is a member of the mixture-exponential family. To be precise, it will be presumed that the assessment P(•) for any finite subsequence (X1,…,Xn) can be represented using the product of an identical non-degenerate parametric measure for each Xi, Pθ (•)= P(•θ=θ), determined by $$ {\rm{d}}{{\rm{P}}_{\rm{\theta }}} = \exp \{ {\rm{\theta x}} - {\rm{M}}({\rm{\theta }})\} {\rm{d\mu }} $$ (1.1) μ being a σ-finite measure on the class B of Borel sets of IR. It will always be assumed that the interior X° of the convex hull X of the support of μ (in symbols:suρp(μ))is a nonempty open set (interval) in IR and that {Pθ;θeΘ} is a regular exponential family (cf. Barndorff - Nielsen 1978, p.116). The latter condition implies that Θ = {θ:M(θ)<∞ } is an open interval in IR. Moreover, we will suppose that the set of the logically possible values of θ coincides with θ. Given such a particular frame, the present paper deals with the choice of a prior for (1.1); an excellent treatment of the same topic is included in Diaconis and Ylvisaker (1979, 1985). Our approach bases itself on the obvious remark that the choice of a prior establishes the strength of the dependence among the elements of the sequence {Xn} and, consequently, the strength of the influence exercised by experience on our future predictions. This subjective standpoint is skilfully expounded in de Finetti (1937).

Liouvillian functions are functions that are built up from rational functions using exponentiation, integration, and algebraic functions. We show that if a system of differential equations has a generic solution that satisfies a liouvillian relation, that is, there is a liouvillian function of several variables vanishing on the curve defined by this solution, then the system has a liouvillian first integral, that is a nonconstant liouvillian function that is constant on solution curves in some nonempty open set. We can refine this result in special cases to show that the first integral must be of a very special form. For example, we can show that if the system $dx/dz = P(x,y)$, $dy/dz = Q(x,y)$ has a solution $(x(z),y(z))$ satisfying a liouvillian relation then either $x(z)$ and $y(z)$ are algebraically dependent or the system has a liouvillian first integral of the form $F(x,y) = \smallint RQ dx - RP dy$ where $R = \exp (\smallint U dx + V dy)$ and $U$ and $V$ rational functions of $x$ and $y$ . We can also reprove an old result of Ritt stating that a second order linear differential equation has a nonconstant solution satisfying a liouvillian relation if and only if all of its solutions are liouvillian.

Definition: A subset Y of a space X is said to be dense in X if every nonempty open set in X contains at least one point in Y.

Bau-Sen Du introduced a notion of chaos which is stronger than Li-Yorke sensitivity. A TDS (X, f) is called chaotic if there is a positive e such that for any x and any nonempty open set V of X there is a point y in V such that the pair (x, y) is proximal but not e-asymptotic. In this article, we show that a TDS (T, f) is transitive but not mixing if and only if (T, f) is Li-Yorke sensitive but not chaotic, where T is a tree. Moreover, we compare such chaos with other notions of chaos.