New type of multi-bump solutions for Schrodinger-Poisson systems

In this paper, we consider the following linearly coupled nonlinear Schrödinger system: where , and are radial, positive, and continuous and satisfying that When and satisfy some weak power decay condition at infinity; we show that problem has infinitely many nonradial positive solutions by using the Lyapunov–Schmidt reduction method.

In this paper, we use the Lyapunov–Schmidt reduction method to obtain the existence of multi-bump solutions for a nonlinear Kirchhoff-type system with the parameter ɛ. As a result, when ɛ → 0, the system has more and more multi-bump positive solutions.

A new experimental characterization is presented of time-, field-, and temperature-dependent dynamic effects in magnetization of a nanocomposite which displays slow decay. Field and temperature variations of irreversible susceptibility, χ irr , decay coefficient, S, fluctuation field, hf , and activation volume, V, have been calculated for the nanocomposite sample (Co80Ni20) using a recently developed modified Preisach–Arrhenius (MPA) model. The sample is composed of non-interacting nanoparticles having negligible reversible magnetization. Non-Arrhenius behavior is observed in both the maximum decay coefficient, S max, and the fluctuation field, hf , as a function of temperature T. The peak of both temperature curves are identical and occur at a critical temperature Tk of ∼50 K, which agrees with our experimental results. Based on the effect of a temperature-dependent chemical potential on energy barrier, hf is studied for T < Tk and T ≥ Tk , respectively. A more complete MPA model that can predict the magnetization as function of time, field and temperature for a magnetic material with slow decay rates is proposed. This model uses a multi-variable analytical formula, , which incorporates the characteristic parameters.

In the paper we prove that the Lagrangian system$ \ddot{q} = \alpha(\omega t) V'(q), \quad t \in \mathbb R, q \in \mathbb R^N,$ $\qquad\qquad (L_\omega)$ has, for some classes of functions $\alpha$, a chaotic behavior---more precisely the system has multi-bump solutions---for all $\omega$ large. These classes of functions include some quasi-periodic and some limit-periodic ones, but not any periodic function. We prove the result using global variational methods.

Multi-bump solutions for nonlinear elliptic equations involving critical Sobolev exponents

In this work we examine the existence of positive classical solutions of { − Δ u + u = | x | α u p − 1 a m p ; in B 1 , u &gt; 0 a m p ; in B 1 , ∂ ν u = 0 a m p ; on ∂ B 1 , \begin{equation*} \begin {cases} -\Delta u +u = |x|^\alpha u^{p-1} &amp; \text { in } B_1, \\ u&gt;0 &amp; \text { in } B_1, \\ \partial _\nu u= 0 &amp; \text { on } \partial B_1, \end{cases} \end{equation*} where p &gt; 1 p&gt;1 , α &gt; 0 \alpha &gt;0 and B 1 B_1 is the unit ball in R N {\mathbb {R}}^N where N ≥ 4 N \ge 4 and is even. Of particular interest is the existence of nonradial position classical solutions. We show that under suitable conditions on p , α p,\alpha and N N there are positive classical nonradial solutions. Our approach is to utilize a variational approach on suitable convex cones.

4 - Recursive Enumerability

We study the critical Neumann problem $ \begin{equation*} \begin{cases} -\Delta u = |u|^{2^*-2}u &amp;\text{in }\Sigma_\omega,\\ \quad\frac{\partial u}{\partial\nu} = 0 &amp;\text{on }\partial\Sigma_\omega, \end{cases} \end{equation*} $ in the unbounded cone $\Sigma_\omega: = \{tx:x\in\omega\text{ and }t&gt;0\}$, where $\omega$ is an open connected subset of the unit sphere $\mathbb{S}^{N-1}$ in $\mathbb{R}^N$ with smooth boundary, $N\geq 3$ and $2^*: = \frac{2N}{N-2}$. We assume that some local convexity condition at the boundary of the cone is satisfied. If $\omega$ is symmetric with respect to the north pole of $\mathbb{S}^{N-1}$, we establish the existence of a nonradial sign-changing solution. On the other hand, if the volume of the unitary bounded cone $\Sigma_\omega\cap B_1(0)$ is large enough (but possibly smaller than half the volume of the unit ball $B_1(0)$ in $\mathbb{R}^N$), we establish the existence of a positive nonradial solution.

Sensitivity of a general class of shape functionals to topological changes

This article concerns the existence of multi-bump positive solutions for the following logarithmic Schrödinger equation: $$\left\{ {\begin{array}{*{20}{l}} { - \Delta u + \lambda V(x)u = u\log {u^2}\;\;\;in\;\;\;{\mathbb{R}^N},} \\ {u \;\in {H^1}({R^N}),} \end{array}} \right.$$ where N ⩾ 1, ⋋ > 0 is a parameter and the nonnegative continuous function V: ℝN → ℝ has potential well Ω:= int V−1(0) which possesses k disjoint bounded components $${\rm{\Omega}}\,{\rm{=}}\, \cup _{j = 1}^k{{\rm{\Omega}}_j}$$ . Using the variational methods, we prove that if the parameter ⋋ > 0 is large enough, then the equation has at least 2k − 1 multi-bump positive solutions.

In this paper we establish the existence and multiplicity of multi-bump nodal solutions for the class of problemswhereλ ∈(0, ∞),fis a continuous function with exponential critical growth andV: ℝ2→ℝ is a continuous function verifying some hypotheses.

In this paper, we establish existence and multiplicity of multi-bump type nodal solutions for the following class of problems $$ -\Delta u + (\lambda V(x)+ 1)u= f(u), \quad u> 0 \quad \text{in } {\mathbb R}^N, $$ where $N \geq 1$, $\lambda \in (0, \infty), f$ is a continuous function with subcritical growth and $V\colon {\mathbb R}^N \rightarrow {\mathbb R} $ is a continuous function verifying some hypotheses.

This paper considers the fractional Schrödinger-Poisson system in ℝ3. We prove that the problem has m-bump solutions under some given conditions which are given in the Introduction. Moreover, the system has more and more multi-bump solutions as ϵ → 0.

Our starting point are several general classes of real functions defined on the unit square satisfying some basic properties such as a boundary condition or several types of monotonicity and continuity. Applying to these functions some parameterized transformations and other constructions such as the transpose and flipping (which describe different aspects of symmetry) and truncation, we ask for conditions yielding (again) a bivariate copula. Some of these transformations are involutive (on one or more classes of functions), others are not even injective, and occasionally they induce additional properties, yielding, e.g., a (quasi-)copula. For several typical scenarios we identify the (not necessarily convex) sets of parameters leading to a copula and conditions imposing a minimal set of parameters.

In this paper, we introduce notable Jensen–Mercer inequality for a general class of convex functions, namely uniformly convex functions. We explore some interesting properties of such a class of functions along with some examples. As a result, we establish Hermite–Jensen–Mercer inequalities pertaining uniformly convex functions by considering the class of fractional integral operators. Moreover, we establish Mercer–Ostrowski inequalities for conformable integral operator via differentiable uniformly convex functions. Finally, we apply our inequalities to get estimations for normal probability distributions (Gaussian distributions).