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In this study, we consider the following sublinear Schrödinger equations −Δu+Vxu=fx,u,for x∈ℝN,ux⟶0,asu⟶∞, where fx,u satisfies some sublinear growth conditions with respect to u and is not required to be integrable with respect to x. Moreover, V is assumed to be coercive to guarantee the compactness of the embedding from working space to LpℝN for all p∈1,2∗. We show that the abovementioned problem admits at least one solution by using the linking theorem, and there are infinitely many solutions when fx,u is odd in u by using the variant fountain theorem.

By introducing a new superquadratic condition, we obtain the existence of two nontrivial homoclinic solutions for a class of perturbed second order Hamiltonian systems which are obtained by the mountain pass theorem and Ekeland’s variational principle.

Similarity measures are very effective and meaningful tool used for evaluating the closeness between any two attributes which are very important and valuable to manage awkward and complex information in real-life problems. Therefore, for better handing of fuzzy information in real life, Ullah et al. (Complex Intell Syst 6(1): 15–27, 2020) recently introduced the concept of complex Pythagorean fuzzy set (CPyFS) and also described valuable and dominant measures, called various types of distance measures (DisMs) based on CPyFSs. The theory of CPyFS is the essential modification of Pythagorean fuzzy set to handle awkward and complicated in real-life problems. Keeping the advantages of the CPyFS, in this paper, we first construct an example to illustrate that a DisM proposed by Ullah et al. does not satisfy the axiomatic definition of complex Pythagorean fuzzy DisM. Then, combining the 3D Hamming distance with the Hausdorff distance, we propose a new DisM for CPyFSs, which is proved to satisfy the axiomatic definition of complex Pythagorean fuzzy DisM. Moreover, similarly to some DisMs for intuitionistic fuzzy sets, we present some other new complex Pythagorean fuzzy DisMs. Finally, we apply our proposed DisMs to a building material recognition problem and a medical diagnosis problem to illustrate the effectiveness of our DisMs. Finally, we aim to compare the proposed work with some existing measures is to enhance the worth of the derived measures.

A new existence of hyperbolic orbits is obtained for a class of singular Hamiltonian systems with prescribed energies by taking the limit for a sequence of approximate solutions. Furthermore, we show that the hyperbolic orbits possess the given directions at infinity.

We study the multiplicity of solutions for a class of semilinear Schrödinger equations: -Δu+V(x)u=gx,u,  for  x∈RN;  u(x)→0,  as  u→∞, where V satisfies some kind of coercive condition and g involves concave-convex nonlinearities with indefinite signs. Our theorems contain some new nonlinearities.

In this paper, we consider the following Klein-Gordon-Maxwell equations \\begin{eqnarray*} \\left\\{ \\begin{array}{ll} -\\Delta u+ V(x)u-(2\\omega+\\phi)\\phi u=f(x,u)+h(x)&\\mbox{in \\(\\mathbb{R}^{3}\\)},\\\ -\\Delta \\phi+ \\phi u^2=-\\omega u^2&\\mbox{in \\(\\mathbb{R}^{3}\\)}, \\end{array} \\right. \\end{eqnarray*} where \\(\\omega>0\\) is a constant, \\(u\\), \\(\\phi : \\mathbb{R}^{3}\\rightarrow \\mathbb{R}\\), \\(V : \\mathbb{R}^{3} \\rightarrow\\mathbb{R}\\) is a potential function. By assuming the coercive condition on \\(V\\) and some new superlinear conditions on \\(f\\), we obtain two nontrivial solutions when \\(h\\) is nonzero and infinitely many solutions when \\(f\\) is odd in \\(u\\) and \\(h\\equiv0\\) for above equations.

In this paper, we consider the existence and nonuniqueness of solutions for the following Schrödinger equations with the nonlinear term being asymptotically linear at infinity, - Δ u + V ( x ) u = f ( x , u ) for x ∈ R N , u ( x ) → 0 as | x | → ∞ . We introduce a new condition on V ( x ) and obtain a new compact embedding theorem. Some new asymptotically linear conditions on f ( x ,  u ) are introduced which are quite different from the previous ones in the references. An existence theorem is obtained using the Generalized Mountain Pass theorem. Furthermore, we obtain the existence of infinitely many solutions for above asymptotically linear Schrödinger equations by the Variant Fountain theorem, which has been considered by only few authors.

In this paper, we show the existence and multiplicity of solutions for the following fourth-order Kirchhoff type elliptic equations \\begin{eqnarray*} \\Delta^{2}u-M(\\|\\nabla u\\|_{2}^{2})\\Delta u+V(x)u=f(x,u),\\ \\ \\ \\ \\ x\\in \\mathbb{R}^{N}, \\end{eqnarray*} where \\(M(t):\\mathbb{R}\\rightarrow\\mathbb{R}\\) is the Kirchhoff function, \\(f(x,u)=\\lambda k(x,u)+ h(x,u)\\), \\(\\lambda\\geq0\\), \\(k(x,u)\\) is of sublinear growth and \\(h(x,u)\\) satisfies some general 3-superlinear growth conditions at infinity. We show the existence of at least one solution for above equations for \\(\\lambda=0\\). For \\(\\lambda>0\\) small enough, we obtain at least two nontrivial solutions. Furthermore, if \\(f(x,u)\\) is odd in \\(u\\), we show that above equations possess infinitely many solutions for all \\(\\lambda\\geq0\\). Our theorems generalize some known results in the literatures even for \\(\\lambda=0\\) and our proof is based on the variational methods.

In this paper, we study the existence and multiplicity of homoclinic solutions for following Hamiltonian systems with asymptotically quadratic nonlinearities at infinity \\begin{eqnarray*} \\ddot{u}(t)-L(t)u+\\nabla W(t,u)=0. {eqnarray*} We introduce a new coercive condition and obtain a new embedding theorem. With this theorem, we show that above systems possess at least one nontrivial homoclinic orbits by Generalized Mountain Pass Theorem. By Variant Fountain Theorem, infinitely many homoclinic orbits are obtained for above problem with symmetric condition. Our asymptotically quadratic conditions are different from previous ones in the references.

In the present study, we constructed a lentivirus, FIV-CMV-GFP-miR-7-3, containing the microRNA-7-3 gene and the green fluorescent protein gene, and used it to transfect human glioma U251 cells. Fluorescence microscopy showed that 80% of U251 cells expressed green fluorescence. Real-time reverse transcription PCR showed that microRNA-7-3 RNA expression in U251 cells was significantly increased. Proliferation was slowed in transfected U251 cells, and most cells were in the G1 phase of the cell cycle. In addition, the expression of the serine/threonine protein kinase 2 was decreased. Results suggested that transfection with a lentivirus carrying microRNA-7-3 can effectively suppress epidermal growth factor receptor pathway activity in U251 cells, arrest cell cycle transition from GI phase to S phase and inhibit glioma cell growth.