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In this paper, we discuss the existence and the concentration of sign-changing solutions to a class of Kirchhoff-type systems with Hartree-type nonlinearity in R 3 . By the minimization argument on the sign-changing Nehari manifold and a quantitative deformation lemma, we prove that the system has a sign-changing solution. Moreover, concentration behaviors of sign-changing solutions are obtained when the coefficient of the potential function tends to infinity. Specially, our results cover general Schrödinger equations, Kirchhoff equations and Schrödinger–Poisson systems. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of generalized quasilinear Schrödinger equations with a Kirchhoff-type perturbation. Under the assumption that the potential may be vanishing at infinity, the existence of both the ground state and the ground state sign-changing solutions is established. Furthermore, the behavior of these solutions is studied when the perturbation vanishes. It is a surprise that we find an interesting phenomenon about the monotonicity for the quotient function as a byproduct. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence and multiplicity of positive radial solutions to some Kirchhoff equations and elliptic equations. By using the Leggett–Williams three solutions theorem skillfully, we obtain two positive solutions to Kirchhoff equations under some appropriate conditions. As a direct corollary, we also obtain the same result to elliptic equations. In order to facilitate the proof of main results, we first establish a new simple three solutions theorems for the equation [ a + b φ ( x ) ] x = A x , where A is a completely continuous operator, a > 0 , b ⩾ 0 and φ ( x ) ⩽ k ( ‖ x ‖ ) for all x ∈ P , where k is a nondecreasing nonnegative continuous function, then apply it to prove the main results. In addition, we discover and prove a new inequality in the article. In the end of the paper, we present an example which makes the elliptic equation has infinite many positive radial solutions and give the approximate images of the nonlinear term. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the positive solutions to the equation φ ( u ) u = λ a A u + B u + u 0 , where A is a positive linear completely continuous operator, B is an α -homogeneous operator defined on a cone in a real Banach space and φ ( u ) = a + b ‖ u ‖ β . By using the fixed point index theory, when u 0 is sufficiently small, the spectral radius λ r ( A ) < 1 and α − γ β > 1 , where γ = sgn b , we obtain a positive solution to the above equation under some appropriate conditions. The new results generalize the previous research about the homogeneous operator equation. As an application, by using our main theorem we can obtain a symmetrical positive solution to the one dimensional Kirchhoff equation. [ABSTRACT FROM AUTHOR]

Abstract: The existence of positive solutions depending on a nonnegative parameter λ to Kirchhoff type problems with zero mass is proved by using variational method, and the new result does not require usual compactness conditions. A priori estimate and a Pohozaev type identity are used to obtain the bounded Palais–Smale sequences for constant coefficient nonlinearity, while a cut-off functional and Pohozaev type identity are utilized to obtain the bounded Palais–Smale sequences for the variable-coefficient case. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss the existence of a positive radial solution to a generalized Schrödinger–Poisson system without compactness conditions. By the method of the combination of a cut-off function, a monotonicity trick and a Pohozaev type identity, we obtain the boundedness of a Palais–Smale sequence. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the nontrivial solutions to a class of systems of second-order ordinary differential equations by using the critical groups and establish the existence results under the resonant case. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the fourth-order Neumann boundary value problem for all and subject to . Using the fixed point index and the critical group, we establish the existence theorem of solutions that guarantees the problem has at least one positive solution and two sign-changing solutions under certain conditions. [Copyright &y& Elsevier]

Abstract: In this paper, the existence and multiplicity of solutions are obtained for the 2mth-order ordinary differential equation two-point boundary value problems for all subject to Dirichlet, Neumann, mixed and periodic boundary value conditions, respectively, where f is continuous, for all . Since these four boundary value problems have some common properties and they can be transformed into the integral equation of form , we firstly deal with this nonlinear integral equation. By using the strongly monotone operator principle and the critical point theory, we establish some conditions on f which are able to guarantee that the integral equation has a unique solution, at least one nonzero solution, and infinitely many solutions. Furthermore, we apply the abstract results on the integral equation to the above four 2mth-order two-point boundary problems and successfully resolve the existence and multiplicity of their solutions. [Copyright &y& Elsevier]

Abstract: In this paper, the existence of sign-changing solutions for nonlinear operator equations is discussed by using the topological degree and fixed point index theory. The main theorems are some new three-solution theorems which are different from the famous Amann''s and Leggett-Williams'' three-solution theorems as well as the results in [F. Li, G. Han, Generalization for Amann''s and Leggett–Williams'' three-solution theorems and applications, J. Math. Anal. Appl. 298 (2004) 638–654]. These three solutions are all nonzero. One of them is positive, another is negative, and the third one is a sign-changing solution. Furthermore, the theoretical results are successfully applied to both integral and differential equations. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence and multiplicity of solutions of the operator equation in the real Hilbert space . Under certain conditions on the linear operator K, we establish the conditions on f which are able to guarantee that the operator equation has at least one solution, a unique solution, and infinitely many solutions, respectively. The monotone operator principle and the critical point theory are employed to discuss this problem, respectively. In argument, quadratic root operator and its properties play an important role. As an application, we investigate the existence and multiplicity of solutions to fourth-order boundary value problems for ordinary differential equations with two parameters, and give some new existence results of solutions. [Copyright &y& Elsevier]

Abstract: In this paper, the existence and multiplicity results of solutions are obtained for the second order two-point boundary value problem for all subject to , where f is continuous. The monotone operator theory and critical point theory are employed to discuss this problem, respectively. In argument, quadratic root operator and its properties play an important role. [Copyright &y& Elsevier]

In this paper, some new three-solution theorems are obtained. Moreover, the famous Amann''s and Leggett–Williams'' three-solution theorems in nonlinear functional analysis can be seen as their special cases, namely these two famous theorems are united. So they are both improved. And that some new three-solution theorems are applied to ordinary differential equations, and some new existence results are obtained. [Copyright &y& Elsevier]

A class of Kirchhoff-type transmission problems with critical perturbation is studied. Unlike other studies on elliptical equations with critical growth, a ground-state solution is obtained using a perturbation method instead of verifying that the mountain pass level is lower than the critical energy, which is used to verify the PS condition. [ABSTRACT FROM AUTHOR]