Search

This paper concerns positive solutions to the boundary value problems of the scalar field equation in the half space with a Sobolev supercritical nonlinearity and an inhomogeneous Dirichlet boundary condition, admitting a nontrivial nonnegative Radon measure as the boundary data. Under a suitable integrability assumption on the boundary data and the Joseph–Lundgren subcritical condition on the nonlinear term, we give a complete classification of the existence/nonexistence of a positive solution with respect to the size of the boundary data. Furthermore, we give a result on the existence of multiple positive solutions via bifurcation theory. [ABSTRACT FROM AUTHOR]

In this paper, we study the fractional Schrödinger–Poisson system with a general nonlinearity as follows: (- Δ) s u + u + l (x) ϕ u = f (u) in R 2 , (- Δ) t ϕ = l (x) u 2 in R 2 , where 1 2 < t ≤ s < 1 , the potential l ∈ C (R 2 , R +) and f ∈ C (R , R) does not require the classical (AR)-condition. When l (x) ≡ μ > 0 is a parameter, by establishing new estimates for the fractional Laplacian, we find two positive solutions, depending on the range of μ . As a result, a positive ground state solution with negative energy exists for the non-autonomous system without any symmetry on l(x). When l(x) is radially symmetric, we show that the symmetry breaking phenomenon can occur, and that a non-radial ground state solution with negative energy exists. Furthermore, under additional assumptions on l(x), three positive solutions are found. The intrinsic differences between the planar SP system and the planar fSP system are analyzed. [ABSTRACT FROM AUTHOR]

This article is concerned with the following Kirchhoff equation: −a+b∫R3∣∇u∣2dxΔu=g(u)+h(x)inR3,-\left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{3}}{| \nabla u| }^{2}{\rm{d}}x\right)\Delta u=g\left(u)+h\left(x)\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{3}, where aa and bb are positive constants and h≠0h\ne 0. Under the Berestycki-Lions type conditions on gg, we prove that the equation has at least two positive solutions by using variational methods. Furthermore, we obtain the existence of ground state solutions.

We study the positive solutions for a class of Kirchhoff‐type problems involving the nonlinearity λf(x)up−1+g(x)u5(2

In this paper, we develop some new multiple fixed point theorems for the sum of two operators T + S where I - T is Lipschitz invertible and S is a k-set contraction on translate of a cone in a Banach space. New existence criteria for multiple positive solutions of a singular generalized Sturm-Liouville multipoint boundary value problem are established. The article ends with an illustrative example. [ABSTRACT FROM AUTHOR]

The existence of multiple pairs of smooth positive solutions for a Carrier problem, driven by a p (x) -Laplacian operator, is studied. The approach adopted combines sub-super solutions, truncation, and variational techniques. In particular, after an explicit computation of a sub-solution, obtained combining a monotonicity type hypothesis on the reaction term and the Giacomoni–Takáč's version of the celebrated Díaz–Saá's inequality, we derive a multiplicity of solution by investigating an associated one-dimensional fixed point problem. The nonlocal term involved may be a sign-changing function and permit us to obtain the existence of multiple pairs of positive solutions, one for each "positive bump" of the nonlocal term. A new result, also for a constant exponent, is established and an illustrative example is proposed. [ABSTRACT FROM AUTHOR]

We investigate the multiplicity results of positive solutions for a Kirchhoff-type problems with supercritical and critical nonlinear terms in a ball. By employing the Nehari method and Lusternik–Schnirelmann category theory to an auxiliary problems, we note that there is a relationship between the number of maxima in the coefficient function of the critical term and the number of positive solutions for the problems. [ABSTRACT FROM AUTHOR]

This article deals with the existence of multiple positive solutions to the following system of nonlinear equations involving Pucci's extremal operators: - M λ 1 , Λ 1 + (D 2 u 1) = f 1 (u 1 , u 2 , ⋯ , u n) in Ω , - M λ 2 , Λ 2 + (D 2 u 2) = f 2 (u 1 , u 2 , ⋯ , u n) in Ω , ⋮ = ⋮ - M λ n , Λ n + (D 2 u n) = f n (u 1 , u 2 , ⋯ , u n) in Ω , u 1 = u 2 = ⋯ = u n = 0 on ∂ Ω , where Ω is a smooth and bounded domain in R N and f i : [ 0 , ∞) × [ 0 , ∞) ⋯ × [ 0 , ∞) → [ 0 , ∞) are C α functions for i = 1 , 2 , ⋯ , n . The multiplicity result in this work is motivated by the work Amann (SIAM Rev 18(4):620–709, 1976), and Shivaji (Nonlinear analysis and applications (Arlington, Tex., 1986), Dekker, New York, 1987), where the three solutions theorem (multiplicity) has been proved for linear equations. Later on, it was extended for a system of equations involving the Laplace operator by Shivaji and Ali (Differ Integr Equ 19(6):669–680, 2006). Thus, the results here can be considered as a nonlinear analog of the results mentioned above. We also have applied the above results to show the existence of three positive solutions to a system of nonlinear elliptic equations having combined sublinear growth by explicitly constructing two ordered pairs of sub and supersolutions. [ABSTRACT FROM AUTHOR]

Based on properties of Green's function and the some conditions of $ f(t, u) $, we found a minimal and a maximal positive solution by the method of sequence approximation. Moreover, based on the properties of Green's function and fixed point index theorem, the existence of multiple positive solutions for a singular $ p $-Laplacian fractional differential equation with infinite-point boundary conditions was obtained and, at last, an example was given to demonstrate the validity of our main results.

Based on properties of Green's function and the some conditions of f (t , u) , we found a minimal and a maximal positive solution by the method of sequence approximation. Moreover, based on the properties of Green's function and fixed point index theorem, the existence of multiple positive solutions for a singular p -Laplacian fractional differential equation with infinite-point boundary conditions was obtained and, at last, an example was given to demonstrate the validity of our main results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the existence of the multiple positive solutions for a class of higher-order fractional differential equations on infinite intervals with infinite-point boundary value conditions is mainly studied. First, we construct the Green function and analyze its properties, and then by using the Leggett–Williams fixed point theorem, some new results on the existence of positive solutions for the boundary value problem are obtained. Finally, we illustrate the application of our conclusion by an example.

Abstract This paper investigates the existence of positive solutions for a class of fractional boundary value problems involving an improper integral and the infinite-point on the half-line by making use of properties of the Green function and Avery–Peterson fixed point theorem. In addition, an example is presented to illustrate the applicability of our main result.

In our previous paper [K. Ishige, S. Okabe, and T. Sato, A supercritical scalar field equation with a forcing term, J. Math. Pures Appl. 128 (2019), pp. 183–212], we proved the existence of a threshold κ∗>0{\kappa }^{\ast }\gt 0 such that the elliptic problem for an inhomogeneous elliptic equation −Δu+u=up+κμ-\Delta u+u={u}^{p}+\kappa \mu in RN{{\bf{R}}}^{N} possesses a positive minimal solution decaying at the space infinity if and only if 01p\gt 1 is in the Joseph-Lundgren subcritical case. In this article, we prove the existence of nonminimal positive solutions to the elliptic problem. Our arguments are also applicable to inhomogeneous semilinear elliptic equations with exponential nonlinearity.

This paper is divided in two parts. In the first part, we prove coercivity results and minimization of the Euler energy functional. In the second part, we focus on the existence and multiplicity of a positive solution of fractional Dirichlet problem involving the γ(ξ)-Laplacian equation with non-negative weight functions in Hα,β;χ γ(ξ) (Λ,R) using some variational techniques and Nehari manifold. [ABSTRACT FROM AUTHOR]

The main aim of the present work is to review and study a variational method in existence and multiplicity of positive solutions for quasilinear elliptic systems with critical Hardy-Sobolev and sign-changing function exponents. [ABSTRACT FROM AUTHOR]

This paper investigates the existence of positive solutions for a class of fractional boundary value problems involving an improper integral and the infinite-point on the half-line by making use of properties of the Green function and Avery–Peterson fixed point theorem. In addition, an example is presented to illustrate the applicability of our main result. [ABSTRACT FROM AUTHOR]

In this paper, the existence of the multiple positive solutions for a class of higher-order fractional differential equations on infinite intervals with infinite-point boundary value conditions is mainly studied. First, we construct the Green function and analyze its properties, and then by using the Leggett–Williams fixed point theorem, some new results on the existence of positive solutions for the boundary value problem are obtained. Finally, we illustrate the application of our conclusion by an example. [ABSTRACT FROM AUTHOR]

In this article we study a nonlocal equation involving singular and critical Hardy-Sobolev non-linearities, (−∆p)su − µ |u|p−2u/ |x|sp = λu−α + |u|p∗s (t)−2u |x|t, in Ω, u > 0, in Ω, u = 0, in RN \ Ω, where Ω ⊂ RN is a bounded domain with Lipschitz boundary and (−∆p) s is the fractional p-Laplacian operator. We combine some variational techniques with a perturbation method to show the existence of multiple solutions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the following fractional Schrödinger-Poisson systems with concave-convex nonlinearities: ..., where 1/2 < t ≤ s < 1, 1 < q < 2 < p < min {4, 2*s}, 2*s = 6/(3 - 2s), and μ > 0 is a parameter, f ∈ C (R³) is sign-changing in R³ and g ∈ Lp/(p-q) (R³). Undersome suitable assumptions on l(x), f(x) and g(x), we explore that the energy functional corresponding to the system is coercive and bounded below on Hα(R³) which gets a positive solution. Furthermore, we constructed some new estimation techniques, and obtained other two positive solutions. Recent results from the literature are generally improved and extended. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the multiplicity of positive solutions for the following Kirchhoff type problem ..., where ε>0 is a small parameter, a,b>0 are constants, 4

In this paper, we study three-dimensional Kirchhoff equations with critical growth and singular nonlinearity. We are concerned with the qualitative analysis of solutions to the following nonlocal problem - a + b ∫ Ω | ∇ u | 2 d x Δ u = λ u - γ + u 5 , in Ω , u > 0 , in Ω , u = 0 , on ∂ Ω , where Ω ⊂ R 3 is a bounded domain with smooth boundary, 0 < γ < 1 , and a , b , λ are positive constants. By combining variational methods with some delicate decomposition techniques, we obtain the existence of two positive solutions in the case of low perturbations of the singular nonlinearity, namely for small values of the parameter λ . [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity results of positive solutions for a class of Kirchhoff type problems with singularity and critical exponents. Combining with the Nehari method and variational method, we prove the existence of positive ground state solutions. Furthermore, we obtain a relationship between the number of positive solutions and the topology of the global maximum set of Q(x).

In this work, we study the multiple results of positive solutions for a class of elliptic problem involving a critical Sobolev exponent − △ u + λ V (x) u = μ f (x) u q − 1 + g (x) u 2 ∗ − 1 in R N , we prove that the problem has at least k distinct positive solutions. The methods used here are based on combining minimax methods, Nehari manifold and Ekeland's variational principle. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of Schrödinger-Poisson (SP) systems with general nonlinearity where the nonlinearity does not require Ambrosetti-Rabinowitz and Nehari monotonic conditions. We establish new estimates and explore the associated energy functional which is coercive and bounded below on Sobolev space. Together with Ekeland variational principle, we prove the existence of ground state solutions. Furthermore, when the 'charge' function is greater than a fixed positive number, the (SP) system possesses only zero solutions. In particular, when 'charge' function is radially symmetric, we establish the existence of three positive solutions and the symmetry breaking of ground state solutions. [ABSTRACT FROM AUTHOR]

We study the following Kirchhoff type equation:

In this paper, we study the following critical nonlocal problem \begin{equation*} \begin{cases} -\left(a-\lambda b\displaystyle\int_{\Omega}|\nabla u|^2dx\right)\Delta u=\lambda |u|^{p-2}u+Q(x)|u|^{2}u, & x\in\Omega,\\ u=0, & x\in\partial\Omega, \end{cases} \end{equation*} where $a>0$, $b\ge0$, $20$ is a parameter, $\Omega$ is a smooth bounded domain in $\mathbb{R}^4$ and $Q(x)\in C({\overline{\Omega}})$ is a nonnegative function. By virtue of variational methods and delicate estimates, we prove that problem admits $k$ positive solutions for $\lambda>0$ sufficiently small, provided that the maximum of $Q(x)$ is achieved at $k$ interior points in $\Omega$.

Based on properties of Green's function and by Avery-Peterson fixed point theorem, the existence of multiple positive solutions are obtained for singular p-Laplacian fractional differential equation with infinite-point boundary conditions, and an example is given to demonstrate the validity of our main results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the following nonlocal problem: { − ( a − b ∫ Ω | ∇ u | 2 d x ) Δ u = λ | u | q − 2 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , $$ \textstyle\begin{cases} - (a-b \int _{\Omega } \vert \nabla u \vert ^{2}\,dx ) \Delta u= \lambda \vert u \vert ^{q-2}u, & x\in \Omega , \\ u=0, & x\in \partial \Omega , \end{cases} $$ where Ω is a smooth bounded domain in R N $\mathbb{R}^{N}$ with N ≥ 3 $N\ge 3$ , a , b > 0 $a,b>0$ , 1 < q < 2 $1< q 0 $\lambda >0$ is a parameter. By virtue of the variational method and Nehari manifold, we prove the existence of multiple positive solutions for the nonlocal problem. As a co-product of our arguments, we also obtain the blow-up and the asymptotic behavior of these solutions as b ↘ 0 $b\searrow 0$ .

In this article, we consider the multiplicity of solutions for nonhomogeneous Schrodinger-Poisson systems under the Berestycki-Lions type conditions. With the aid of Ekeland's variational principle, the mountain pass theorem and a Pohozaev type identity, we prove that the system has at least two positive solutions.

In this work we consider the fractional Kirchhoff equations with singular nonlinearity, ... where Ω is a bounded domain in RN with smooth boundary, N > ps, s ∈ (0, 1), 0 < α < 1, 0 < β < 1, 2 - α - β < p ≤ pθ < q < p*s, p *s = 2N/N-2s is the fractional Sobolev exponent, λ, µ are two parameters, a, b, c ∈ C(Ω) are nonnegative weight functions, M(t) = k + ltθ-1 with k > 0, l, θ ≥ 1, and (-∆)s p is the fractional p-laplacian operator. We prove the existence of multiple nonnegative solutions by studying the nature of the Nehari manifold with respect to the parameters λ and µ. [ABSTRACT FROM AUTHOR]

In this paper, we study the Schrödinger equation − Δ u = f (u) + h (x) i n R N , where N ≥ 3 , h (x) ≢ 0 and f satisfies the conditions of Berestycki-Lions type. On the basis of variational methods, by using Ekeland's variational principle, the mountain pass theorem and a Pohožaev type identity, we obtain the existence of ground state solutions and the multiplicity of positive solutions under suitable assumptions on h. [ABSTRACT FROM AUTHOR]

In this work, we study the fractional Laplacian equation with singular nonlinearity: (− Δ) s u = λ a (x) | u | q − 2 u + 1 − α 2 − α − β c (x) | u | − α | v | 1 − β in Ω , (− Δ) s v = μ b (x) | v | q − 2 v + 1 − β 2 − α − β c (x) | u | 1 − α | v | − β in  Ω , u = v = 0 in ℝ N ∖ Ω , where Ω is a bounded domain in ℝ n with smooth boundary ∂ Ω , N > 2 s , s ∈ (0 , 1) , 0 < α < 1 , 0 < β < 1 , 1 < q < 2 < 2 s ∗ , 2 s ∗ = 2 N N − 2 s is the fractional Sobolev exponent, λ , μ are two parameters, a , b , c ∈ C (Ω ¯) are nonnegative weight functions, and (− Δ) s is the fractional Laplace operator. We use the Nehari manifold approach and some variational techniques in order to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter λ and μ. [ABSTRACT FROM AUTHOR]

The purpose of this work is to study the following singular problem: ( P λ) − div (a (x , ∇ u)) = u − α + λ u q in Ω ; u > 0 , in Ω ; u = 0 , in ℝ n ∖ Ω ; where Ω ⊂ ℝ N , N ≥ 2 be a bounded smooth domain, λ is a positive parameter, p ≥ 2 such that N ≥ p and 0 < α ≤ 1 , p − 1 < q ≤ p ∗ − 1 , where p ∗ = N p N − p. We employ the Nehari manifold approach and the fibering maps in order to show the existence of T q , α such that for all λ ∈ (0 , T q , α) , problem (P λ) has at least two solutions. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Abstract In this paper, we study the existence of multiple positive solutions for boundary value problems of high-order Riemann–Liouville fractional differential equations involving the p-Laplacian operator. Not only new existence conclusions of two positive solutions are obtained by employing functional-type cone expansion-compression fixed point theorem, but also some sufficient conditions for existence of at least three positive solutions are established by applying the Leggett–Williams fixed point theorem. In addition, we demonstrate the effectiveness of the main result by using an example.

Abstract We consider the combined effect of concave–convex nonlinearities on the number of solutions for an indefinite truncated Kirchhoff-type system involving the weight functions. When α + β < 4 $\alpha+ \beta

Abstract We prove the existence of multiple positive solutions of fractional Laplace problems with critical growth by using the method of monotonic iteration and variational methods.

Abstract This paper is focused on researching a class of mixed fractional differential system with p-Laplacian operators. Based on the properties of the corresponding Green’s function, different combinations of superlinearity or sublinearity for the nonlinearities and other appropriate conditions, the existence of multiple positive solutions are derived via the Guo–Krasnosel’skii fixed point theorem. An example is then given to illustrate the usability of the main results.

Abstract We establish the existence of positive solutions to a class of a singular nonlinear Hadamard-type fractional differential equations with infinite-point boundary conditions (BCs) or integral BCs. Our analysis is based on Leray–Schauder type continuation. Several examples are given to illustrate our results.

In this paper we investigate the existence, multiplicity and asymptotic behavior of positive solution for the nonlocal nonlinear Schrödinger equations. We exploiting the relationship between the Nehari manifold and eigenvalue problems to discuss how the Nehari manifold changes as parameters μ, λ changes and show how existence, multiplicity and asymptotic results for positive solutions of the equation are linked to properties of the manifold.

In this article, we study the existence of multiple positive solutions for Schrodinger-Poisson systems involving concave-convex nonlinearities and sign-changing weight potentials. With the help of Nehari manifold and Ljusternik-Schnirelmann category theory, we investigate how the coefficient g(x) of the critical nonlinearity affects the number of positive solutions. Furthermore, we obtain a relationship between the number of positive solutions and the topology of the global maximum set of g.

Abstract We prove the existence of multiple positive solutions for a fractional Laplace problem with critical growth and sign-changing weight in non-contractible domains.

In this article we prove a three solution type theorem for the following boundary value problem:{−M λ,Λ+ u = 0 (D2u) = f(u) in Ω, on ∂Ω, where Ω is a bounded smooth domain in RN and f : [0, ∞] → [0, ∞] is a Cα function. This is motivated by the work of Amann [3] and Shivaji [27], where a three solutions theorem has been established for the Laplace operator. Furthermore, using this result we show the existence of three positive solutions to above boundary value by explicitly constructing two ordered pairs of sub and supersolutions when f has a sublinear growth and f(0) = 0. [ABSTRACT FROM AUTHOR]

In this paper, we study the following nonlocal problem: { − (a − b ∫ Ω | ∇ u | 2 d x) Δ u = λ | u | q − 2 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where Ω is a smooth bounded domain in R N with N ≥ 3 , a , b > 0 , 1 < q < 2 and λ > 0 is a parameter. By virtue of the variational method and Nehari manifold, we prove the existence of multiple positive solutions for the nonlocal problem. As a co-product of our arguments, we also obtain the blow-up and the asymptotic behavior of these solutions as b ↘ 0 . [ABSTRACT FROM AUTHOR]