Existence Of Positive Solutions Research Articles

Existence of positive solutions for fractional Laplacian systems with critical growth

Existence of positive solutions for fractional Laplacian systems with critical growth

Positive Solutions for a Class of Integral Boundary Value Problem of Fractional q-Difference Equations

This paper studies a class of integral boundary value problem of fractional q-difference equations. We first give an explicit expression for the associated Green’s function and obtain an important property of the function. The new property allows us to prove sufficient conditions for the existence of positive solutions based on the associated parameter. The results are derived from the application of a fixed point theorem on order intervals.

Existence of Positive Solutions for the Nonlinear Kirchhoff Type Equations in $${\mathbb {R}}^3$$

Existence of Positive Solutions for the Nonlinear Kirchhoff Type Equations in $${\mathbb {R}}^3$$

Existence of Positive Solution for Second-Order Singular Semipositone Difference Equation with Nonlinear Boundary Conditions

Existence of Positive Solution for Second-Order Singular Semipositone Difference Equation with Nonlinear Boundary Conditions

Positive Solutions for a High-Order Riemann-Liouville Type Fractional Integral Boundary Value Problem Involving Fractional Derivatives

In this paper, under some super- and sub-linear growth conditions, we study the existence of positive solutions for a high-order Riemann–Liouville type fractional integral boundary value problem involving fractional derivatives. Our analysis methods are based on the fixed point index and nonsymmetric property of the Green function. Additionally, we provide some valid examples to illustrate our main results.

Positive solutions for a planar Schrödinger–Poisson system with prescribed mass

In this paper, we investigate a planar Schrödinger–Poisson system with prescribed mass on a bounded domain. Under certain assumptions, we prove the existence of positive solutions to the above system by using a fixed point theorem in Carl and Heikkilä (2002). In addition, by employing Moser’s iterative technique, we establish the L∞-bounds on the positive solution of the system.

A class of piecewise fractional functional differential equations with impulsive

Abstract In this paper, we study a class of piecewise fractional functional differential equations with impulsive and integral boundary conditions. By using Schauder fixed point theorem and contraction mapping principle, the results for existence and uniqueness of solutions for the piecewise fractional functional differential equations are established. And by using cone stretching and cone contraction fixed point theorems in norm form, the existence of positive solutions for the equations are also obtained. Finally, an example is given to illustrate the effectiveness of the conclusion.

On the generalised Brézis–Nirenberg problem

For $$ p \in (1,N)$$ and a domain $$\Omega $$ in $$\mathbb {R}^N$$ , we study the following quasi-linear problem involving the critical growth: $$\begin{aligned} -\Delta _p u - \mu g|u|^{p-2}u = |u|^{p^{*}-2}u \ \text{ in } \mathcal {D}_p(\Omega ), \end{aligned}$$ where $$\Delta _p$$ is the p-Laplace operator defined as $$\Delta _p(u) = \text {div}(|{\nabla u}|^{p-2} \nabla u),$$ $$p^{*}= \frac{Np}{N-p}$$ is the critical Sobolev exponent and $$\mathcal {D}_p(\Omega )$$ is the Beppo-Levi space defined as the completion of $$\text {C}_c^{\infty }(\Omega )$$ with respect to the norm $$\Vert u\Vert _{\mathcal {D}_p}:= \left[ \displaystyle \int _{\Omega } |\nabla u|^p \mathrm{d}x\right] ^ \frac{1}{p}.$$ In this article, we provide various sufficient conditions on g and $$\Omega $$ so that the above problem admits a positive solution for certain range of $$\mu $$ . As a consequence, for $$N \ge p^2$$ , if g is such that $$g^+ \ne 0$$ and the map $$u \mapsto \displaystyle \int _{\Omega } |g||u|^p \mathrm{d}x$$ is compact on $$\mathcal {D}_p(\Omega )$$ , we show that the problem under consideration has a positive solution for certain range of $$\mu $$ . Further, for $$\Omega =\mathbb {R}^N$$ , we give a necessary condition for the existence of positive solution.

Systems of Riemann–Liouville Fractional Differential Equations with ρ-Laplacian Operators and Nonlocal Coupled Boundary Conditions

In this paper, we study the existence of positive solutions for a system of fractional differential equations with ρ-Laplacian operators, Riemann–Liouville derivatives of diverse orders and general nonlinearities which depend on several fractional integrals of differing orders, supplemented with nonlocal coupled boundary conditions containing Riemann–Stieltjes integrals and varied fractional derivatives. The nonlinearities from the system are continuous nonnegative functions and they can be singular in the time variable. We write equivalently this problem as a system of integral equations, and then we associate an operator for which we are looking for its fixed points. The main results are based on the Guo–Krasnosel’skii fixed point theorem of cone expansion and compression of norm type.

Existence and asymptotic behavior of positive solutions for a class of locally superlinear Schrödinger equation

This paper treats the existence of positive solutions of $-\Delta u + V(x) u = \lambda f(u)$ in $\mathbb{R}^N$. Here $N \geq 1$, $\lambda > 0$ is a parameter and $f(u)$ satisfies conditions only in a neighborhood of $u=0$. We shall show the existence of positive solutions with potential of trapping type or $\mathcal{G}$-symmetric potential where $\mathcal{G} \subset O(N)$. Our results extend previous results as well as we also study the asymptotic behavior of a family $(u_\lambda)_{\lambda \geq \lambda_0}$ of positive solutions as $\lambda \to \infty$.

Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function

Abstract In this article, we consider a discrete nonlinear third-order boundary value problem Δ 3 u ( k − 1 ) = λ a ( k ) f ( k , u ( k ) ) , k ∈ [ 1 , N − 2 ] Z , Δ 2 u ( η ) = α Δ u ( N − 1 ) , Δ u ( 0 ) = − β u ( 0 ) , u ( N ) = 0 , \left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}{\Delta }^{3}u\left(k-1)=\lambda a\left(k)f\left(k,u\left(k)),\hspace{1em}k\in {\left[1,N-2]}_{{\mathbb{Z}}},\hspace{1.0em}\\ {\Delta }^{2}u\left(\eta )=\alpha \Delta u\left(N-1),\Delta u\left(0)=-\beta u\left(0),\hspace{1em}u\left(N)=0,\hspace{1.0em}\end{array}\right. where N > 4 N\gt 4 is an integer, λ > 0 \lambda \gt 0 is a parameter. f : [ 1 , N − 2 ] Z × [ 0 , + ∞ ) → [ 0 , + ∞ ) f:{\left[1,N-2]}_{{\mathbb{Z}}}\times \left[0,+\infty )\to \left[0,+\infty ) is continuous, a : [ 1 , N − 2 ] Z → ( 0 , + ∞ ) a:{\left[1,N-2]}_{{\mathbb{Z}}}\to \left(0,+\infty ) , α ∈ 0 , 1 N − 1 \alpha \in \left[0,\frac{1}{N-1}\right) , β ∈ 0 , 2 ( 1 − α ( N − 1 ) ) N ( 2 − α ( N − 1 ) ) \beta \in \left[0,\frac{2\left(1-\alpha \left(N-1))}{N\left(2-\alpha \left(N-1))}\right) , and η ∈ N − 2 2 + 1 , N − 2 Z \eta \in {\left[\left[\frac{N-2}{2}\right]+1,N-2\right]}_{{\mathbb{Z}}} . With the sign-changing Green’s function, we obtain not only the existence of positive solutions but also the multiplicity of positive solutions to this problem.

Existence and concentration of positive solutions for a fractional Schrödinger logarithmic equation

ABSTRACT In this paper, we study the existence and concentration of positive solutions for the following fractional Schrödinger logarithmic equation: { ε 2 s ( − Δ ) s u + V ( x ) u = u log ⁡ u 2 , x ∈ R N , u ∈ H s ( R N ) , where 0 $ ]]> ε > 0 is a small parameter, 2s, $ ]]> N > 2 s , s ∈ ( 0 , 1 ) , ( − Δ ) s is the fractional Laplacian, the potential V is a continuous function having a global minimum. Using variational method to modify the nonlinearity with the sum of a C 1 functional and a convex lower semicontinuous functional, we prove the existence of positive solutions and concentration around of a minimum point of V when ε tends to zero.

The existence of positive solutions to the Choquard equation with critical exponent and logarithmic term

We investigate the existence of solutions for the following nonlinear critical Choquard equation with the logarithmic term:{−Δu=(∫Ω|u(y)|2μ⁎|x−y|μdy)|u|2μ⁎−2u+βu+λulog⁡u2,x∈Ω,u=0,x∈∂Ω, where Ω is a bounded domain of RN with smooth boundary. λ, β are real parameters and 2μ⁎=2N−μN−2(N≥3) is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. We will show that there exists at last one positive solution to the above problem under some appropriate assumptions of λ and β.

Existence of vortices for Schrödinger equations with logarithmic and saturable nonlinearity

In this paper, we study the existence of stationary vortex wave solutions of two kinds of nonlinear Schrödinger equations. For the first one, which is equipped with logarithmic nonlinearity arising from Bose–Einstein condensation, we consider two types of boundary value problems. In both cases, we establish the existence of positive solutions through a direct minimization method. For the second one, with a saturable nonlinearity originating from geometric optics, we use a constrained minimization approach to establish the existence of vortex wave solutions. Moreover, some explicit estimates for the bound of the wave propagation constant are derived.

Supercritical elliptic problems on nonradial domains via a nonsmooth variational approach

In this paper we are interested in positive classical solutions of(1){−Δu=a(x)up−1 in Ω,u>0 in Ω,u=0 on ∂Ω, where Ω is a bounded annular domain (not necessarily an annulus) in RN(N≥3) and a(x) is a nonnegative continuous function. We show the existence of a classical positive solution for a range of supercritical values of p when the problem enjoys certain mild symmetry and monotonicity conditions. As a consequence of our results, we shall show that (1) has ⌊N2⌋ (the floor of N2) positive nonradial solutions when a(x)=1 and Ω is an annulus with certain assumptions on the radii. We also obtain the existence of positive solutions in the case of toroidal domains. Our approach is based on a new variational principle that allows one to deal with supercritical problems variationally by limiting the corresponding functional on a proper convex subset instead of the whole space at the expense of a mild invariance property.

A class of supercritical Sobolev type inequalities with logarithm and related elliptic equations

In this paper, we first deduce the following Sobolev inequality with logarithmic term:(0.1)sup⁡{∫B|u|2⁎|ln⁡(τ+|u|)||x|βdx:u∈H0,rad1(B),‖∇u‖L2(B)=1}<∞, where β>0, τ≥0 are constants, B is the unit ball in RN, N≥3, and 2⁎=2N/(N−2) is the critical Sobolev exponent. Then we show that the supremum in (0.1) is attained when 0<β<min⁡{N/2,N−2} and 1≤τ<∞. The inequality (0.1) can be used to prove the existence of positive solution for the following supercritical problem:(0.2){−Δu=u2⁎−1(ln⁡(τ+u))|x|β+g(|x|,u),u>0inB,u=0on∂B, where g(r,u)∈C([0,1)×R) is a subcritical perturbation. As a consequence, we can deduce the existence of positive solution for the supercritical problem with non-power nonlinearity:(0.3){−Δu=u2⁎−1(ln⁡(τ+u))|x|β,u>0inB,u=0on∂B. This is somewhat surprising, because the problem (0.3) has no nontrivial solution by Pohozaev's identity if the variable exponent |x|β is replaced by any non-negative constant.

A fixed point index approach to Krasnosel’skiĭ-Precup fixed point theorem in cones and applications

We present an alternative approach to the vector version of Krasnosel’skiĭ compression–expansion fixed point theorem due to Precup, which is based on the fixed point index. It allows us to obtain new general versions of this fixed point theorem and also multiplicity results. We emphasize that all of them are coexistence fixed point theorems for operator systems, that means that every component of the fixed points obtained is non-trivial. Finally, these coexistence fixed point theorems are applied to obtain results concerning the existence of positive solutions for systems of Hammerstein integral equations and radially symmetric solutions of (p1,p2)-Laplacian systems.

Positive solutions for the Schrödinger–Poisson system with steep potential well

In this paper, we consider the following Schrödinger–Poisson system [Formula: see text] where [Formula: see text] are real parameters and [Formula: see text]. Suppose that [Formula: see text] represents a potential well with the bottom [Formula: see text], the system has been widely studied in the case [Formula: see text]. In contrast, no existence result of solutions is available for the case [Formula: see text] due to the presence of the nonlocal term [Formula: see text]. With the aid of the truncation technique and the parameter-dependent compactness lemma, we first prove the existence of positive solutions for [Formula: see text] large and [Formula: see text] small in the case [Formula: see text]. Then we obtain the nonexistence of nontrivial solutions for [Formula: see text] large and [Formula: see text] large in the case [Formula: see text]. Finally, we explore the decay rate of the positive solutions as [Formula: see text] as well as their asymptotic behavior as [Formula: see text] and [Formula: see text].

Existence of positive solutions for p-Laplacian boundary value problems of fractional differential equations

In this paper, we study the existence and multiplicity of ρ-concave positive solutions for a p-Laplacian boundary value problem of two-sided fractional differential equations involving generalized-Caputo fractional derivatives. Using Guo–Krasnoselskii fixed point theorem and under some additional assumptions, we prove some important results and obtain the existence of at least three solutions. To establish the results, Green functions are used to transform the considered two-sided generalized Katugampola and Caputo fractional derivatives. Finally, applications with illustrative examples are presented to show the validity and correctness of the obtained results.

A class of quasilinear elliptic equations with singular and exponential terms

We use a scheme of approximation together with Brouwer's theorem to establish the existence of positive solutions for a class of quasilinear elliptic problems. More precisely, we study the class of equations: (1) { − Div ( a ( | ∇u | p ) | ∇u | p − 2 ∇u ) = η u − γ + λ u s + f ( u ) in Ω , u = 0 on ∂Ω , where Ω ⊂ R N ( N ≥ 3 ) is a bounded domain with smooth boundary, 2 ≤ p < N , 0 < γ < 1 , 0<s<N−1, η and λ are positive parameters, a : R + → R + is a function of class C 1 ( R + ) . The nonlinear term f involves three different types of Trudinger–Moser growth: subcritical, critical or supercritical. We also study the asymptotic behavior of the solutions with respect to the parameters.