Multiple Positive Solutions Research Articles

Multiple positive solutions for singular elliptic problems involving concave-convex nonlinearities and sign-changing potential

In this paper, we are interested in considering the following singular elliptic problem with concaveconvex nonlinearities $$\left\{ {\begin{array}{*{20}{l}} { - \Delta u - \frac{\mu }{{|x{|^2}}}u = f(x)|u{|^{p - 2}}u + g(x)|u{|^{q - 2}}u,}&{in\;\;\Omega \backslash \{ 0\} ,} \\ {u = 0,}&{on\;\;2\Omega ,} \end{array}} \right.$$ where Ω ⊂ ℝN(N ≥ 3) is a smooth bounded domain with 0 ∈ Ω, $$0 < \mu < \bar\mu = \frac{{{{(N - 2)}^2}}}{4}$$, 1 < q < 2 < p < 2* and $$2* = \frac{{2N}}{{N - 2}}$$ is the Sobolev critical exponent, the coefficient functions f, g may change sign on Ω. By the Nehari method, we obtain two solutions, and one of them is a ground state solution. Under some stronger conditions, we point that the two solutions are positive solutions by the strong maximum principle.

Multiple Positive Solutions for the Fractional Schrödinger–Poisson Systems Involving Singular Terms

In this paper, we study the existence of multiple positive solutions for a class of fractional Schrodinger–Poisson systems involving singular terms and critical nonlinearities. Using the Nehari manifold method, we obtain a multiplicity result for them.

Multiple Positive Solutions for a System of Nonlinear Caputo-Type Fractional Differential Equations

By using fixed-point index theory, we consider the existence of multiple positive solutions for a system of nonlinear Caputo-type fractional differential equations with the Riemann-Stieltjes boundary conditions.

Multiple positive solutions for biharmonic equation of Kirchhoff type involving concave-convex nonlinearities

In this article, we study the multiplicity of positive solutions for the biharmonic equation of Kirchhoff type involving concave-convex nonlinearities, $$ \Delta^2u-\Big(a+b\int_{\mathbb{R}^N}|\nabla u|^2dx\Big)\Delta u+V(x)u =\lambda f_1(x)|u|^{q-2}u+f_2(x)|u|^{p-2}u. $$ Using the Nehari manifold, Ekeland variational principle, and the theory of Lagrange multipliers, we prove that there are at least two positive solutions, one of which is a positive ground state solution. For more information see https://ejde.math.txstate.edu/Volumes/2020/44/abstr.html

Global continuum and multiple positive solutions to one-dimensional p-Laplacian boundary value problem

We show the global structure of the set of positive solutions of a discrete Dirichlet problem involving the p-Laplacian difference operator suggesting suitable conditions on the weight function and nonlinearity. We obtain existence and multiplicity of positive solutions for λ lying in various intervals in mathbb {R} by using the directions of a bifurcation and the Picone-type identity for discrete p-Laplacian operators.

Eigenvalue Criteria for Existence of Positive Solutions to Fractional Boundary Value Problem

The existence and multiplicity of positive solutions for the nonlinear fractional differential equation boundary value problem (BVP) DC0+αyx+fx,yx=0, 0&lt;x&lt;1, y0=y′1=y″0=0 is established, where 2&lt;α≤3, CD0+α is the Caputo fractional derivative, and f:0,1×0,∞⟶0,∞ is a continuous function. The conclusion relies on the fixed-point index theory and the Leray-Schauder degree theory. The growth conditions of the nonlinearity with respect to the first eigenvalue of the related linear operator is given to guarantee the existence and multiplicity.

Multiplicity of Positive Solutions for a Nonlocal Elliptic Problem Involving Critical Sobolev-Hardy Exponents and Concave-Convex Nonlinearities

In this article, we study the following critical problem involving the fractional Laplacian: $$\begin{cases}(-\Delta)^{\frac{\alpha}{2}}u-\gamma\frac{u}{|x|^\alpha}=\lambda\frac{|u|^{q-2}}{|x|^s}+\frac{u^{2_\alpha^*(t)-2}u}{|x|^t} & in\;\Omega, \\u=0 & in\;\mathbb{R}^N\backslash\Omega,\end{cases}$$ where Ω ⊂ ℝN (N > α) is a bounded smooth domain containing the origin, α ∈ (0,2), 0 ≤ s, t 0, $$2_\alpha^*(t)=\frac{2(N-t)}{N-\alpha}$$ is the fractional critical Sobolev-Hardy exponent, 0 ≤ γ < γH, and γH is the sharp constant of the Sobolev-Hardy inequality. We deal with the existence of multiple solutions for the above problem by means of variational methods and analytic techniques.

Positive Solutions for Elliptic Problems with the Nonlinearity Containing Singularity and Hardy-Sobolev Exponents

In this paper, we study multiplicity of positive solutions for a class of semilinear elliptic equations with the nonlinearity containing singularity and Hardy-Sobolev exponents. Using variational methods, we establish the existence and multiplicity of positive solutions for the problem.

Existence of multiple positive periodic solutions for discrete hematopoiesis models with a unimodal production function

This paper is devoted to elucidating a sufficient condition under which Mackey-Glass type discrete hematopoiesis models have at least two positive periodic solutions. This model has periodic coefficients and time delays, and includes several production function terms that act as feedback. Our result is obtained by applying the Krasnosel’skii fixed point theorem and is represented by a relationship between period, coefficients and the production function. Example and its simulations are attached to show how to apply our result. In this example, there are exactly two positive 3-periodic solutions in the hematopoiesis model. Simulation shows that one periodic solution is stable and the other is unstable. It also shows that our result can be improved by weakening assumptions about the production function.

Evolutionary behavior of rational wave solutions to the (4 + 1)-dimensional Boiti–Leon–Manna–Pempinelli equation

A nonlinear integrable model known as the (4 + 1)-dimensional Boiti–Leon–Manna–Pempinelli (4D-BLMP) equation is studied in the present paper. To this end, by considering the Hirota bilinear form of the model and utilizing the linear superposition method (LSM) along with symbolic computations, a group of rational wave solutions including multiple wave and positive (non-singular) compelexiton solutions is formally derived. The dynamical behavior of the solutions is also analyzed graphically by considering the special values of the involved parameters. The results of the current work reveal the existence of different wave structures to the 4D-BLMP equation and distinguish it from other models that do not possess non-singular compelexiton solutions.

Existence of Positive Solutions to Singular φ-Laplacian Nonlocal Boundary Value Problems when φ is a Sup-multiplicative-like Function

In this paper, using a fixed point index theorem on a cone, we present some existence results for one or multiple positive solutions to φ -Laplacian nonlocal boundary value problems when φ is a sup-multiplicative-like function and the nonlinearity may not satisfy the L 1 -Carath e ´ odory condition.

On a Schr\xf6dinger\u2013Poisson system with singularity and critical nonlinearities

In this paper, we study the Schrödinger–Poisson system with singularity and critical growth terms. By means of variational methods with an appropriate truncation argument, the existence and multiplicity of positive solutions are obtained.

Multiple positive solutions for nonlocal elliptic problems involving the Hardy potential and concave–convex nonlinearities

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave–convex nonlinearities: [Formula: see text] where [Formula: see text] is a smooth bounded domain in [Formula: see text] containing [Formula: see text] in its interior, and [Formula: see text] with [Formula: see text] which may change sign in [Formula: see text]. We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for [Formula: see text] sufficiently small. The variational approach requires that [Formula: see text] [Formula: see text] [Formula: see text], and [Formula: see text], the latter being the best fractional Hardy constant on [Formula: see text].

Multiplicity of positive solutions for Hadamard fractional differential equations with p-Laplacian operator

In this paper, we investigate the multiplicity results of some positive solutions for a system of Hadamard fractional differential equations with parameters and p-Laplacian operator subject to three-point boundary conditions which contains fractional derivatives. The proofs of our main result, multiplicity of positive solutions, are derived in terms of different values of parameters by using Guo–Krasnosel’skii’s fixed point theorem.

Multiple positive solutions for nonhomogeneous Schrödinger–Poisson system in ℝ3*

In this paper, we study the multiplicity of solutions for a class of nonhomogeneous Schrodinger–Poisson systems under general superlinear conditions at infinity. With the aid of Ekeland’s variational principle, Jeanjean’s monotone method, Pohožaev’s identity, and the mountain pass theorem, we prove that a Schrodinger–Poisson system has at least two positive solutions, which generalizes and improves some recent results in the literature.

Existence of multiple positive solutions for a truncated Kirchhoff-type system involving weight functions and concave\u2013convex nonlinearities

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 alpha+ beta<4, since the concave-convex nonlinearities do not satisfy the mountain pass geometry, it is difficult to obtain a bounded Palais–Smale sequence by the usual mountain pass theorem. To overcome the problem, we properly introduce a method of Nehari manifold and then establish the existence of multiple positive solutions when the pair of the parameters is under a certain range.

Multiple positive solutions for nonlinear high-order Riemann\u2013Liouville fractional differential equations boundary value problems with p-Laplacian operator

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.

Existence and bifurcation of nontrivial solutions for the coupled nonlinear Schrödinger–Korteweg–de Vries system

In the present paper, we study the existence and bifurcation of nontrivial solutions of the nonlinear Schrodinger–Korteweg–de Vries (NLS–KdV) and Schrodinger–Korteweg–de Vries–Korteweg–de Vries (NLS–KdV–KdV) systems which arise from fluid mechanics. On the one hand, for both the three-wave system and the two-wave system, the existence/nonexistence, continuous dependence and asymptotic behavior of positive ground state solutions are established. On the other hand, multiple positive solutions are found via a combination of Nehari manifold and bifurcation methods for the attractive interaction case, which has not been found for the conventional nonlinear Schrodinger systems with cubic nonlinearity.

Multiplicity of Weak Positive Solutions for Fractional p &amp; q Laplacian Problem with Singular Nonlinearity

In this paper, we prove the existence and multiplicity of positive solutions for a class of fractional p &amp; q Laplacian problem with singular nonlinearity. Our approach relies on the variational method, some analysis techniques, and the method of Nehari manifold.

Existence and Multiplicity of Positive Solutions for Kirchhoff-Type Equations with the Critical Sobolev Exponent

In this paper, we consider the following Kirchhoff-type problems involving critical exponent −a+b∫Ω∇u2dxΔu+Vxu=μu2∗−1+λgx,u, x∈Ωu&gt;0, x∈Ωu=0, x∈∂Ω. The existence and multiplicity of positive solutions for Kirchhoff-type equations with a nonlinearity in the critical growth are studied under some suitable assumptions on Vx and gx,u. By using the mountain pass theorem and Brézis–Lieb lemma, the existence and multiplicity of positive solutions are obtained.