Multiple Positive Solutions Research Articles

Multiple positive solutions to critical p-Laplacian equations with vanishing potential

Multiple positive solutions to critical p-Laplacian equations with vanishing potential

System of fractional boundary value problem with p-Laplacian and advanced arguments

In this paper, we discuss the existence and multiplicity of positive solutions for a system of fractional differential equations with boundary condition and advanced arguments. The existence result is proved via Leray–Schauder’s fixed point theorem type in a vector Banach space. Further, by using a new fixed point theorem in order Banach space, we study the multiplicity of positive solutions. Finally, some examples are given to illustrate our results.

Multiple positive solutions for degenerate Kirchhoff equations with singular and Choquard nonlinearity

In this paper we study the existence, multiplicity, and regularity of positive weak solutions for the following Kirchhoff–Choquard problem: where Ω is open bounded domain of with C2 boundary, N > 2s and s ∈ (0, 1). M models Kirchhoff‐type coefficient in particular, the degenerate case where Kirchhoff coefficient M is zero at zero. (− Δ)s is fractional Laplace operator, λ > 0 is a real parameter, γ ∈ (0, 1) and is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality. We prove that each positive weak solution is bounded and satisfy Hölder regularity of order s. Furthermore, using the variational methods and truncation arguments, we prove the existence of two positive solutions.

Multiple Positive Solutions of Second-Order Nonlinear Difference Systems with Repulsive Singularities

We study the existence of positive solutions for second-order nonlinear repulsive singular difference systems with periodic boundary conditions. Our nonlinearity may be singular in its dependent variable. The proof of the main result relies on a fixed point theorem in cones and a nonlinear alternative principle of Leray-Schauder; the result is applicable to the case of a weak singularity as well as the case of a strong singularity. An example is given; some recent results in the literature are improved and generalized.

New multiplicity of positive solutions for some class of nonlocal problems

In this paper, we study the following nonlocal problem: {−(a−b∫Ω|∇u|2dx)Δu=λ|u|q−2u,x∈Ω,u=0,x∈∂Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} - (a-b \\int _{\\Omega } \\vert \\nabla u \\vert ^{2}\\,dx ) \\Delta u= \\lambda \\vert u \\vert ^{q-2}u, & x\\in \\Omega , \\\\ u=0, & x\\in \\partial \\Omega , \\end{cases} $$\\end{document} where Ω is a smooth bounded domain in mathbb{R}^{N} with Nge 3, a,b>0, 1< q<2 and 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 bsearrow 0.

Σ-Shaped Bifurcation Curves

AbstractWe study positive solutions to the steady state reaction diffusion equation of the form:−Δu=λf(u); Ω∂u∂η+λu=0; ∂Ω$$\begin{array}{} \displaystyle \left\lbrace \begin{matrix} -{\it\Delta} u =\lambda f(u);~ {\it\Omega} \\ \frac{\partial u}{\partial \eta}+ \sqrt{\lambda} u=0;~\partial {\it\Omega}\end{matrix} \right. \end{array}$$whereλ&gt; 0 is a positive parameter,Ωis a bounded domain in ℝNwhenN&gt; 1 (with smooth boundary∂ Ω) orΩ= (0, 1), and∂u∂η$\begin{array}{} \displaystyle \frac{\partial u}{\partial \eta} \end{array}$is the outward normal derivative ofu. Heref(s) =ms+g(s) wherem≥ 0 (constant) andg∈C2[0,r) ∩C[0, ∞) for somer&gt; 0. Further, we assume thatgis increasing, sublinear at infinity,g(0) = 0,g′(0) = 1 andg″(0) &gt; 0. In particular, we discuss the existence of multiple positive solutions for certain ranges ofλleading to the occurrence ofΣ-shaped bifurcation diagrams. We establish our multiplicity results via the method of sub-supersolutions.

A system of nonlinear fractional BVPs with ϕ-Laplacian operators and nonlocal conditions

This work investigates the existence of multiple positive solutions for a system of two nonlinear higher-order fractional differential equations with ϕ-Laplacian operators and nonlocal conditions. A degenerate nonlinearity which obeys some general growth conditions is considered. The singularities are dealt with by approximating the fixed point operator. New existence results are presented by using the fixed point index theory. Examples of applications illustrate the theoretical results.

Existence of multiple positive solutions for fractional Laplace problems with critical growth

We prove the existence of multiple positive solutions of fractional Laplace problems with critical growth, we consider the concave power case or the convex power case. We establish the relationship between the number of the local maximum points of the coefficient function of the critical nonlinearity and the number of the positive solutions of the equation&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2021/23/abstr.html

Multiple positive solutions for four-point boundary value problem of fractional delay differential equations with p-Laplacian operator

In this work, a class of fractional delay differential equations with four-point boundary condition and p-Laplacian operator is discussed. Based on the Avery-Peterson theorem, the existence of at least triple positive solutions is derived. A simple example is given to show the validity of the conditions of our main theorem.

New multiple positive solutions for Hadamard-type fractional differential equations with nonlocal conditions on an infinite interval

This paper is concerned with the existence of multiple positive solutions for a nonlinear Hadamard-type fractional differential equation with nonlocal boundary conditions on an infinite interval. We first transform the given problem into an equivalent fixed point problem on cone, and then solve it by using the generalized Avery–Henderson fixed point theorem. Finally, an example is constructed to illustrate the application of the obtained results.

Existence and multiplicity of positive solutions to fractional Laplacian systems with combined critical Sobolev terms

We deal with the existence and multiplicity of solutions for a non-local fractional system involving combined critical non-linearities. Some existence results are obtained by splitting the Nehari manifold and applying the Mountain–Pass theorem of Ambrosetti and Rabinowitz.

Multiple positive solutions for a class of Kirchhoff equation on bounded domain

ABSTRACT In this paper, we study the existence of multiple positive solutions of the following Kirchhoff equation where Ω is a regular bounded domain of We obtain the existence of positive solutions by using the variational methods and Lusternik–Schnirelmann theory.

Multiple positive solutions for an elliptic problem involving a critical Sobolev exponent

In this work, we study the multiple results of positive solutions for a class of elliptic problem involving a critical Sobolev exponent 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.

S-Shaped Connected Component of Positive Solutions for a Minkowski-Curvature Dirichlet Problem with Indefinite Weight

In this paper, we investigate the existence of an S-shaped connected component in the set of positive solutions for a Minkowski-curvature Dirichlet problem with indefinite weight. By figuring the shape of unbounded continua of solutions, we show the existence and multiplicity of positive solutions with respect to the parameter $$\lambda $$ . In particular, we obtain the existence of at least three positive solutions for $$\lambda $$ being in a certain interval.

Positive solutions for a critical elliptic problem involving singular nonlinearity

We study the existence of positive solutions for a semi-linear elliptic equation involving both critical growth and singular nonlinearity. By applying the Nehari manifold and Lusternik-Schnirelmann category theory to an auxiliary problem, we prove the existence of multiple positive solutions.

Multiple positive solutions for critical elliptic problem with singularity

This paper investigates the following critical elliptic problem with singularity $$\begin{aligned} \left\{ \begin{array}{lll} -\Delta u=\lambda f(x)|u|^{-\gamma }+Q(x)|u|^{4}u, &{}&{}\quad x\in {\mathbb {R}}^3,\\ u>0,&{}&{}\quad x\in {\mathbb {R}}^3, \end{array}\right. \end{aligned}$$ where $$0<\gamma <1$$ , $$\lambda >0$$ , $$f\in L^{\frac{6}{5+\gamma }}({\mathbb {R}}^3)$$ is a positive function and $$0<Q(x)\in C({\mathbb {R}}^3)$$ . By variational method and Nehari type constraint, we establish a relation between the number of maxima of Q(x) and the number of positive solutions.

Non-local second-order boundary value problems with derivative-dependent nonlinearity.

We prove the existence of multiple positive solutions of nonlinear second-order nonlocal boundary value problems with nonlinear term having derivative dependence. We allow the nonlinearity to grow quadratically with respect to derivatives. We obtain a priori bounds for norms of derivatives by using a recently obtained Gronwall-type inequality. Three examples illustrate some of the results. This article is part of the theme issue 'Topological degree and fixed point theories in differential and difference equations'.

Multiple positive solutions for nonlinear fractional qq-difference equation with pp-Laplacian operator

In this paper, we investigate a class of four-point boundary value problems of fractional $q$-difference equation with $p$-Laplacian operator which is the first time to be studied and is extended from a bending elastic beam equation. By Avery-Peterson theorem and the method of lower and upper solutions associated with monotone iterative technique, we obtain some sufficient conditions for the existence of multiple positive solutions. As applications, examples are presented to illustrate the main results.

Multiple positive solutions for a logarithmic Schrödinger–Poisson system with singular nonelinearity

In this article, we devote ourselves to investigate the following logarithmic Schrödinger–Poisson systems with singular nonlinearity { − Δ u + ϕ u = | u | p−2 u log ⁡ | u | + λ u γ , i n Ω , − Δ ϕ = u 2 , i n Ω , u = ϕ = 0 , o n ∂ Ω , where Ω is a smooth bounded domain with boundary 0 &lt; γ &lt; 1 , p ∈ ( 4 , 6 ) and λ &gt; 0 is a real parameter. By using the critical point theory for nonsmooth functional and variational method, the existence and multiplicity of positive solutions are established.

Multiple positive bound state solutions for a critical Choquard equation

<p style='text-indent:20px;'>In this paper we consider the problem <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ (P_{\lambda})\ \ \ \ \ \ \left\{ \begin{array}{rcl} -\Delta u+V_{\lambda}(x)u = (I_{\mu}*|u|^{2^{*}_{\mu}})|u|^{2^{*}_{\mu}-2}u \ \ \mbox{in} \ \ \mathbb{R}^{N}, \\ u&gt;0 \ \ \mbox{in} \ \ \mathbb{R}^{N}, \end{array} \right. $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ V_{\lambda} = \lambda+V_{0} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ \lambda \geq 0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ V_0\in L^{N/2}({\mathbb{R}}^N) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ I_{\mu} = \frac{1}{|x|^\mu} $\end{document}</tex-math></inline-formula> is the Riesz potential with <inline-formula><tex-math id="M5">\begin{document}$ 0&lt;\mu&lt;\min\{N, 4\} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ 2^{*}_{\mu} = \frac{2N-\mu}{N-2} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M7">\begin{document}$ N\geq 3 $\end{document}</tex-math></inline-formula>. Under some smallness assumption on <inline-formula><tex-math id="M8">\begin{document}$ V_0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> we prove the existence of two positive solutions of <inline-formula><tex-math id="M10">\begin{document}$ (P_\lambda) $\end{document}</tex-math></inline-formula>. In order to prove the main results, we used variational methods combined with degree theory.