Existence Of Positive Solutions Research Articles

Existence and concentration of positive solutions to generalized Chern-Simons-Schrödinger system with critical exponential growth

We are concerned with a class of generalized Chern-Simons-Schrödinger systems{−Δu+λV(x)u+A0u+∑j=12Aj2u=f(u),∂1A2−∂2A1=−12|u|2,∂1A1+∂2A2=0,∂1A0=A2|u|2,∂2A0=−A1|u|2, where λ>0 denotes a sufficiently large parameter, V:R2→R admits a potential well Ω≜intV−1(0) and the nonlinearity f fulfills the critical exponential growth in the Trudinger-Moser sense at infinity. Under some suitable assumptions on V and f, based on variational method together with some new technical analysis, we are able to get the existence of positive solutions for some large λ>0, and the asymptotic behavior of the obtained solutions is also investigated when λ→+∞.

Existence of Positive Solutions for Singular Difference Equations with Nonlinear Boundary Conditions

In this paper, we delve into a discrete nonlinear singular semipositone problem, characterized by a nonlinear boundary condition. The nonlinearity, given by f(u)−auα with α>0, exhibits a singularity at u=0 and tends towards −∞ as u approaches 0+. By constructing some suitable auxiliary problems, the difficulty that arises from the singularity and semipositone of nonlinearity and the lack of a maximum principle is overcome. Subsequently, employing the Krasnosel’skii fixed-point theorem, we determine the parameter range that ensures the existence of at least one positive solution and the emergence of at least two positive solutions. Furthermore, based on our existence results, one can obtain the symmetry of the solutions after adding some symmetric conditions on the given functions by using a standard argument.

On the existence of a positive solution to a boundary value problem for a third-order nonlinear functional differential equation with an integral boundary condition at one of the ends of the segment

The article studies the existence of positive solutions on the segment [0,1] of a two-point boundary value problem for one nonlinear third-order functional differential equation with an integral boundary condition at one of the ends of the segment. Using the Go–Krasnoselsky fixed point theorem and some properties of the Green's function of the corresponding differential operator, sufficient conditions for the existence of at least one positive solution to the problem under consideration are obtained. An example is given to illustrate the results obtained.

On a planar equation involving [formula omitted]-Laplacian with zero mass and Trudinger–Moser nonlinearity

In this work, we study existence of positive solutions to a class of (2,q)-equations in the zero mass case in R2. We establish a weighted Sobolev embedding and we introduce a new Trudinger–Moser type inequality. Moreover, since we work on a suitable radial Sobolev space, we prove an appropriate version of the well-known Symmetric Criticality Principle by Palais. Finally, we study regularity of solutions applying Moser iteration scheme.

Existence of positive solutions for generalized fractional Brézis-Nirenberg problem

In this article, we study the fractional Brézis-Nirenberg type problem on whole domain $\R^N$ associated with the fractional $p$-Laplace operator. To be precise, we want to study the following problem: \begin{equation} \label{Pr1} (-\Delta_{p})^{s}u - \lambda w |u|^{p-2}u= |u|^{p_{s}^{*}-2}u \quad \text{in } \mathcal{D}^{s,p}\big(\mathbb{R}^{N}\big) , \tag{P} \end{equation} where $s\in (0,1), p\in (1, {N}/{s})$, $p_{s}^{*}={Np}/({N-sp})$ and the operator $(-\Delta_{p})^{s}$ is the fractional $p$-Laplace operator. The space $\mathcal{D}^{s,p}\big(\mathbb{R}^{N}\big)$ is the completion of $C_c^\infty\big(\R^N\big)$ with respect to the Gagliardo semi-norm. In this article, we prove the existence of a positive solution to problem \eqref{Pr1} by allowing the Hardy weight function $w$ to change its sign.

Existence of positive solution for a class of quasilinear Schrödinger equations with potential vanishing at infinity on nonreflexive Orlicz-Sobolev spaces

In this paper we investigate the existence of positive solution for a class of quasilinear problem on an Orlicz-Sobolev space that can be nonreflexive $$ - \Delta_{\Phi} u +V(x)\phi(|u|)u= K(x)f(u)\quad\mbox{in } \mathbb{R}^{N}, $$ where $ N \geq 2 $, $ V, K $ are nonnegative continuous functions and $f$ is a continuous function with a quasicritical growth. Here we extend the Hardy-type inequalities presented in \cite{AlvesandMarco} to nonreflexive Orlicz spaces. Through inequalities together with a variational method for non-differentiable functionals we will obtain a ground state solution. We analyze also the problem with $V=0$.

Existence of positive solutions to impulsive nonlinear differential systems of second order with two point boundary conditions

Existence of positive solutions to impulsive nonlinear differential systems of second order with two point boundary conditions

Existence of Positive Solutions for Hadamard-Type Fractional Boundary Value Problems at Resonance on an Infinite Interval

This paper investigates a class of resonance boundary value problems for Hadamard-type fractional differential equations on an infinite interval. Utilizing the Leggett-Williams norm-type theorem proposed by O’Regan and Zima, the existence of positive solutions is established. The main conclusions are illustrated with an example.

Existence of Positive Solutions of a Fractional Dynamic Equation Involving Integral Boundary Conditions on Time Scales

Existence of Positive Solutions of a Fractional Dynamic Equation Involving Integral Boundary Conditions on Time Scales

Existence of positive solutions of symmetric variational eigenvalue problems

Existence of positive solutions of symmetric variational eigenvalue problems

Existence results for a fourth order problem with functional perturbed clamped beam boundary conditions

Abstract In this paper, we study the existence of positive solutions for a fourth order boundary value problem coupled to functional perturbed clamped beam boundary conditions. Our main ingredient is the classical fixed point index. The problem investigated is an extension of other problems studied in previous papers by covering very general nonlocal perturbed conditions on the boundary.

Structure of positive solutions for a reaction-diffusion model with additional food and protection zone

The additional food and protection zone provided to predator and prey respectively are introduced into a reaction-diffusion predation model with nonlinear functional response in this paper. We mainly concentrate on the combined effects of additional food and protection zone on the existence and uniqueness of solution for the model. It is shown that, compared with the models with classical Beddington-DeAngelis functional response, the integration of additional food and protection zone into Beddington-DeAngelis functional response causes the model to produce new critical values for the existence of positive solutions. Moreover, we derive that the model permits a unique positive solution when the strength of mutual interference among predators or the quantity of additional food is large, where, the effect of quantity of additional food on the positive solution is a new finding. The results shed light on that the combination of additional food and protection zone is an effective and environmentally-friendly strategy in preserving biodiversity.

Existence of positive solutions for a class of p-Laplacian fractional differential equations with nonlocal boundary conditions

This article is devoted to proving the uniqueness of positive solutions for p-Laplacian equations with Caputo and Riemann-Liouville fractional derivative. The uniqueness result and the dependence of the solution on a parameter are established based on the fixed point point theorem of mixed monotone operators. In the end, a numerical simulation is given to verify the main results.

On positive solutions of the Cauchy problem for doubly nonlocal equations

We study the Cauchy problem for the doubly nonlocal equation where and for some m ∈ ℝ and 0 < α ≤ +∞, here α = +∞ means that J is compactly supported. The problem not only describes nonlocal diffusions and nonlocal interactions, but also it is a nonlocal counterpart of the nonlocal heat equation in [10] where and m > 0. As for the local existence of positive solutions, we find that problem (0.1) is quite different from the nonlocal heat equation. That is, problem (0.1) admits positive solutions if and only if m > n − q(n + α), however the nonlocal heat equation has positive solutions for any m ∈ ℝ. For m ≥ 0 or n(1 − q) < m < 0 with α = +∞, we determine the critical Fujita curve of problem (0.1) as F(p, q, n, m, r, α) := (p − 1)[nq − (n − m)+] − q(β − nr) = 0 with β = min{α, 2}. That is, if F ≤ 0, every positive solution of (0.1) blows up in finite time, whereas if F > 0, there exist both nonglobal solutions and global solutions. In the supercritical case F > 0, we further discuss the second critical exponent which describes the critical decay rate of initial data to distinguish both solutions. Particularly, we find a new phenomenon, namely, if with α = +∞, the second critical exponent is independent of p and r.

Effect of density-dependent diffusion on a diffusive predator–prey model in spatially heterogeneous environment

The paper presents a class of predator–prey model with density-dependent diffusion in the spatially heterogeneous environment. We first provide the global existence and boundedness of the solution for the model. Then, by taking a variable transformation, the difficulty brought by the cross-diffusion can be overcome, and the existence, stability and local bifurcation of semi-trivial steady-state solutions for the equivalent system are further studied. Finally, the existence of positive solutions of the system is also given by using the Leray–Schauder degree theory and the method of principle eigenvalue, especially for the limit cases when the diffusion coefficient tends to zero or infinite.

Existence of positive solutions for a class of singular elliptic problems with convection term and critical exponential growth

This paper uses the Galerkin method to investigate the existence of positive solution to a class of singular elliptic problems given by {−Δu=λ0uβ0+Λ0|∇u|γ0+f0(u)|x|α0+h0(x),u>0inΩ,u=0on∂Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} \ extstyle\\begin{cases} -\\Delta u= \\displaystyle \\frac {\\lambda _{0}}{u^{\\beta _{0}}} + \\Lambda _{0} |\ abla u|^{\\gamma _{0}}+ \\frac{f_{0}(u)}{|x|^{\\alpha _{0}}}+ h_{0}(x), \\ \\ u>0 \\ \\ \ ext{in} \\ \\Omega , \\\\ u=0 \\ \ ext{on} \\ \\ \\partial \\Omega , \\end{cases}\\displaystyle \\end{aligned}$$ \\end{document} where Ω⊂R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\Omega \\subset \\mathbb{R}^{2}$\\end{document} is a bounded smooth domain, 0<β0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$0<\\beta _{0}$\\end{document}, γ0≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\gamma _{0} \\leq 1$\\end{document}, α0∈[0,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\alpha _{0} \\in [0,2)$\\end{document}, h0(x)≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$h_{0}(x)\\geq 0$\\end{document}, h0≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$h_{0}\ eq 0$\\end{document}, h0∈L∞(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$h_{0}\\in L^{\\infty}(\\Omega )$\\end{document}, 0<∥h0∥∞<λ0<Λ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$0<\\|h_{0}\\|_{\\infty} < \\lambda _{0} < \\Lambda _{0}$\\end{document}, and f0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f_{0}$\\end{document} are continuous functions. More precisely, f0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f_{0}$\\end{document} has a critical exponential growth, that is, the nonlinearity behaves like exp(ϒ‾s2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\exp (\\overline{\\Upsilon}s^{2})$\\end{document} as |s|→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$|s| \ o \\infty $\\end{document}, for some ϒ‾>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\overline{\\Upsilon}>0$\\end{document}.

Neutral Emden–Fowler Differential Equation of Second Order: Oscillation Criteria of Coles Type

In this work, we study the asymptotic and oscillatory behavior of solutions to the second-order general neutral Emden–Fowler differential equation (avηxvz′v)′ + qvFxgv = 0, where v≥v0 and the corresponding function z = x + px∘h. Besides the importance of equations of the neutral type, studying the qualitative behavior of solutions to these equations is rich in analytical points and interesting issues. We begin by finding the monotonic features of positive solutions. The new properties contribute to obtaining new and improved relationships between x and z for use in studying oscillatory behavior. We present new conditions that exclude the existence of positive solutions to the examined equation, and then we establish oscillation criteria through the symmetry property between non-oscillatory solutions. We use the generalized Riccati substitution method, which enables us to apply the results to a larger area than the special cases of the considered equation. The new results essentially improve and extend previous results in the literature. We support this claim by applying the results to an example and comparing them with previous findings. Moreover, the reduction of our results to Euler’s differential equation introduces the well-known sharp oscillation criterion.

A discussion on some new boundary value problems involving the q-derivative operator via the existence of positive solutions

A discussion on some new boundary value problems involving the q-derivative operator via the existence of positive solutions

Schrödinger–Poisson systems with zero mass in the Sobolev limiting case

AbstractWe study the existence of positive solutions for a class of systems which strongly couple a quasilinear Schrödinger equation driven by a weighted ‐Laplace operator and without the mass term, and a higher‐order fractional Poisson equation. Since the system is set in , the limiting case for the Sobolev embedding, we consider nonlinearities with exponential growth. Existence is proved relying on the study of a corresponding Choquard equation in which the Riesz kernel is a logarithm, hence sign‐changing and unbounded from above and below. This is in turn solved by means of a variational approximating procedure for an auxiliary Choquard equation, where the logarithm is uniformly approximated by polynomial kernels. Our results are new even in the planar case .

Existence of positive solutions for an elliptic system with sign-changing weights

Using the method of sub-super solutions and comparison principle, we study the existence of positive solutions for a class of quasilinear elliptic systems with sign-changing weights.