Singular Nonlinearity Research Articles

Comparison Principle for Elliptic Equations with Mixed Singular Nonlinearities

We deal with existence and uniqueness of positive solutions of an elliptic boundary value problem modeled by−Δpu=fuγ+guqinΩ,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}$$ \\left \\{\\begin {array}{ll} \\displaystyle -{\\Delta }_{p} u= \\frac {f}{u^{\\gamma }} + g u^{q} & \ ext { in } {\\Omega }, \\\\ u = 0 & \ ext {on } \\partial {\\Omega }, \\end {array}\\right . $$\\end{document} where Ω is an open bounded subset of mathbb {R}^{N} where Ω is an open bounded subset of mathbb {R}^{N}, Δpu := ÷(|∇u|p− 2∇u) is the usual p-Laplacian operator, γ ≥ 0 and 0 ≤ q ≤ p − 1; f and g are nonnegative functions belonging to suitable Lebesgue spaces.

A critical fractional choquard problem involving a singular nonlinearity and a radon measure

This article concerns about the existence of a positive SOLA (Solutions Obtained as Limits of Approximations) for the following singular critical Choquard problem involving fractional power of Laplacian and a critical Hardy potential. 0.1 $$\begin{aligned} \begin{aligned} (-\Delta )^su-\alpha \frac{u}{|x|^{2s}}&=\lambda u+ u^{-\gamma }+\beta \left( \int _{\Omega }\frac{u^{2_b^*}(y)}{|x-y|^b}dy\right) u^{2_b^*-1}+\mu ~\text {in}~\Omega ,\\ u&>0~\text {in}~\Omega ,\\ u&= 0~\text {in}~\mathbb {R}^N{\setminus }\Omega . \end{aligned} \end{aligned}$$ Here, $$\Omega $$ is a bounded domain of $$\mathbb {R}^N$$ , $$s\in (0,1)$$ , $$\alpha ,\lambda $$ and $$\beta $$ are positive real parameters, $$N>2s$$ , $$\gamma \in (0,1)$$ , $$0<b<\min \{N,4s\}$$ , $$2_b^*=\frac{2N-b}{N-2s}$$ is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality and $$\mu $$ is a bounded Radon measure in $$\Omega $$ .

Existence and Hölder regularity of infinitely many solutions to a p-Kirchhoff-type problem involving a singular nonlinearity without the Ambrosetti–Rabinowitz (AR) condition

We carry out an investigation of the existence of infinitely many solutions to a fractional p-Kirchhoff-type problem with a singularity and a superlinear nonlinearity with a homogeneous Dirichlet boundary condition. Further, the solution(s) will be proved to be bounded and a weak comparison principle has also been proved. A ‘\(C^1\) versus \(W_0^{s,p}\)’ analysis has also been discussed.

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.

Positive Periodic Solutions for a Class of Strongly Coupled Differential Systems with Singular Nonlinearities

This article studies the existence of positive periodic solutions for a class of strongly coupled differential systems. By applying the fixed point theory, several existence results are established. Our main findings generalize and complement those in the literature studies.

Interval Type-2 Fuzzy Passive Filtering for Nonlinear Singularly Perturbed PDT-Switched Systems and Its Application

The problem of designing a passive filter for nonlinear switched singularly perturbed systems with parameter uncertainties is explored in this paper. Firstly, the multiple-time-scale phenomenon is settled effectively by introducing a singular perturbation parameter in the plant. Secondly, the interval type-2 fuzzy set theory is employed where parameter uncertainties are expressed in membership functions rather than the system matrices. It is worth noting that interval type-2 fuzzy sets of the devised filter are different from the plant, which makes the design of the filter more flexible. Thirdly, the persistent dwell-time switching rule, as a kind of time-dependent switching rules, is used to manage the switchings among nonlinear singularly perturbed subsystems, and this rule is more general than dwell-time and average dwell-time switching rules. Next, sufficient conditions are provided for guaranteeing that the filtering error system is globally uniformly exponentially stable with a passive performance. Furthermore, on the basis of the linear matrix inequalities, the explicit expression of the designed filter can be obtained. Finally, a tunnel diode electronic circuit is rendered as an example to confirm the correctness and the validity of the developed filter.

Existence of Solution for Infinite System of Nonlinear Singular Integral Equations and Semi-Analytic Method to Solve it

Existence of Solution for Infinite System of Nonlinear Singular Integral Equations and Semi-Analytic Method to Solve it

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.

Nonlinear elliptic equations with variable exponents involving singular nonlinearity

In this paper, we prove the existence and regularity of weak positive solutions for a class of nonlinear elliptic equations with a singular nonlinearity, lower order terms and L1 datum in the setting of Sobolev spaces with variable exponents. We will prove that the lower order term has some regularizing effects on the solutions. This work generalizes some results given in [1–3].

Existence of solution for Kirchhoff model problems with singular nonlinearity

We study the fourth order Kirchhoff equation Δ2u−(a+b∫Ω|∇u|2)γΔu=f(u) in Ω with −Δu&gt;0 and u&gt;0 in Ω, and Δu=u=0 on ∂Ω, where f(t)=α1tθ+λtq+μt+g(t) for t≥0, g has subcritical growth, α&gt;0, λ&gt;0, μ≥0, 0&lt;θ&lt;1, 0&lt;q&lt;1, γ≥0, a&gt;0, b≥0. We use the Galerkin projection method to show the existence of solution under some boundedness restriction on α,λ,μ. In some cases we study the behavior of the norm of the solution u as λ→0 and as λ→∞. Similar issues are addressed for the equation (a+b∫Ω|∇u|2)γΔ2u−ϱΔu=f(u), ϱ≥0.

Combined effects of Choquard and singular nonlinearities in fractional Kirchhoff problems

AbstractThe aim of this paper is to study the existence and multiplicity of solutions for a class of fractional Kirchho problems involving Choquard type nonlinearity and singular nonlinearity. Under suitable assumptions, two nonnegative and nontrivial solutions are obtained by using the Nehari manifold approach combined with the Hardy-Littlehood-Sobolev inequality.

Solutions for a Singular Hadamard-Type Fractional Differential Equation by the Spectral Construct Analysis

In this paper, we consider the existence of positive solutions for a Hadamard-type fractional differential equation with singular nonlinearity. By using the spectral construct analysis for the corresponding linear operator and calculating the fixed point index of the nonlinear operator, the criteria of the existence of positive solutions for equation considered are established. The interesting point is that the nonlinear term possesses singularity at the time and space variables.

(p,\xa0q)-Equations with Singular and Concave Convex Nonlinearities

We consider a nonlinear Dirichlet problem driven by the (p, q)-Laplacian with 1<q<p. The reaction is parametric and exhibits the competing effects of a singular term and of concave and convex nonlinearities. We are looking for positive solutions and prove a bifurcation-type theorem describing in a precise way the set of positive solutions as the parameter varies. Moreover, we show the existence of a minimal positive solution and we study it as a function of the parameter.

On Singular Equations Involving Fractional Laplacian

We study the existence and the regularity of solutions for a class of nonlocal equations involving the fractional Laplacian operator with singular nonlinearity and Radon measure data.

Existence and nonexistence results for anisotropic p-Laplace equation with singular nonlinearities

Let and consider the following anisotropic p-Laplace equation Under suitable hypothesis on the weight function g we present an existence result for in a bounded smooth domain Ω and nonexistence results for or with respectively.

Correction to: Multiplicity of Solution for a Quasilinear Equation with Singular Nonlinearity

Correction to: Multiplicity of Solution for a Quasilinear Equation with Singular Nonlinearity

The impact of a lower order term in a Dirichlet problem with a singular nonlinearity

In this paper we study the existence and regularity of solutions to the following Dirichlet problem \begin{cases} -\operatorname{div}(a(x)|\nabla u|^{p-2} \nabla u) + u|u|^{r-1} = \frac{f(x)}{u^{\theta}} &amp; \text{in }\Omega, \\ u &gt; 0 &amp; \text{in }\Omega, \\ u = 0 &amp; \text{on }\partial\Omega \end{cases} proving that the lower order term u|u|^{r-1} has some regularizing effects on the solutions.

Infinitely many small solutions to an elliptic PDE of variable exponent with a singular nonlinearity

ABSTRACT We prove the existence of infinitely many nonnegative solutions to the following nonlocal elliptic partial differential equation involving singularity where is a smooth, bounded domain, , , for all , for all and is the fractional -Laplacian operator with variable exponent. The nonlinear function f satisfies certain growth conditions. Moreover, we establish a uniform estimate of the solution(s) by the Moser iteration technique.

Weak Periodic Solution for Semilinear Parabolic Problem with Singular Nonlinearities and $$L^{1}$$ Data

We consider a periodic parabolic problem involving singular nonlinearity and homogeneous Dirichlet boundary condition modeled by $$\begin{aligned} \dfrac{\partial u}{\partial t}-\Delta u =\dfrac{f}{u^{\gamma }} \text { in }Q_{T}, \end{aligned}$$ where $$T>0$$ is a period, $$\Omega $$ is an open regular bounded subset of $$\mathbb {R}^{N}$$ , $$Q_{T}=]0,T[\times \Omega $$ , $$\gamma \in \mathbb {R}$$ and f is a nonnegative integrable function periodic in time with period T. Under a suitable assumptions on f, we establish the existence of a weak T-periodic solution for all ranges of value of $$\gamma $$ .

On a Class of Weighted p-Laplace Equation with Singular Nonlinearity

This article deals with the existence of the following quasilinear degenerate singular elliptic equation \begin{equation*} (P_\la)\left\{ \begin{split} -\text{div}(w(x)|\nabla u|^{p-2}\nabla u) &= g_{\la}(u),\;u>0\; \text{in}\; \Om, u&=0 \; \text{on}\; \partial \Om, \end{split}\right. \end{equation*} where $ \Om \subset \mb R^n$ is a smooth bounded domain, $n\geq 3$, $\la>0$, $p>1$ and $w$ is a Muckenhoupt weight. Using variational techniques, for $g_{\la}(u)= \la f(u)u^{-q}$ and certain assumptions on $f$, we show existence of a solution to $(P_\la)$ for each $\la>0$. Moreover when $g_{\la}(u)= \la u^{-q}+ u^{r}$ we establish existence of atleast two solutions to $(P_\la)$ in a suitable range of the parameter $\la$. Here we assume $q\in (0,1)$ and $r \in (p-1,p^*_s-1)$.