Singular Nonlinearity Research Articles

Positive Solutions of a Fractional Boundary Value Problem with Sequential Derivatives

We investigate the existence of positive solutions of a Riemann-Liouville fractional differential equation with sequential derivatives, a positive parameter and a nonnegative singular nonlinearity, supplemented with integral-multipoint boundary conditions which contain fractional derivatives of various orders and Riemann-Stieltjes integrals. Our general boundary conditions cover some symmetry cases for the unknown function. In the proof of our main existence result, we use an application of the Krein-Rutman theorem and two theorems from the fixed point index theory.

Nonlocal Lazer\u2013McKenna-type problem perturbed by the Hardy\u2019s potential and its parabolic equivalence

In this paper, we study the effect of Hardy potential on the existence or nonexistence of solutions to the following fractional problem involving a singular nonlinearity: \t\t\t{(−Δ)su=λu|x|2s+μuγ+fin Ω,u>0in Ω,u=0in (RN∖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} $$\\begin{aligned} \\textstyle\\begin{cases} (-\\Delta )^{s} u = \\lambda \\frac{u}{ \\vert x \\vert ^{2s}} + \\frac{\\mu }{u^{\\gamma }}+f & \\text{in } \\Omega, \\\\ u>0 & \\text{in } \\Omega, \\\\ u=0 & \\text{in } (\\mathbb{R}^{N} \\setminus \\Omega ). \\end{cases}\\displaystyle \\end{aligned}$$ \\end{document} Here 0 < s<1, lambda >0, gamma >0, and Omega subset mathbb{R}^{N} (N > 2s) is a bounded smooth domain such that 0 in Omega . Moreover, 0 leq mu,f in L^{1}(Omega ). For 0< lambda leq Lambda _{N,s}, Lambda _{N,s} being the best constant in the fractional Hardy inequality, we find a necessary and sufficient condition for the existence of a positive weak solution to the problem with respect to the data μ and f. Also, for a regular datum of f, under suitable assumptions, we obtain some existence and uniqueness results and calculate the rate of growth of solutions. Moreover, we mention a nonexistence and a complete blowup result for the case lambda > Lambda _{N,s}. Besides, we consider the parabolic equivalence of the above problem in the case mu equiv 1 and some suitable f(x,t), that is, \t\t\t{ut+(−Δ)su=λu|x|2s+1uγ+f(x,t)in Ω×(0,T),u>0in Ω×(0,T),u=0in (RN∖Ω)×(0,T),u(x,0)=u0in RN,\\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} $$\\begin{aligned} \\textstyle\\begin{cases} u_{t}+(-\\Delta )^{s} u = \\lambda \\frac{u}{ \\vert x \\vert ^{2s}} + \\frac{1}{u^{\\gamma }}+f(x,t) & \\text{in } \\Omega \\times (0,T), \\\\ u>0 & \\text{in } \\Omega \\times (0,T), \\\\ u =0 & \\text{in } (\\mathbb{R}^{N} \\setminus \\Omega ) \\times (0,T), \\\\ u(x,0)=u_{0} & \\text{in } \\mathbb{R}^{N}, \\end{cases}\\displaystyle \\end{aligned}$$ \\end{document} where u_{0} in X_{0}^{s}(Omega ) satisfies an appropriate cone condition. In the case 0<gamma leq 1 or gamma >1 with 2s(gamma -1)<(gamma +1), we show the existence of a unique solution for any 0< lambda < Lambda _{N,s} and prove a stabilization result for certain range of λ.

On a class of N/s-fractional Hardy–Schrödinger equations with singular exponential nonlinearity in $${\mathbb {R}}^{N}$$

On a class of N/s-fractional Hardy–Schrödinger equations with singular exponential nonlinearity in $${\mathbb {R}}^{N}$$

Multiple solutions of a nonlocal system with singular nonlinearities

In this work, we study the fractional Laplacian equation with singular nonlinearity: [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] with smooth boundary [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] [Formula: see text] [Formula: see text] is the fractional Sobolev exponent, [Formula: see text] are two parameters, [Formula: see text] are nonnegative weight functions, and [Formula: see text] is the fractional Laplace operator. We use the Nehari manifold approach and some variational techniques in order to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter [Formula: see text] and [Formula: see text].

Nonlocal semilinear elliptic problems with singular nonlinearity

Nonlocal semilinear elliptic problems with singular nonlinearity

Periodic parabolic equation involving singular nonlinearity with variable exponent

Periodic parabolic equation involving singular nonlinearity with variable exponent

Positive solution to singular semilinear elliptic problems

In this paper, we study the existence and regularity of a non-trivial weak solution to the following semilinear elliptic problem with singular weights and singular nonlinearity: under assumptions and .

Sobolev and Hölder regularity results for some singular nonhomogeneous quasilinear problems

This article deals with the study of the following singular quasilinear equation: $$\begin{aligned} (P) \left\{ \ -\Delta _{p}u -\Delta _{q}u = f(x) u^{-\delta },\; u>0 \text { in }\; \Omega ; \; u=0 \text { on } \partial \Omega , \right. \end{aligned}$$where \(\Omega \) is a bounded domain in \({\mathbb {R}}^N\) with \(C^2\) boundary \(\partial \Omega \), \(1< q< p<\infty \), \(\delta >0\) and \(f\in L^\infty _{loc}(\Omega )\) is a non-negative function which behaves like \(\text {dist}(x,\partial \Omega )^{-\beta },\) \(\beta \ge 0\) near the boundary of \(\Omega \). We prove the existence of a weak solution in \(W^{1,p}_{loc}(\Omega )\) and its behaviour near the boundary for \(\beta <p\). Consequently, we obtain optimal Sobolev regularity of weak solutions. By establishing the comparison principle, we prove the uniqueness of weak solution for the case \(\beta <2-\frac{1}{p}\). Subsequently, for the case \(\beta \ge p\), we prove the non-existence result. Moreover, we prove Hölder regularity of the gradient of weak solution to a more general class of quasilinear equations involving singular nonlinearity as well as lower order terms (see (1.6)). This result is completely new and of independent interest. In addition to this, we prove Hölder regularity of minimal weak solutions of (P) for the case \(\beta +\delta \ge 1\) that has not been fully answered in former contributions even for p-Laplace operators.

Evolutionary Integrated Heuristic with Gudermannian Neural Networks for Second Kind of Lane–Emden Nonlinear Singular Models

In this work, a new heuristic computing design is presented with an artificial intelligence approach to exploit the models with feed-forward (FF) Gudermannian neural networks (GNN) accomplished with global search capability of genetic algorithms (GA) combined with local convergence aptitude of active-set method (ASM), i.e., FF-GNN-GAASM to solve the second kind of Lane–Emden nonlinear singular models (LE-NSM). The proposed method based on the computing intelligent Gudermannian kernel is incorporated with the hidden layer configuration of FF-GNN models of differential operatives of the LE-NSM, which are arbitrarily associated with presenting an error-based objective function that is used to optimize by the hybrid heuristics of GAASM. Three LE-NSM-based examples are numerically solved to authenticate the effectiveness, accurateness, and efficiency of the suggested FF-GNN-GAASM. The reliability of the scheme via statistical valuations is verified in order to authenticate the stability, accuracy, and convergence.

On singular quasilinear elliptic equations with data measures

AbstractThe aim of this work is to study a quasilinear elliptic equation with singular nonlinearity and data measure. Existence and non-existence results are obtained under necessary or sufficient conditions on the data, where the main ingredient is the isoperimetric inequality. Finally, uniqueness results for weak solutions are given.

Existence of positive solution for elliptic equations with singular terms and combined nonlinearities

We use an approximation scheme, sub-super solution and Perron's method to show the existence and nonexistence of positive solutions for a boundary value problem with a singular nonlinearity in the equation and on the boundary condition. We also study the asymptotic behavior of the solutions with respect to the parameters.

Existence and Regularity of Solutions for Unbounded Elliptic Equations with Singular Nonlinearities

For q , γ &gt; 0 , we study existence and regularity of solutions for unbounded elliptic problems whose simplest model is − div 1 + u q ∇ u = f / u γ in Ω u = 0 on ∂ Ω , where f ∈ L m Ω , m ≥ 1 .

Regularity results for a class of nonlinear fractional Laplacian and singular problems

In this article, we investigate the existence, uniqueness, nonexistence, and regularity of weak solutions to the nonlinear fractional elliptic problem of type (P) (see below) involving singular nonlinearity and singular weights in smooth bounded domain. We prove the existence of weak solution in $$W_{loc}^{s,p}(\Omega )$$ via approximation method. Establishing a new comparison principle of independent interest, we prove the uniqueness of weak solution for $$0 \le \delta < 1+s- \frac{1}{p}$$ and furthermore the nonexistence of weak solution for $$\delta \ge sp.$$ Moreover, by virtue of barrier arguments we study the behavior of weak solutions in terms of distance function. Consequently, we prove Hölder regularity up to the boundary and optimal Sobolev regularity for weak solutions.

On a singular Kirchhoff type problems driven by p(⋅)-Laplacian operator

ABSTRACT This work is devoted to introduce a new Hardy-type inequality related to -Laplacian operator. Moreover, by using the Mountain Pass theorem and the Krasnoselskii's genus theory, we show the existence and infinite solutions to the -Kirchhoff type problem with singular nonlinearity.

Positive Solutions for a System of Coupled Semipositone Fractional Boundary Value Problems with Sequential Fractional Derivatives

We study the existence and multiplicity of positive solutions for a system of Riemann–Liouville fractional differential equations with sequential derivatives, positive parameters and sign-changing singular nonlinearities, subject to nonlocal coupled boundary conditions which contain Riemann–Stieltjes integrals and various fractional derivatives. In the proof of our main existence results we use the nonlinear alternative of Leray–Schauder type and the Guo–Krasnosel’skii fixed point theorem.

A Kirchhoff $p(x)$-Biharmonic Problem Involving Singular Nonlinearities and Navier Boundary Conditions

The aim of this work is to study the existence of weak solutions for a nonhomogeneous singular p(x) -Kirchhoff problem of the following form (\textbf{P}_{\pm \lambda}) \quad \left\{ \begin{aligned} M(t)\Delta(|\Delta u|^{p(x)-2}\Delta u) &amp;=a(x) u^{-\gamma (x)}\pm \lambda u^{q(x)-2}u, &amp;\ &amp;\quad \mathrm{in} \quad \Omega, \\ \Delta u&amp;=u=0, &amp;\ &amp;\quad \mathrm{on} \quad\partial\Omega, \end{aligned} \right. by using variational techniques and monotonicity arguments combined with the theory of the generalized Lebesgue Sobolev spaces.

Weighted and anisotropic Sobolev inequality with extremal

For a bounded smooth domain $$\Omega \subset {\mathbb {R}}^N$$ with $$N\ge 2$$ , we establish a weighted and an anisotropic version of Sobolev inequality related to the embedding $$W_{0}^{1,p}(\Omega )\hookrightarrow L^q(\Omega )$$ for $$1<p<\infty $$ and $$2\le p<\infty $$ respectively. Our main emphasize is the case of $$0<q<1$$ and we deal with a class of Muckenhoupt weights. Moreover, we obtain existence results for weighted and anisotropic p-Laplace equation with mixed singular nonlinearities and observe that the extremals of our inequalities are associated to such singular problems.

Existence and uniqueness of positive solutions to fractional Laplacians with singular nonlinearities

In this paper we prove the existence and uniqueness of positive classical solutions of fractional Laplacians with singular nonlinearities in a smooth bounded domain with zero Dirichlet boundary conditions. By the sub-supersolution method, we derive the existence of positive classical solutions to the approximation problems. In order to obtain the regularity, we first establish the existence of weak solutions for the fraction Laplacians. Thanks to Xia and Yang (2013), the regularity follows from the boundedness of weak solutions.

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 $$\Omega $$ .