Singular Nonlinearity Research Articles

Existence of multiple positive solutions for a class of Quasilinear Schrödinger-Poisson systems with p-Laplacian and singular nonlinearity terms in $${\mathbb {R}}^N$$

Existence of multiple positive solutions for a class of Quasilinear Schrödinger-Poisson systems with p-Laplacian and singular nonlinearity terms in $${\mathbb {R}}^N$$

Anisotropic degenerate elliptic problem with a singular nonlinearity

Anisotropic degenerate elliptic problem with a singular nonlinearity

Multiplicity of solutions of Kirchhoff-type fractional Laplacian problems with critical and singular nonlinearities

In this ariticle, the following Kirchhoff-type fractional Laplacian problem with singular and critical nonlinearities is studied: { ( a + b ‖ u ‖ 2 μ − 2 ) ( − Δ ) s u = λ l ( x ) u 2 s ∗ − 1 + h ( x ) u − γ , in Ω , u > 0 , in Ω , u = 0 , in R N ∖ Ω , where s ∈ ( 0 , 1 ) , N > 2 s , ( − Δ ) s is the fractional Laplace operator, 2 s ∗ = 2 N / ( N − 2 s ) is the critical Sobolev exponent, Ω ⊂ R N is a smooth bounded domain, l ∈ L ∞ ( Ω ) is a non-negative function and max { l ( x ) , 0 } ≢ 0 , h ∈ L 2 s ∗ 2 s ∗ + γ − 1 ( Ω ) is positive almost everywhere in Ω , γ ∈ ( 0 , 1 ) , a > 0 , b > 0 , μ ∈ [ 1 , 2 s ∗ / 2 ) and parameter λ is a positive constant. Here we utilize a special method to recover the lack of compactness due to the appearance of the critical exponent. By imposing appropriate constraint on λ , we obtain two positive solutions to the above problem based on the Ekeland variational principle and Nehari manifold technique.

ON A SYSTEM OF COUPLED NONLOCAL SINGULAR FRACTIONAL BOUNDARY VALUE PROBLEMS WITH <i>δ</i>-LAPLACIAN OPERATORS

We investigate the existence of at least one or two positive solutions of a system of two Riemann-Liouville fractional differential equations with $ \delta $-Laplacian operators and singular nonlinearities, supplemented with coupled nonlocal boundary conditions which contain Riemann-Stieltjes integrals and several fractional derivatives of different orders. We apply the Guo-Krasnosel'skii fixed point theorem in the proof of our main existence results.

Multiple positive solutions for a fractional Kirchhoff type equation with logarithmic and singular nonlinearities

In this paper, we study the following fractional Kirchhoff type equation { ( a + b ∫ R N ∫ R N | u ( x ) − u ( y ) | p | x − y | N + p s d x d y ) ( − Δ ) p s u = | u | q − 2 u ln ⁡ | u | 2 + λ u γ , i n Ω , u > 0 , i n Ω , u = 0 , i n R N ∖ Ω , where Ω ⊂ R N is a bounded domain with Lipschitz boundary, 0 < s < 1 < p , 0 < γ < 1 , a > 0 , b ≥ 0 , N > p s , 2 p < q < q + 2 < p s ∗ , p s ∗ = N p N − p s is the fractional critical exponent, λ > 0 is a real parameter. By using the critical point theory for nonsmooth functionals and analytic techniques, the existence and multiplicity of positive solutions are obtained.

Existence results for $ p(x) $-biharmonic problems involving a singular and a Hardy type nonlinearities

<abstract><p>In this paper, we proved the existence and the multiplicity of solutions for some $ p(x) $-biharmonic problems involving singular nonlinearity and a Hardy potential. More precisely, by the use of the min-max method, we proved the existence of a nontrivial solution for such a problem. Next, diversions of the mountain pass theorem were used to prove the multiplicity of solutions.</p></abstract>

A critical elliptic problem involving exponential and singular nonlinearities

We study the existence of a solution to an elliptic problem driven by a fractional p-Laplacian and a nonlinearity which is a product of an exponential and a singular nonlinearity. The exponential singularity is of Trudinger-Moser type and the singularity is of the order \(0<\beta <1\).

Kirchhoff systems involving fractional p-Laplacian and singular nonlinearity

Kirchhoff systems involving fractional p-Laplacian and singular nonlinearity

Low Perturbations and Combined Effects of Critical and Singular Nonlinearities in Kirchhoff Problems

Low Perturbations and Combined Effects of Critical and Singular Nonlinearities in Kirchhoff Problems

Asymptotic Solutions of Resonant Nonlinear Singularly Perturbed Problems in the Case of Intersecting Eigenvalues of the Limit Operator

Lomov’s regularization method is generalized to resonant, weakly nonlinear, singularly perturbed systems in the case of intersecting roots of the characteristic equation of the limit operator. For constructing asymptotic solutions, the regularization of the original problem by using normal forms developed by the authors is performed. In the absence of resonance, the regularizing normal form is linear, whereas in the presence of resonances, it is nonlinear. In this paper, the resonant case of a weakly nonlinear problem is considered. By using an algorithm of normal forms, we construct an asymptotic solution of any order (with respect to a parameter) and justify this algorithm.

Multiplicity of Solutions to a p-q Fractional Laplacian System with Concave Singular Nonlinearities

Multiplicity of Solutions to a p-q Fractional Laplacian System with Concave Singular Nonlinearities

Fractional heat equation with singular nonlinearity

Fractional heat equation with singular nonlinearity

Onp-Laplace Equations with Singular Nonlinearities and Critical Sobolev Exponent

In this paper, we deal withp-Laplace equations with singular nonlinearities and critical Sobolev exponent. By using the Nehari manifold, Mountain Pass theorem, and Maximum principle theorem, we prove the existence of at least four distinct nontrivial solutions.

Multiple solutions for weighted fractional $p$-Laplace equations involving singular nonlinearity

This paper is concerned with the existence of solutions for the following weighted fractional $p$-Laplace problem involving singular nonlinearity: $$ \begin{cases} \displaystyle 2\int_{\mathbb{R}^{N}}\frac{|x|^{\alpha p} |y|^{\alpha p}|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}dy\\ \qquad \qquad \qquad \qquad \qquad \qquad = a(x)u^{-q}+\lambda |x|^{\beta}u^{r-1}\,\, \ & {\rm in}\ \Omega,\\ u=0\ \ \ & {\rm in}\ \mathbb{R}^N\setminus \Omega, \end{cases} $$ where $s\in(0,1)$, $\alpha \in\mathbb{R}$, $\Omega\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $1 < p < N/s$, $0 < q < 1, p < r < p^{\ast}_s$, $\lambda>0$, $p_s^{\ast}=Np/(N-sp)$ is the critical Sobolev exponent. By the Nehari manifold method and the fractional Caffarelli--Kohn--Nirenberg inequalities, we show that there exists $\lambda^* > 0$ such that the problem admits at least two positive solutions as $\lambda\in(0,\lambda^*)$.

Positive solutions for a class of nonlocal problems with possibly singular nonlinearity

We study a class of elliptic problems with homogeneous Dirichlet boundary condition and a nonlinear reaction term f which is nonlocal depending on the \(L^{p}\)-norm of the unknown function. The nonlinearity f can make the problem degenerate since it may even have multiple singularities in the nonlocal variable. We use fixed point arguments for an appropriately defined solution map, to produce multiplicity of classical positive solutions with ordered norms.

Positive Solutions for a Fractional Differential Equation with Sequential Derivatives and Nonlocal Boundary Conditions

We study the existence of positive solutions for a Riemann–Liouville fractional differential equation with sequential derivatives, a positive parameter and a sign-changing singular nonlinearity, subject to nonlocal boundary conditions containing varied fractional derivatives and general Riemann–Stieltjes integrals. We also present the associated Green functions and some of their properties. In the proof of the main results, we apply the Guo–Krasnosel’skii fixed point theorem. Two examples are finally given that illustrate our results.

Topological structure of the solution set for a fractional p-Laplacian problem with singular nonlinearity

We establish the existence of connected components of positive solutions for the equation \( (-\Delta_p)^s u = \lambda f(u)\), under Dirichlet boundary conditions, where the domain is a bounded in \(\mathbb{R}^N\) and has smooth boundary, \((-\Delta_p)^s\) is the fractional p-Laplacian operator, and \(f:(0,\infty) \to \mathbb{R}\) is a continuous function which may blow up to \(\pm \infty\) at the origin.

A weighted fractional problem involving a singular nonlinearity and an L 1 datum

In this article, we show the existence of a unique entropy solution to the following problem: where the domain is bounded and contains the origin, , , , sp<N, , for q>1 and h is a general singular function with singularity at 0. Further, the fractional p-Laplacian with weight α is given by

An elliptic system with singular nonlinearities: Existence via non variational arguments

In this paper we are concerned with existence of positive solutions for an elliptic system with singular or strongly-singular terms. We overcome the difficulty given by the singular terms through an approximation scheme and controlling the approximated sequences of solutions with suitable barriers from above and from below. Besides this, in some particular case, we show that the unique energy solution of the singular system is a saddle point of a suitable functional.

On critical double phase Kirchhoff problems with singular nonlinearity

The paper deals with the following double phase problem -m∫Ω|∇u|pp+a(x)|∇u|qqdxdiv|∇u|p-2∇u+a(x)|∇u|q-2∇u=λu-γ+up∗-1inΩ,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}$$\\begin{aligned} \\begin{aligned}&-m \\left[ \\int _\\Omega \\left( \\frac{|\ abla u|^p}{p} + a(x) \\frac{|\ abla u|^q}{q}\\right) \\,\\mathrm {d}x\\right] {\ ext{div}} \\left( |\ abla u|^{p-2}\ abla u + a(x) |\ abla u|^{q-2}\ abla u \\right) \\\\&\\quad = \\lambda u^{-\\gamma } +u^{p^*-1}&\\quad \ ext {in } \\Omega ,\\\\&u > 0&\\quad \ ext {in } \\Omega ,\\\\&u = 0&\\quad \ ext {on } \\partial \\Omega , \\end{aligned} \\end{aligned}$$\\end{document}where Omega subset {mathbb {R}}^N is a bounded domain with Lipschitz boundary partial Omega , Nge 2, m represents a Kirchhoff coefficient, 1<p<q<p^* with p^*=Np/(N-p) being the critical Sobolev exponent to p, a bounded weight a(cdot )ge 0, lambda >0 and gamma in (0,1). By the Nehari manifold approach, we establish the existence of at least one weak solution.