Sign-changing Weight Function Research Articles

Sub-elliptic systems involving critical Hardy-Sobolev exponents and sign-changing weight functions on Carnot groups

Sub-elliptic systems involving critical Hardy-Sobolev exponents and sign-changing weight functions on Carnot groups

Existence of at least four solutions for Schrodinger equations with magnetic potential involving and sign-changing weight function

We consider the elliptic problem $$ - \Delta_A u + u = a_{\lambda}(x) |u|^{q-2}u+b_{\mu}(x) |u|^{p-2}u , $$ for \(x \in \mathbb{R}^N\), \( 1 < q < 2 < p < 2^*= 2N/(N-2)\), \(a_{\lambda}(x)\) is a sign-changing weight function, \(b_{\mu}(x)\) satisfies some additional conditions, \(u \in H^1_A(\mathbb{R}^N)\) and \(A:\mathbb{R}^N \to \mathbb{R}^N\) is a magnetic potential. Exploring the Bahri-Li argument and some preliminary results we will discuss the existence of a four nontrivial solutions to the problem in question.

Nehari manifold for a Schrödinger equation with magnetic potential involving sign-changing weight function

We consider the following class of elliptic problems − Δ A u + u = a λ ( x ) | u | q − 2 u + b μ ( x ) | u | p − 2 u , x ∈ R N , where 1 < q < 2 < p < 2 ∗ = 2 N N − 2 N ≥ 3 , a λ ( x ) is a sign-changing weight function, b μ ( x ) is continuous, 0 $ ]]> λ > 0 and 0 $ ]]> μ > 0 are real parameters, u ∈ H A 1 ( R N ) and A : R N → R N is a magnetic potential. Exploring the relationship between the Nehari manifold and fibering maps, we will discuss the existence, multiplicity and regularity of solutions.

Fourth Order Elliptic Equation Involving Sign-Changing Weight Function in $$\mathbb {R}^{N}$$

Fourth Order Elliptic Equation Involving Sign-Changing Weight Function in $$\mathbb {R}^{N}$$

Shooting from singularity to singularity and a quasilinear &lt;i&gt;p&lt;/i&gt;-Laplace-Beltrami equation with indefinite weight

For surfaces of revolution we prove the existence of infinitely many sign-changing rotationally symmetric solutions to a wide class of $ p $-Laplace-Beltrami equations with polynomial like nonlinearities. We combine the methods in [10] where the semilinear case $ (p = 2) $ with positive weight was studied with those in [5] and [6] where radial solutions to a $ p $-Laplacian Dirichlet problem in a ball was considered. Our equations lead to ordinary differential equations with two singularities and a sign changing weight function which leads to an initial value problem that may blow up in the region of definition of the equation.

Biharmonic system with Hartree-type critical nonlinearity

In this article, we investigate the multiplicity results of the following biharmonic Choquard system involving critical nonlinearities with sign-changing weight function: { Δ 2 u = λ F ( x ) | u | r − 2 u + H ( x ) ( ∫ Ω H ( y ) | v ( y ) | 2 α ∗ | x − y | α d y ) | u | 2 α ∗ − 2 u in Ω , Δ 2 v = μ G ( x ) | v | r − 2 v + H ( x ) ( ∫ Ω H ( y ) | u ( y ) | 2 α ∗ | x − y | α d y ) | v | 2 α ∗ − 2 v in Ω , u = v = ∇ u = ∇ v = 0 on ∂ Ω , where Ω is a bounded domain in R N with smooth boundary ∂ Ω , N≥ 5, 1&lt;r&lt;2, 0&lt;α &lt;N, 2α ∗ =2 N − α N − 4 is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality and Δ 2 denotes the biharmonic operator. The functions F , G and H : Ω ¯ → R are sign-changing weight functions satisfying F , G ∈ L 2 ∗ 2 ∗ − r ( Ω ) and H ∈ L ∞ ( Ω ) respectively. By adopting Nehari manifold and fibering map technique, we prove that the system admits at least two nontrivial solutions with respect to parameter ( λ , μ ) ∈ R + 2 ∖ { ( 0 , 0 ) } .

Infinitely many solutions for a class of fractional Schrödinger equations with sign-changing weight functions

In this paper, we study the fractional Schrödinger equation {(−Δ)su+u=a(x)|u|p−2u+b(x)|u|q−2u,u∈Hs(RN),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extstyle\\begin{cases} (-\\Delta )^{s}u+u=a(x) \\vert u \\vert ^{p-2}u+b(x) \\vert u \\vert ^{q-2}u, \\\\ u\\in H^{s}(\\mathbb{R}^{N}), \\end{cases} $$\\end{document} where (-Delta )^{s} denotes the fractional Laplacian of order sin (0,1), N>2s, 2< p< q<2^{*}_{s}, and 2^{*}_{s} is the fractional critical Sobolev exponent. The weight potentials a or b is a sign-changing function and satisfies some valid assumptions. We obtain the existence of infinitely many solutions to the problem by the Nehari manifold.

On subhomogeneous indefinite p-Laplace equations in the supercritical spectral interval

We study the existence, multiplicity, and certain qualitative properties of solutions to the zero Dirichlet problem for the equation $$-\Delta _p u = \lambda |u|^{p-2}u + a(x)|u|^{q-2}u$$ in a bounded domain $$\Omega \subset {\mathbb {R}}^N$$ , where $$1<q<p$$ , $$\lambda \in {\mathbb {R}}$$ , and a is a sign-changing weight function. Our primary interest concerns ground states and nonnegative solutions which are positive in $$\{x\in \Omega : a(x)>0\}$$ , when the parameter $$\lambda $$ lies in a neighborhood of the critical value $$\lambda ^* := \inf \left\{ \int _\Omega |\nabla u|^p \, dx/\int _\Omega |u|^p \, dx: u\in W_0^{1,p}(\Omega ) {\setminus } \{0\},\ \int _\Omega a|u|^q\,dx \ge 0\,\right\} $$ . Among main results, we show that if $$p>2q$$ and either $$\int _\Omega a\varphi _p^q\,dx=0$$ or $$\int _\Omega a\varphi _p^q\,dx>0$$ is sufficiently small, then such solutions do exist in a right neighborhood of $$\lambda ^*$$ . Here $$\varphi _p$$ is the first eigenfunction of the Dirichlet p-Laplacian in $$\Omega $$ . This existence phenomenon is of a purely subhomogeneous and nonlinear nature, since either in the superhomogeneous case $$q>p$$ or in the sublinear case $$q<p=2$$ the nonexistence takes place for any $$\lambda \ge \lambda ^*$$ . Moreover, we prove that if $$p>2q$$ and $$\int _\Omega a\varphi _p^q\,dx>0$$ is sufficiently small, then there exist three nonzero nonnegative solutions in a left neighborhood of $$\lambda ^*$$ , two of which are strictly positive in $$\{x\in \Omega : a(x)>0\}$$ .

Shooting method in the application of boundary value problems for differential equations with sign-changing weight function

Abstract In this paper, we use the shooting method to study the solvability of the boundary value problem of differential equations with sign-changing weight function: u ″ ( t ) + ( λ a + ( t ) − μ a − ( t ) ) g ( u ) = 0 , 0 &lt; t &lt; T , u ′ ( 0 ) = 0 , u ′ ( T ) = 0 , \left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}{u}^{^{\prime\prime} }\left(t)+\left(\lambda {a}^{+}\left(t)-\mu {a}^{-}\left(t))g\left(u)=0,\hspace{1.0em}0\lt t\lt T,\\ u^{\prime} \left(0)=0,\hspace{1.0em}u^{\prime} \left(T)=0,\end{array}\right. where a ∈ L [ 0 , T ] a\in L\left[0,T] is sign-changing and the nonlinearity g : [ 0 , ∞ ) → R g:{[}0,\infty )\to {\mathbb{R}} is continuous such that g ( 0 ) = g ( 1 ) = g ( 2 ) = 0 g\left(0)=g\left(1)=g\left(2)=0 , g ( s ) &gt; 0 g\left(s)\gt 0 for s ∈ ( 0 , 1 ) s\in \left(0,1) , g ( s ) &lt; 0 g\left(s)\lt 0 for s ∈ ( 1 , 2 ) s\in \left(1,2) .

Multiplicity of solutions for a singular system involving the fractional $p$-$q$-Laplacian operator and sign-changing weight functions

This paper deals with the following singular system: $$ \begin{cases} (-\Delta)^s\_p u+ (-\Delta)^s\_q u = \lambda f(x)|{u}|^{r-2}u + \frac{1-\alpha}{2-\alpha-\beta}h(x) |{u}|^{-\alpha}|{v}|^{1-\beta} & \quad \text{in}\ \Omega,\ (-\Delta)^s\_p v+ (-\Delta)^s\_q v = \mu g(x)|{v}|^{r-2}v + \frac{1-\beta}{2-\alpha-\beta}h(x) |{u}|^{1-\alpha}|{v}|^{-\beta} & \quad \text{in}\ \Omega,\ u = v = 0 & \quad \text{in}\ \mathbb{R}^N\setminus\Omega, \end{cases} $$ where $\Omega\subset \mathbb{R}^N$ is a bounded smooth domain, $\lambda, \mu$ are positive parameters, $s\in (0,1)$, $1< p < N/s$, $0<\alpha, \beta<1$, $2-\alpha-\beta < q < p < r< p^\ast\_s = Np/(N-sp)$, and $(-\Delta)^s\_\sigma u$ denotes the fractional $\sigma$-Laplacian, $\sigma = p,q$. Under appropriate conditions on the weight functions $f, g, h$ which may change sign in $\Omega$, we establish the existence of multiple solutions by using the Nehari manifold method. Our paper is one of the first attempts to study the existence of solutions for fractional singular systems involving sign-changing weight functions.

Positive Solutions for Slightly Subcritical Elliptic Problems Via Orlicz Spaces

This paper concerns semilinear elliptic equations involving sign-changing weight function and a nonlinearity of subcritical nature understood in a generalized sense. Using an Orlicz–Sobolev space setting, we consider superlinear nonlinearities which do not have a polynomial growth, and state sufficient conditions guaranteeing the Palais–Smale condition. We study the existence of a bifurcated branch of classical positive solutions, containing a turning point, and providing multiplicity of solutions.

Existence of Multiple Solutions for a Quasilinear Elliptic System Involving Sign-Changing Weight Functions and Variable Exponent

Existence of Multiple Solutions for a Quasilinear Elliptic System Involving Sign-Changing Weight Functions and Variable Exponent

Multiple solutions for biharmonic critical Choquard equation involving sign-changing weight functions

Multiple solutions for biharmonic critical Choquard equation involving sign-changing weight functions

Infinitely many solutions for the fractional $p$&amp;$q$ problem with critical Sobolev-Hardy exponents and sign-changing weight functions

In this paper, we study the Kirchhoff type problems involving fractional $p$&$q$ problem with critical Sobolev-Hardy exponents and sign-changing weight functions. By using the fractional version of concentration-compactness principle together with Krasnoselskii's genus, we obtain the multiplicity of solutions for this kind problem. The main feature and difficulty of our equations arise in the fact that the Kirchhoff term $M$ could vanish at zero, that is, the problem is degenerate.

Infinitely many solutions for a nonlocal type problem with sign-changing weight function

In this article, we study the existence of weak solutions for a fractional type problem driven by a nonlocal operator of elliptic type $$\displaylines{ (-\Delta)^s_{a_1} u -\lambda a_2(|u|)u = f(x,u)+g(x)|u|^{q(x)-2}u \quad \text{in } \Omega \cr u = 0 \quad \text{in } \mathbb{R}^N\setminus \Omega. }$$ Our approach is based on critical point theorems and variational methods.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2021/16/abstr.html

Positive solutions for fractional Laplacian system involving concave-convex nonlinearities and sign-changing weight functions

Positive solutions for fractional Laplacian system involving concave-convex nonlinearities and sign-changing weight functions

Multiple solutions for a coupled Kirchhoff system with fractional p-Laplacian and sign-changing weight functions

In this paper, we investigate the multiplicity of solutions for Kirchhoff fractional p-Laplacian system in bounded domains: By using the Nehari manifold method, together with Ekeland's variational principle, we show that there exist two distinct solutions under suitable conditions on weight functions and h. Our results extend and generalize the main results in Xiang et al. [Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional p-Laplacian. Nonlinearity. 2016;29:3186–3205] in Nonlinearity 2016, in which are constants.

Multiplicity results for critical fractional equations with sign-changing weight functions

Multiplicity results for critical fractional equations with sign-changing weight functions

Multiplicity and concentration of positive solutions to the fractional Kirchhoff type problems involving sign-changing weight functions

&lt;p style='text-indent:20px;'&gt;The aim of this paper is to study the multiplicity and concentration of positive solutions to the fractional Kirchhoff type problems involving sign-changing weight functions and concave-convex nonlinearities with subcritical or critical growth. Applying Nehari manifold, fibering maps and Ljusternik-Schnirelmann theory, we investigate a relationship between the number of positive solutions and the topology of the global maximum set of &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ K $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;.&lt;/p&gt;

Polyharmonic systems involving critical nonlinearities with sign-changing weight functions

This article concerns the existence of multiple solutions of the polyharmonic system involving critical nonlinearities with sign-changing weight functions $$\displaylines{ (-\Delta)^mu = \lambda f(x) |u|^{r-2}u+ \frac{\beta}{\beta+\gamma} h(x) |u|^{\beta-2}u |v|^{\gamma}\quad \text{in }\Omega,\cr (-\Delta)^mv = \mu g(x) |v|^{r-2}v+ \frac{\gamma}{\beta+\gamma} h(x) |u|^{\beta} |v|^{\gamma-2} v \quad \text{in }\Omega, \cr D^ku=D^kv=0\quad \text{for all }|k|\leq m-1\quad \text{on }\partial\Omega, }$$ where \((-\Delta)^m\) denotes the polyharmonic operators, \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial \Omega\), \(m\in \mathbb N\), \(N\geq {2m+1}\), \(1&lt;r&lt;2\) and \(\beta&gt;1\), \(\gamma&gt;1\) satisfying \(2&lt;\beta+\gamma\leq 2_{m}^{*}\) with \(2_{m}^{*}=\frac{2N}{N-2m}\) as a critical Sobolev exponent and \(\lambda\), \(\mu&gt;0\). The functions f, g and \(h:\overline{\Omega}\to \mathbb R\) are sign-changing weight functions satisfying f, \(g\in L^{\alpha}(\Omega)\) and \(h\in L^{\infty}(\Omega)\) respectively. Using the variational methods and Nehari manifold, we prove that the system admits at least two nontrivial solutions with respect to parameter \((\lambda, \mu)\in \mathbb R^2_{+} \setminus \{(0, 0)\}\).&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2020/119/abstr.html