Sign-changing Weight Research Articles

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.

Multiple Solutions for Singular Systems with Sign-Changing Weight, Nonlinear Singularities and Critical Exponent

This paper is an attempt to establish the existence and multiplicity results of nontrivial solutions to singular systems with sign-changing weight, nonlinear singularities, and critical exponent. By using variational methods, the Nehari manifold, and under sufficient conditions on the parameter η which represent some physical meanings, we prove some existing results by researching the critical points as the minimizers of the energy functional associated with the proposed problem (2) on the constraint defined by the Nehari manifold, which are solutions of our system, under some sufficient conditions on the parameters α, β, μ, and η. To the best of our knowledge, this paper is one of the first contributions to the study of singular systems with sign-changing weight, nonlinear singularities, and critical exponent.

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

Spectral optimization for weighted anisotropic problems with Robin conditions

We study a weighted eigenvalue problem with anisotropic diffusion in bounded Lipschitz domains Ω⊂RN, N≥1, under Robin boundary conditions, proving the existence of two positive principal eigenvalues λ± respectively associated with a positive and a negative eigenfunction. Next, we analyze the minimization of λ± with respect to the sign-changing weight, showing that the optimal eigenvalues Λ± are equal if the domain has a center of symmetry and the optimal weights are of bang-bang type, namely piece-wise constant functions, each one taking only two values. As a consequence, the problem is equivalent to the minimization with respect to the subsets of Ω satisfying a volume constraint. Then, we completely solve the optimization problem in one dimension, in the case of homogeneous Dirichlet or Neumann conditions, showing new phenomena induced by the presence of the anisotropic diffusion. The optimization problem for λ+ naturally arises in the study of the optimal spatial arrangement of resources for a species to survive in an heterogeneous habitat.

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.

Principal eigenvalues for the fractional p-Laplacian with unbounded sign-changing weights

Let \(\Omega\) be a bounded regular domain of \( \mathbb{R}^N\), \(N\geqslant 1\), \(p\in (1,+\infty)\), and \( s\in (0,1) \). We consider the eigenvalue problem $$\displaylines{ (-\Delta_p)^s u + V|u|^{p-2}u= \lambda m(x)|u|^{p-2}u \quad\hbox{in } \Omega \cr u=0 \quad \hbox{in } \mathbb{R}^N \setminus \Omega, }$$ where the potential V and the weight m are possibly unbounded and are sign-changing. After establishing the boundedness and regularity of weak solutions, we prove that this problem admits principal eigenvalues under certain conditions. We also show that when such eigenvalues exist, they are simple and isolated in the spectrum of the operator.

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

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

Existence and multiplicity of positive solutions for one-dimensional $ p $-Laplacian problem with sign-changing weight

&lt;abstract&gt;&lt;p&gt;In this paper, we show the positive solutions set for one-dimensional $ p $-Laplacian problem with sign-changing weight contains a reversed $ S $-shaped continuum. By figuring the shape of unbounded continuum of positive solutions, we identify the interval of bifurcation parameter in which the $ p $-Laplacian problem has one or two or three positive solutions according to the asymptotic behavior of nonlinear term at 0 and $ \infty $. The proof of the main result is based upon bifurcation technique.&lt;/p&gt;&lt;/abstract&gt;

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 ) } .

On the maximum principle for the discrete p-laplacian with sign-changing weight

This work deals with the maximum principle for the discrete Neumann or Dirichlet problem-Δφp(Δu(k - 1)) = λm(k)φp(u(k))+ h(k) in [1, n].We study the existence and nonexistence of positive solution and its uniqueness.

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} &amp; \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} &amp; \quad \text{in}\ \Omega,\\ u = v = 0 &amp; \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&lt; p &lt; N/s , 0&lt;\alpha, \beta&lt;1 , 2-\alpha-\beta &lt; q &lt; p &lt; r&lt; 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

Global structure for a fourth-order boundary value problem with sign-changing weight

Abstract We study the fourth-order boundary value problem with a sign-changing weight function: u ⁗ = λ m ( t ) u + f 1 ( t , u , u ′ , u ″ , u ‴ , λ ) + f 2 ( t , u , u ′ , u ″ , u ‴ , λ ) , t ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 , $$\left\{\begin{array}{ll} u''''=\lambda m(t)u+f_1(t,u,u',u'',u''',\lambda)+f_2(t,u,u',u'',u''',\lambda),\qquad t\in(0,1),\\[1.3ex] u(0)=u(1)=u''(0)=u''(1)=0, \end{array}\right.$$ where λ ∈ ℝ is a parameter, f 1, f 2 ∈ C([0, 1] × ℝ5, ℝ), f 1 is not differentiable at the origin and infinity. Under some suitable conditions on nonlinear terms, we prove the existence of unbounded continua of positive and negative solutions of this problem which bifurcating from intervals of the line of trivial solutions or from infinity, respectively.