Hardy Potential Research Articles

Infinitely many solutions for fractional elliptic systems involving critical nonlinearities and Hardy potentials

This paper is dedicated to studying the existence of multiple nontrivial solutions for a class of singular fractional elliptic systems involving critical nonlinearities and Hardy potentials in RN. Based upon the Caffarelli–Silvestre extension method and the Krasnoselskii genus theory, together with the symmetric criticality principle of Palais, we establish several existence results of infinitely many solutions under certain appropriate hypotheses on the weights and the parameters.

Solutions to a (p_1, ldots ,p_n)-Laplacian Problem with Hardy Potentials

We are concerned with the existence and multiplicity of weak solutions for a general form of a (p_1, ldots ,p_n)-Laplacian elliptic problem including singular terms. Our approaches are mainly based on critical points theory.

Ground state solution for the Schrödinger equation with Hardy potential and critical Sobolev exponent

In this paper, we consider the following Schrödinger equation (0.1) − Δ u − μ u | x | 2 + V ( x ) u = K ( x ) | u | 2 ∗ − 2 u + f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) , where N ⩾ 4, 0 ⩽ μ < μ ‾, μ ‾ = ( N − 2 ) 2 4 , V is periodic in x, K and f are asymptotically periodic in x, we take advantage of the generalized Nehari manifold approach developed by Szulkin and Weth to look for the ground state solution of (0.1).

Ground state solutions for critical Schrödinger equations with Hardy potential

In this paper, we investigate the following Schrödinger equation 0.1 −Δu−μ|x|2u=g(u)+|u|2*−2uinRN\\{0}, where N ⩾ 3, , is called the critical Sobolev exponent and g satisfies some appropriate subcritical conditions. For any , we prove that problem (0.1) has a positive radial ground state solution, which possesses exponential decaying property at infinity and blow-up property at origin. Moreover, for any sequence {μ n } ⊂ (0, +∞) satisfying μ n → 0+, the sequence of ground state solutions to problem (0.1) converges to a ground state solution of 0.2 −Δu=g(u)+|u|2*−2uinRN. when μ < 0, we prove that the mountain pass level of problem (0.1) in cannot be achieved. Further, we obtain a ground state radial solution of problem (0.1) whose energy is strictly greater than the mountain pass level in . Also, for any sequence {μ n } ⊂ (0, +∞) satisfying μ n → 0+, the sequence of ground state radial solutions to problem (0.1) converges to a ground state radial solution of the limiting problem as n → ∞.

A special self-similar solution and existence of global solutions for a reaction-diffusion equation with Hardy potential

Existence and uniqueness of a specific self-similar solution is established for the following reaction-diffusion equation with Hardy singular potential∂tu=Δum+|x|−2up,(x,t)∈RN×(0,∞), in the range of exponents 1≤p<m and dimension N≥3. The self-similar solution is unbounded at x=0 and has a logarithmic vertical asymptote, but it remains bounded at any x≠0 and t∈(0,∞) and it is a weak solution in L1 sense, which moreover satisfies u(t)∈Lp(RN) for any t>0 and p∈[1,∞). As an application of this self-similar solution, it is shown that there exists at least a weak solution to the Cauchy problem associated to the previous equation for any bounded, nonnegative and compactly supported initial condition u0, contrasting with previous results in literature for the critical limit p=m.

Existence and asymptotic behavior of solutions for a quasilinear Schrödinger equation with Hardy potential

In this paper, we discuss the quasilinear Schrödinger equation with a Hardy potential (Pμ)−Δu−μu|x|2−Δ(u2)u=g(u),x∈RN∖{0},u∈H1(RN),where N≥3, μ<μ̄=(N−2)24, 1|x|2 is called the Hardy potential and g∈C(R,R) satisfies the Berestycki–Lions type conditions. When 0<μ<(N−2)24, we show that the above problem has a positive and radial solution ū∈H1(RN). At the same time, we prove that the solution ū together with its derivatives up to order 2 have exponential decay at infinity and the solution has the possibility of blow-up at the origin. When μ<0, we obtain a solution ũ of (Pμ) in the radial space Hr1(RN) and we also prove that the solution ũ together with its derivatives up to order 2 have exponential decay at infinity and the solution has the possibility of blow-up at the origin. Furthermore, we construct a family of solutions which converge to a solution of the limiting problem as μ→0.

Weak solutions for some fractional singular (p, q)-Laplacian nonlocal problems with Hardy potential

Weak solutions for some fractional singular (p, q)-Laplacian nonlocal problems with Hardy potential

Existence of positive solutions to nonlinear singular parabolic equations with Hardy potential

Existence of positive solutions to nonlinear singular parabolic equations with Hardy potential

A Double Phase Problem Involving Hardy Potentials

In this paper, we deal with the following double phase problem $$\begin{aligned} \left\{ \begin{array}{lll} -\text{ div }\left( |\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u\right) =&{} \gamma \left( \displaystyle \frac{|u|^{p-2}u}{|x|^p}+a(x)\displaystyle \frac{|u|^{q-2}u}{|x|^q}\right) \\ &{}+f(x,u) &{} \text{ in } \Omega ,\\ u=0&{} &{} \text{ in } \partial \Omega , \end{array} \right. \end{aligned}$$where \(\Omega \subset {\mathbb {R}}^N\) is an open, bounded set with Lipschitz boundary, \(0\in \Omega \), \(N\ge 2\), \(1<p<q<N\), weight \(a(\cdot )\ge 0\), \(\gamma \) is a real parameter and f is a subcritical function. By variational method, we provide the existence of a non-trivial weak solution on the Musielak-Orlicz-Sobolev space \(W^{1,{\mathcal {H}}}_0(\Omega )\), with modular function \({\mathcal {H}}(t,x)=t^p+a(x)t^q\). For this, we first introduce the Hardy inequalities for space \(W^{1,{\mathcal {H}}}_0(\Omega )\), under suitable assumptions on \(a(\cdot )\).

Global $$W^{1,p}$$ regularity for elliptic problem with measure source and Leray–Hardy potential

In this paper, we develop the Littman–Stampacchia–Weinberger duality approach to obtain global \(W^{1,p}\) estimates for a class of elliptic problems involving Leray–Hardy operators and measure sources in a distributional framework associated to a dual formulation with a specific weight function.

Finite Morse Index Solutions of the Fractional Henon–Lane–Emden Equation with Hardy Potential

In this paper, we study the fractional Henon–Lane–Emden equation associated with Hardy potential \[ (-\Delta)^{s} u - \gamma |x|^{-2s} u = |x|^a |u|^{p-1} u \quad \textrm{in $\mathbb{R}^{n}$}. \] Extending the celebrated result of [14], we obtain a classification result on finite Morse index solutions to the fractional elliptic equation above with Hardy potential. In particular, a critical exponent $p$ of Joseph–Lundgren type is derived in the supercritical case studying a Liouville type result for the $s$-harmonic extension problem.

Bound state solutions for a class of Nonlinear Elliptic Equations with Hardy potential and Berestycki–Lions type conditions

By using the variational methods, a class of Nonlinear Elliptic Equations with the Hardy potential and Berestycki-Lions type conditions is studied and a bound state solution is obtained.

Some existence results for a fractional problem involving Hardy potential and variable exponent singular term

Some existence results for a fractional problem involving Hardy potential and variable exponent singular term

Non-Nehari Manifold Method for Hamiltonian Elliptic System with Hardy Potential: Existence and Asymptotic Properties of Ground State Solution

Non-Nehari Manifold Method for Hamiltonian Elliptic System with Hardy Potential: Existence and Asymptotic Properties of Ground State Solution

Ground states for a system of nonlinear Schrödinger equations with singular potentials

&lt;p style='text-indent:20px;'&gt;In this paper, we consider the existence and asymptotic behavior of ground state solutions for a class of Hamiltonian elliptic system with Hardy potential. The resulting problem engages three major difficulties: one is that the associated functional is strongly indefinite, the second difficulty we must overcome lies in verifying the link geometry and showing the boundedness of Cerami sequences when the nonlinearity is different from the usual global super-quadratic condition. The third difficulty is singular potential, which does not belong to the Kato's class. These enable us to develop a direct approach and new tricks to overcome the difficulties caused by singularity of potential and the dropping of classical super-quadratic assumption on the nonlinearity. Our approach is based on non-Nehari method which developed recently, we establish some new existence results of ground state solutions of Nehari-Pankov type under some mild conditions, and analyze asymptotical behavior of ground state solutions.&lt;/p&gt;

Qualitative properties of solutions to the Dirichlet problem for a Laplace equation involving the Hardy potential with possibly boundary singularity

&lt;abstract&gt;&lt;p&gt;We consider positive solutions to semilinear elliptic problems with Hardy potential and a first order term in bounded smooth domain $ \Omega $ with $ 0\in \overline \Omega $. We deduce symmetry and monotonicity properties of the solutions via the moving plane procedure under suitable assumptions on the nonlinearity.&lt;/p&gt;&lt;/abstract&gt;

Fractional KPZ equations with fractional gradient term and Hardy potential

&lt;abstract&gt;&lt;p&gt;In this work we address the question of existence and non existence of positive solutions to a class of fractional problems with non local gradient term. More precisely, we consider the problem&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \left\{ \begin{array}{rcll} (-\Delta )^s u &amp;amp; = &amp;amp;\lambda \dfrac{u}{|x|^{2s}}+ (\mathfrak{F}(u)(x))^p+ \rho f &amp;amp; \text{ in } \Omega,\\ u&amp;amp;&amp;gt;&amp;amp;0 &amp;amp; \text{ in }\Omega,\\ u&amp;amp; = &amp;amp;0 &amp;amp; \text{ in }(\mathbb{R}^N\setminus\Omega), \end{array}\right. $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;where $ \Omega\subset \mathbb{R}^N $ is a $ C^{1, 1} $ bounded domain, $ N &amp;gt; 2s, \rho &amp;gt; 0 $, $ 0 &amp;lt; s &amp;lt; 1 $, $ 1 &amp;lt; p &amp;lt; \infty $ and $ 0 &amp;lt; \lambda &amp;lt; \Lambda_{N, s} $, the Hardy constant defined below. We assume that $ f $ is a non-negative function with additional hypotheses. Here $ \mathfrak{F}(u) $ is a nonlocal "gradient" term. In particular, if $ \mathfrak{F}(u)(x) = |(-\Delta)^{\frac s2}u(x)| $, then we are able to show the existence of a critical exponents $ p_{+}(\lambda, s) $ such that: 1) if $ p &amp;gt; p_{+}(\lambda, s) $, there is no positive solution, 2) if $ p &amp;lt; p_{+}(\lambda, s) $, there exists, at least, a positive supersolution solution for suitable data and $ \rho $ small. Moreover, under additional restriction on $ p $, there exists a solution for general datum $ f $.&lt;/p&gt;&lt;/abstract&gt;

Solutions for a nonlocal problem involving a Hardy potential and critical growth

Solutions for a nonlocal problem involving a Hardy potential and critical growth

Multiplicity results for nonlocal critical problems involving Hardy potential in the whole space

The aim of this paper is to study a nonlocal elliptic problem in (denoted as below) involving the fractional Laplacian, a linear Hardy potential term and a critical nonlinear term. According to suitable assumptions on the set of extremal points of the functional coefficients, we prove that has multiple positive solutions, and we determine their precise behavior near the extremal points. This work extends a previous article by the first author et al. on the local case.

The Brezis–Nirenberg problem for fractional systems with Hardy potentials

In this work, we study the existence of positive solutions to the following fractional elliptic systems with Hardy‐type singular potentials and coupled by critical homogeneous nonlinearities where (− Δ)s denotes the fractional Laplace operator, is a smooth bounded domain such that 0 ∈ Ω, μ1, μ2 ∈ [0, ΛN, s) with ΛN, s the sharp constant of the fractional Hardy inequality, and is the fractional critical Sobolev exponent. In order to prove the main result, we study the related fractional Hardy–Sobolev type inequalities and then establish the existence of positive solutions to the systems through variational methods.