Hardy Terms Research Articles

Global compactness results for quasilinear elliptic problems with combined critical Sobolev–Hardy terms

In this paper, we study the global compactness results for quasilinear elliptic problems involving combined critical Sobolev–Hardy terms on the whole space and a bounded smooth domain, respectively. That is, we give the complete descriptions for the Palais–Smale (PS) sequences of the corresponding energy functionals. By using these descriptions, the existence results of solutions are also obtained.

Solutions for the quasilinear elliptic problems involving critical Hardy–Sobolev exponents

In this article, we study the quasilinear elliptic problem involving critical Hardy–Sobolev exponents and Hardy terms. By variational methods and analytic techniques, we obtain the existence of sign–changing solutions to the problem.

Multiple positive solutions for singular elliptic equations with weighted Hardy terms and critical Sobolev–Hardy exponents

Variational methods are used to prove the multiplicity of positive solutions for the following singular elliptic equation:where 0 ∈ Ω ⊂ ℝN, N ≥ 3, is a bounded domain with smooth boundary ∂ Ω, λ > 0 , $1\le q<2$, $0\le\mu<\bar{\mu}=(N-2)^2/4$, 0 ≤ s < 2, 2*(s)=2(N−s)/(N−2) and f and g are continuous functions on $\bar{\varOmega}$, that change sign on Ω.

Quasilinear elliptic problems with critical exponents and Hardy terms in ℝN

AbstractWe deal with a singular quasilinear elliptic problem, which involves critical Hardy-Sobolev exponents and multiple Hardy terms. Using variational methods and analytic techniques, the existence of ground state solutions to the problem is obtained.

Hardy–Sobolev critical singular elliptic equations with mixed Dirichlet–Neumann boundary conditions

Three important inequalities (the Poincaré, Hardy and generalized Poincaré inequalities) on the mixed boundary conditions are firstly established by some analytical techniques. Then the existence and multiplicity of positive solutions are studied for a class of semilinear elliptic equations with mixed Dirichlet–Neumann boundary conditions involving Hardy terms and Hardy–Sobolev critical exponents by using the variational methods.

Asymptotic behavior of the positive solutions for an elliptic equation with Hardy term

We give an explicit formula for exponential decay properties of positive solutions for a class of semilinear elliptic equations with Hardy term in the whole space Rn.

Quasilinear elliptic problems involving multiple critical Hardy–Sobolev exponents

In this paper, we study a kind of quasilinear elliptic problem which involves multiple critical Hardy–Sobolev exponents and Hardy terms. By employing the variational methods and analytical techniques, the existence of sign-changing solutions to the problem is obtained.

Nonexistence of radial node solutions for elliptic problems with critical Sobolev exponents

Let Ω ⊂ R n be a ball centered at the origin with radius 1 and 3 ≤ n ≤ 6 , 2 ∗ = 2 n n − 2 and μ ∈ [ 0 , ( n − 2 2 ) 2 ) . By using a well-known Pohozaev-type identity and some ODE techniques, we obtained the nonexistence of sign-changing radial solutions for elliptic problems with critical Sobolev and Hardy terms − Δ u = μ | x | 2 u + | u | 2 ∗ − 2 u + λ u on Ω , u ∈ H 0 1 ( Ω ) for suitable small positive number λ . Meanwhile, the nonexistence of radial sign-changing solutions for p -Laplace problems with critical Sobolev exponent is also obtained.

Infinitely many radial solutions to elliptic problems with critical Sobolev and Hardy terms

Let Ω ⊂ ℝN be a ball centered at the origin with radius R > 0 and N ⩾ 7, 2* = \( \frac{{2N}} {{N - 2}} \). We obtain the existence of infinitely many radial solutions for the Dirichlet problem −Δu = \( \frac{\mu } {{|x|^2 }}u + |u|^{2^* - 2} u + \lambda u \) in Ω, u = 0 on ∂Ω for suitable positive numbers μ and λ. Such solutions are characterized by the number of their nodes.

Multiple solutions for the Brezis–Nirenberg problem with a Hardy potential and singular coefficients

By energy estimates and by establishing a local (PS) condition, we establish the multiplicity of solutions to a class of semi-linear Brezis–Nirenberg type problems with a Hardy term and singular coefficients via the pseudo-index theory.

Positive solutions for Neumann elliptic problems involving critical Hardy–Sobolev exponent with boundary singularities

In this paper we consider the existence of positive solution for some semilinear elliptic equations with Neumann boundary condition involving a critical Hardy–Sobolev exponent and Hardy terms with boundary singularities. Using mountain pass lemma without (PS) condition and the strong maximum principle, we get the existence of a positive solution.

Existence of multiple positive solutions for a nonlinear elliptic problem with the critical exponent and a Hardy term

In this paper, we show that if $\mu>0$ is small enough, the problem \begin{equation*} \left\{ \begin{aligned} -\Delta u -\mu\frac{u}{|x|^2} & =|u|^{2^\ast-2}u & & \text{in $\Omega$,}\\ u & =0 & & \text{on $\partial\Omega$} \end{aligned} \right. \end{equation*} has at least cat $\Omega -1$ positive solutions, where $\Omega$ is a noncontractible, bounded domain in $\mathbb R^N (N\geq 4)$ such that its boundary is smooth and $0 \in \Omega$.

On the quasilinear elliptic problem with a critical Hardy–Sobolev exponent and a Hardy term

In the present paper, a quasilinear elliptic problem with a critical Sobolev exponent and a Hardy-type term is considered. By means of a variational method, the existence of nontrivial solutions for the problem is obtained. The result depends crucially on the parameters p , t , s , λ and μ .

Existence and multiplicity of positive solutions for a class of semilinear elliptic equations involving Hardy term and Hardy–Sobolev critical exponents

Existence and multiplicity of positive solutions are studied for a class of semilinear elliptic equations with Hardy term, Hardy–Sobolev critical exponents and sublinear nonlinearity by variational methods and some analysis techniques.

Existence of sign-changing solutions for a nonlinear elliptic problem on [formula omitted] with a Hardy term

Let N ⩾ 3 , 1 < p < ( N + 2 ) / ( N − 2 ) and 0 < λ < ( N − 2 ) 2 / 4 . We study the existence of multiple sign-changing solutions of u ∈ H 1 ( R N ) , − Δ u + u − λ u / | x | 2 = | u | p − 1 u in R N , which are not radially symmetric.

Existence and multiplicity of solutions for semilinear elliptic equations with Hardy terms and Hardy–Sobolev critical exponents

Some existence and multiplicity results are obtained for solutions of semilinear elliptic equations with Hardy terms, Hardy–Sobolev critical exponents and superlinear nonlinearity by the variational methods and some analysis techniques.

On the quasilinear elliptic problems with critical Sobolev–Hardy exponents and Hardy terms

Let Ω ⊂ R N be a smooth bounded domain such that 0 ∈ Ω , N ≥ 3 . In this paper, we study the critical quasilinear elliptic problems − Δ p u − μ | u | p − 2 u | x | p = | u | p ∗ ( t ) − 2 | x | t u + λ | u | q − 2 | x | s u , u ∈ W 0 1 , p ( Ω ) with Dirichlet boundary condition, where − Δ p u = − div ( | ∇ u | p − 2 ∇ u ) , 1 < p < N , 0 ≤ μ < μ ̄ ≔ ( N − p p ) p , λ > 0 , 0 ≤ s , t < p , p ≤ q < p ∗ ( s ) ≔ p ( N − s ) N − p , p ∗ ( t ) ≔ p ( N − t ) N − p , p ∗ ( s ) and p ∗ ( t ) are the critical Sobolev–Hardy exponents. Via variational methods, we deal with the conditions that ensure the existence of positive solutions for the equation. The results depend crucially on the parameters p , q , s , λ and μ .

On a class of critical heat equations with an inverse square potential

In this paper, we study a class of parabolic equations with critical Sobolev exponents and Hardy terms. Using Moser-type iteration, we characterize the asymptotic behavior of solutions at singular points. By means of critical point theory and the potential well method, we prove both global existence and finite-time blow-up depending on the initial datum.

Existence of positive solutions for a semilinear elliptic problem with critical Sobolev and Hardy terms

Let N ≥ 4 N\geq 4 , let 2 ∗ = 2 N / ( N − 2 ) 2^{\ast }=2N/(N-2) and let Ω ⊂ R N \Omega \subset \mathbb {R}^{N} be a bounded domain with a smooth boundary ∂ Ω \partial \Omega . Our purpose in this paper is to consider the existence of solutions of the problem: \[ { − Δ u − μ u | x | 2 a m p ; = | u | 2 ∗ − 1 a m p ; a m p ; in Ω , u a m p ; &gt; 0 a m p ; a m p ; in Ω , u a m p ; = 0 a m p ; a m p ; on ∂ Ω , \left \{ \begin {aligned} -\Delta u - \mu \frac {u}{\vert x\vert ^2} &amp;= \vert u\vert ^{2^\ast -1} &amp;&amp; \text {in $\Omega $}, \\ u &amp; &gt; 0 &amp;&amp; \text {in $\Omega $},\\ u &amp; = 0 &amp;&amp; \text {on $\partial \Omega $}, \end {aligned} \right . \] where 0 &gt; μ &gt; ( N − 2 2 ) 2 . 0&gt;\mu &gt;(\frac {N-2}{2})^{2}.

An elliptic equation with combined critical Sobolev–Hardy terms

Let Ω be a bounded domain in R N ( N ≥ 3 ) with the origin 0 ∈ Ω , μ < ( ( N − 2 ) / 2 ) 2 , 2 ∗ ( s ) = 2 ( N − s ) / ( N − 2 ) ; K ( x ) ≥ 0 and Q ( x ) ≥ 0 are two smooth functions on Ω . In this paper, we investigate the singular elliptic equation − Δ u = μ u | x | 2 + K ( x ) u 2 ∗ ( s ) − 1 | x | s + Q ( x ) u 2 ∗ ( t ) − 1 | x − x 0 | t + f ( x , u ) with Dirichlet boundary conditions. We study the limit behavior of the ( P.S.) sequence of the corresponding energy functional and give a global compactness theorem, and then give some existence results.