Critical Sobolev Hardy Exponents Research Articles

Existence and Multiplicity of Nontrivial Solutions for Semilinear Elliptic Equations Involving Hardy–Sobolev Critical Exponents

A class of semi-linear elliptic equations with the critical Hardy–Sobolev exponent has been considered. This model is widely used in hydrodynamics and glaciology, gas combustion in thermodynamics, quantum field theory, and statistical mechanics, as well as in gravity balance problems in galaxies. The PSc sequence of energy functional was investigated, and then the mountain pass lemma was used to prove the existence of at least one nontrivial solution. Also a multiplicity result was obtained. Some known results were generalized.

Existence and Nonexistence of Positive Solutions for Semilinear Elliptic Equations Involving Hardy–Sobolev Critical Exponents

Existence and Nonexistence of Positive Solutions for Semilinear Elliptic Equations Involving Hardy–Sobolev Critical Exponents

Multiple high‐energy solutions for an elliptic system with critical Hardy–Sobolev nonlinearity

This paper discusses the existence of multiple high‐energy solutions for a ‐Laplacian system involving critical Hardy‐Sobolev nonlinearity in . Considering that the “double” lack of compactness in the system is caused by the unboundedness of and the presence of the critical Hardy–Sobolev exponent, we demonstrate the version to of Struwe's classical global compactness result for double ‐Laplace operator. In virtue of the quantitative deformation lemma, a barycenter function, and the Brouwer degree theory, the existence of multiple high‐energy solutions is established. The results of this paper extend and complement the recent work.

Mass effect on an elliptic PDE involving two Hardy-Sobolev critical exponents

Mass effect on an elliptic PDE involving two Hardy-Sobolev critical exponents

On a nonlocal p(x)-Laplacian Dirichlet problem involving several critical Sobolev-Hardy exponents

On a nonlocal p(x)-Laplacian Dirichlet problem involving several critical Sobolev-Hardy exponents

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

Singular p-biharmonic problems involving the Hardy-Sobolev exponent

This article concerns the existence and multiplicity of solutions for the singular p-biharmonic problem involving the Hardy potential and the critical Hardy-Sobolev exponent. To this end we use variational methods combined with the Mountain pass theorem and the Ekeland variational principle. We illustrate the usefulness of our results with and example.
 For mote information see https://ejde.math.txstate.edu/Volumes/2023/61/abstr.html

Two Disjoint and Infinite Sets of Solutions for an Elliptic Equation Involving Critical Hardy-Sobolev Exponents

Two Disjoint and Infinite Sets of Solutions for an Elliptic Equation Involving Critical Hardy-Sobolev Exponents

Classification of positive solutions to [formula omitted]-Laplace equations with critical Hardy–Sobolev exponent

Let u∈D1,p(RN) be a positive solution to the critical equation −Δpu=upa∗−1|x|ain RN,where 1<p<N, 0<a<p and pa∗=(N−a)pN−p. By exploiting the method of moving planes, we show that u is radial and strictly radially decreasing about the origin. Consequently, u must have the form u(x)=(N−a)1pN−pp−1p−1pλλp+|x|p−ap−1N−pp−afor some λ>0.

Multiplicity results of nonlocal singular PDEs with critical Sobolev-Hardy exponent

In this article we study a nonlocal equation involving singular and critical Hardy-Sobolev non-linearities, \[\displaylines{(-\Delta_p)^su-\mu \frac{|u|^{p-2}u}{|x|^{sp}}=\lambda u^{-\alpha}+\frac{|u|^{p_s^*(t)-2}u}{|x|^t}, \quad\hbox{in }\Omega, \\ u&gt;0,\quad\text{in }\Omega,\\ \quad u=0, \quad\text{in } \mathbb{R}^N \setminus\Omega }\] where \(\Omega \subset \mathbb{R}^N\) is a bounded domain with Lipschitz boundary and\( (-\Delta_p)^s\) is the fractional p-Laplacian operator.We combine some variational techniques with a perturbation method to show the existenceof multiple solutions.

Infinitely Many Solutions for a Nonlinear Elliptic PDE with Multiple Hardy–Sobolev Critical Exponents

Infinitely Many Solutions for a Nonlinear Elliptic PDE with Multiple Hardy–Sobolev Critical Exponents

Ground state solutions for Schrödinger–Poisson systems on ℝ3$$ {\mathbb{R}}&amp;#x0005E;3 $$ with a weighted critical exponent

In this paper, we are interested in the existence of ground state solutions for the weighted critical Schrödinger–Poisson systems. The weighted critical exponent may be the Hénon–Sobolev, the Sobolev or the Hardy–Sobolev critical exponent. Moreover, the potentials are power type, which may be singular at origin or unbounded. Due to the presence of Possion term, we can not directly obtain the boundedness of the Palais–Smale sequences of the associated functional for some cases. The Pohozaev identity, the minimax principle and some techniques are used to overcome the difficulties.

On the existence and multiplicity of solutions for a class of sub-Laplacian problems involving critical Sobolev–Hardy exponents on Carnot groups

ABSTRACT In this work, we study the following sub-elliptic equations on Carnot group with Hardy-type singularity and critical Sobolev–Hardy exponents where stands for the sub-Laplacian operator on Carnot group , , and is the critical Sobolev–Hardy exponent, Q is the homogeneous dimension with respect to the dilation naturally associated with , d is the natural gauge associated with the fundamental solution of on , and is the horizontal gradient associated with . Through variational methods combined with the theory of genus, we prove that our problems admit infinitely many solutions.

Sub-elliptic problems with multiple critical Sobolev-Hardy exponents on Carnot groups

Sub-elliptic problems with multiple critical Sobolev-Hardy exponents on Carnot groups

Existence of multiple solutions for a nonhomogeneous p-Laplacian elliptic equation with critical Sobolev-Hardy exponent

This paper is concerned with the existence of multiple nontrivial solutions for nonhomogeneous p-Laplacain elliptic problems involving the critical Hardy-Sobolev exponent. The method used here is based on the Nehari manifold.

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.

Multiplicity and asymptotic behavior of solutions to fractional (p,q)-Kirchhoff type problems with critical Sobolev-Hardy exponent

Let \(\Omega\subset\mathbb{R}^{N}\) be a bounded domain with smooth boundary and \(0\in\Omega\). For \(0&lt;s&lt;1\), \(1\le r&lt;q&lt;p\), \(0\le\alpha&lt;ps&lt;N\) and a positive parameter \(\lambda\), we consider the fractional \((p,q)\)-Laplacian problems involving a critical Sobolev-Hardy exponent. This model comes from a nonlocal problem of Kirchhoff type $$\displaylines{ \big(a+b[u]_{s,p}^{(\theta-1)p}\big)(-\Delta)_{p}^{s}u+(-\Delta)_{q}^{s}u =\frac{|u|^{p_{s}^{*}(\alpha)-2}u}{|x|^{\alpha}}+\lambda f(x)\frac{|u|^{r-2}u}{|x|^{c}}\quad \hbox{in }\Omega,\cr u=0\quad\text{in }\mathbb{R}^{N}\setminus\Omega, }$$ where \(a,b&gt;0\), \(c&lt;sr+N(1-r/p)\), \(\theta\in(1,p_{s}^{*}(\alpha)/p)\) and \(p_{s}^{*}(\alpha)\) is critical Sobolev-Hardy exponent. For a given suitable \(f(x)\), we prove that there are least two nontrivial solutions for small \(\lambda\), by way of the mountain pass theorem and Ekeland's variational principle. Furthermore, we prove that these two solutions converge to two solutions of the limiting problem as \(a\to 0^{+}\). For the limiting problem, we show the existence of infinitely many solutions, and the sequence tends to zero when \(\lambda\) belongs to a suitable range. For more information see https://ejde.math.txstate.edu/Volumes/2021/66/abstr.html

Mountain pass solution to a perturbated Hardy–Sobolev equation involving p-Laplacian on compact Riemannian manifolds

ABSTRACT In this paper, we consider the following quasilinear equation: where M is a compact Riemannian manifold with dimension without boundary, and . Here , and are continuous functions on M satisfying some further conditions. The operator is the p-Laplace–Beltrami operator on M associated with the metric g, and is the Riemannian distance on . Moreover, we assume , , and with . The notion is the critical Hardy–Sobolev exponent. With the help of Mountain Pass Theorem, we get the existence results under different assumptions.

Multiple solutions for a quasilinear Schrödinger equation involving critical Hardy–Sobolev exponent with Robin boundary condition

In this paper, we consider the existence of multiple solutions for a quasilinear Schrödinger equation with Robin boundary condition involving critical Hardy–Sobolev exponent as follows: where satisfies suitable condition. Using variable substitution and Mountain Pass Theorem, we prove that the above equation admits at least two nontrivial solutions.

Existence of solutions to Kirchhoff type equations involving the nonlocal p 1& … &p m fractional Laplacian with critical Sobolev-Hardy exponent

The aim of this paper is to study the existence of solutions for Kirchhoff type equations involving the nonlocal fractional Laplacian with critical Sobolev-Hardy exponent where are nonnegative constants and is called the critical Sobolev-Hardy exponent, . Here with is the fractional r-Laplace operator. Ω is an open bounded subset of with smooth boundary and . are continuous functions and f is a Carathéodory function which does not satisfy the Ambrosetti-Rabinowitz condition. By using the Mountain Pass Theorem, we obtain the existence of solutions for the above problem. Furthermore, using Fountain Theorem, we get the existence of infinitely many solutions for the above problem when . We also study the existence of two nontrivial solutions for the Kirchhoff type equation involving the fractional p-Laplacian via Morse theory. Finally, we consider the case and study a degenerate Kirchhoff equation involving Trudinger-Moser nonlinearity. In our best knowledge, it is the first time our problems are studied in this area.