Hardy Sobolev Exponent 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.

Results on the Existence and Multiple Solutions of a p-Kirchhoff Equation with a Subcritical Hardy-Sobolev Exponent and Concave-Convex Nonlinearities

Results on the Existence and Multiple Solutions of a p-Kirchhoff Equation with a Subcritical Hardy-Sobolev Exponent and Concave-Convex Nonlinearities

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.

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.

Multiple solutions for a boundary singular semilinear equation with sublinear term involving Hardy potential and Hardy–Sobolev exponent

Multiple solutions for a boundary singular semilinear equation with sublinear term involving Hardy potential and Hardy–Sobolev exponent

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.

Multiple solutions for a fractional p&amp;q-Laplacian system involving Hardy-Sobolev exponent

In this paper, we prove the existence of infinitely many solutions for a fractional p&amp;q-Laplacian system involving Hardy-Sobolev exponents and obtain new conclusion under different conditions. The methods used here are based on variational methods and LjusternikSchnirelmann theory.

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.

Multiple solutions for the fractional $p$-Laplacian equation with Hardy–Sobolev exponents

We study the fractional p-Laplacian problem with Hardy–Sobolev exponents. We prove: there is a λ0>0 such that for any λ∈(0,λ0), the above problem possesses infinitely many solutions. We achieve our goal by making use of variational methods, more specifically, the Nehari manifold and Lusternik–Schnirelmann theory.

Positive solution for a boundary singular semilinear equation with Hardy–Sobolev exponent

ABSTRACT In this paper, we establish the existence of at least one positive solution to a singular elliptic equation with zero boundary data and critical Hardy–Sobolev exponent. We show that the existence of the positive solution in high dimension depends on the sign of the mean curvature of the boundary near zero and in the low dimensions, it depends on the mass of the domain.

Multiple solutions for weighted Kirchhoff equations involving critical Hardy-Sobolev exponent

Abstract In this article, we consider a class of Kirchhoff equations with critical Hardy-Sobolev exponent and indefinite nonlinearity, which has not been studied in the literature. We prove very nicely that this equation has at least two solutions in ℝ3. And some known results in the literature are improved.

Existence of solutions to elliptic problems with fractional p-Laplacian and multiple critical nonlinearities in the entire space [formula omitted

Using the Mountain-Pass Theorem of Ambrosetti and Rabinowitz we prove that the following fractional p-Laplacian equation with double critical nonlinearities (−Δp)su=|u|ps∗−2u+|u|ps∗(α)−2u|x|αinRN,admits a positive solution in the class Ẇs,p(RN). In the above, (−Δp)s is the fractional p-Laplacian, s∈(0,1), p>1, 0<α<ps<N, ps∗=NpN−ps is the critical fractional Sobolev exponent and ps∗(α)=p(N−α)N−ps is the critical Hardy–Sobolev exponent, Ẇs,p(RN) denotes the completion of C0∞(RN) with respect to Gagliardo norm [u]s,pp≔∫RN∫RN|u(x)−u(y)|p|x−y|N+psdxdy.Our method is based on the existence of extremals of some fractional Hardy–Sobolev type inequalities, and coupled with some intricate estimates for the nonlocal (s,p)-gradient. Moreover, we also establish the existence of a nontrivial solution to an elliptic system which involves fractional p-Laplacian and critical Hardy–Sobolev exponents in RN.

Positive Solutions for Elliptic Problems with the Nonlinearity Containing Singularity and Hardy-Sobolev Exponents

In this paper, we study multiplicity of positive solutions for a class of semilinear elliptic equations with the nonlinearity containing singularity and Hardy-Sobolev exponents. Using variational methods, we establish the existence and multiplicity of positive solutions for the problem.

A system involving fractional Hardy-Schr&amp;ouml;dinger operators and critical Hardy-Sobolev exponents

We study a system involving fractional Hardy-Schrodinger operators and critical Hardy-Sobolev exponents. Under appropriate conditions, the existence and correlation results of ground state solutions to this system are obtained. To overcome the lack of compactness, we consider a proper approximate problem with subcritical nonlinear terms defined in a bounded domain. Moreover, a compact embedding result for the approximate problem is proved.

Hardy-Sobolev equation on compact Riemannian manifolds involving p-Laplacian

In this paper, we apply the Hardy-Sobolev inequality and minimization method in the spirit of Aubin to obtain the existence theorem for a class of quasilinear elliptic equations with critical Hardy-Sobolev exponent on compact Riemannian manifolds.

Existence and nonexistence results for elliptic equations involving two critical singular nonlinearities at the same pole

In this article, we use variational methods to prove the existence of positive solutions for an elliptic equation with two critical Hardy Sobolev exponents at the same pole and a certain nonlinear perturbation. Some parameters play a crucial role in our work.