Critical Sobolev Hardy Research Articles

The Choquard Equation with Weighted Terms and Sobolev-Hardy Exponent

We study a nonlinear Choquard equation with weighted terms and critical Sobolev-Hardy exponent. We apply variational methods and Lusternik-Schnirelmann category to prove the multiple positive solutions for this problem.

$$\varvec{p}$$-Laplacian problems involving critical Hardy–Sobolev exponents

We prove existence, multiplicity, and bifurcation results for p-Laplacian problems involving critical Hardy–Sobolev exponents. Our results are mainly for the case $$\lambda \ge \lambda _1$$ and extend results in the literature for $$0< \lambda < \lambda _1$$ . In the absence of a direct sum decomposition, we use critical point theorems based on a cohomological index and a related pseudo-index.

On a Degenerate p-Fractional Kirchhoff Equations Involving Critical Sobolev–Hardy Nonlinearities

In this paper, we study a class of degenerate p-fractional Kirchhoff equations with critical Hardy–Sobolev nonlinearities. By means of the Kajikiya’s new version of the symmetric mountain pass lemma, we obtain the existence of infinitely many solutions which tend to zero under a suitable value of \(\lambda \). The main feature and difficulty of our equations is the fact that the Kirchhoff term M could be zero at zero, that is the equation is degenerate. To our best knowledge, our results are new even in the Laplacian and p-Laplacian cases.

Existence of infinitely many solutions for degenerate p-fractional Kirchhoff equations with critical Sobolev–Hardy nonlinearities

In this paper, we study a class of degenerate p-fractional Kirchhoff equations with critical Hardy–Sobolev nonlinearities $$\begin{aligned} M\left( \iint _{\mathbb {R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\mathrm{d}x\mathrm{d}y\right) (-\Delta )_p^s u = \lambda w(x)|u|^{q-2}u + \frac{|u|^{p_s^*(\alpha )-2}u}{|x|^\alpha },\quad x\in \mathbb {R}^N, \end{aligned}$$ where $$(-\Delta )^s_p$$ is the fractional p-Laplacian operator with $$0<s<1<p<\infty $$ , dimension $$N>ps$$ , $$1<q<p^{*}_{s}(\alpha )$$ , $$p^*_s(\alpha ) = p(N-\alpha )/(N-ps)$$ is the critical exponent of the fractional Hardy–Sobolev exponent with $$\alpha \in [0,ps)$$ , $$\lambda $$ is a positive parameter, M is a nonnegative function, while w is a positive weight. By means of the Kajikiya’s new version of the symmetric mountain pass lemma, we obtain the existence of infinitely many solutions which tend to zero under a suitable value of $$\lambda $$ . The main feature and difficulty of our equations is the fact that the Kirchhoff term M could be zero at zero, that is the equation is degenerate. To our best knowledge, our results are new even in the Laplacian and p-Laplacian cases.

On a class of (p; q)-Laplacian problems involving the critical Sobolev-Hardy exponents in starshaped domain

Abstract Let Ω ⊂ ℝn be a bounded starshaped domain and consider the (p; q)-Laplacian problem -∆pu - ∆pu = λ(x)|u|p*-2u + μ|u|r-2u where μ is a positive parameter, 1 &lt; q ≤ p &lt; n, r ≥ p* and is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the (p; q)-Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.

Existence and multiplicity of positive solutions for a perturbed semilinear elliptic equation with two Hardy–Sobolev critical exponents

In this paper, by applying Ekeland's variational principle and strong maximum principle, we investigate the existence and multiplicity of positive solutions for a perturbed semilinear elliptic equation with two critical Hardy–Sobolev exponents and boundary singularities.

Positive Radial Solutions for a Class of Semilinear Elliptic Problems Involving Critical Hardy-Sobolev Exponent and Hardy Terms

In this paper, we investigate the solvability of a class of semilinear elliptic equations which are perturbation of the problems involving critical Hardy-Sobolev exponent and Hardy singular terms. The existence of at least a positive radial solution is established for a class of semilinear elliptic problems involving critical Hardy-Sobolev exponent and Hardy terms. The main tools are variational method, critical point theory and some analysis techniques.

Formula omitted]-fractional Kirchhoff equations involving critical nonlinearities

The paper deals with Kirchhoff type equations on the whole space RN, driven by the p-fractional Laplace operator, involving critical Hardy–Sobolev nonlinearities and nonnegative potentials. We present different variational approaches to overcome the lack of compactness at critical levels, due to the presence of critical terms as well as the possibly degenerate nature of the Kirchhoff problem.

On symmetric solutions of a critical semilinear elliptic system involving the Caffarelli-Kohn-Nirenberg inequality in R N $\mathbb{R}^{N}$

This paper deals with the existence and multiplicity of symmetric solutions for a weighted semilinear elliptic system with multiple critical Hardy-Sobolev exponents and singular potentials in $\mathbb{R}^{N}$ . Applying the symmetric criticality principle and Caffarelli-Kohn-Nirenberg inequality, we establish several existence and multiplicity results of G-symmetric solutions under certain appropriate hypotheses on the parameters and the weighted functions.

A Note on the Sign-Changing Solutions for a Double Critical Hardy–Sobolev–Maz’ya Problem

Abstract In this paper, we investigate the following double critical Hardy–Sobolev–Maz’ya problem: { - Δ ⁢ u = μ ⁢ | u | 2 * ⁢ ( t ) - 2 ⁢ u | y | t + | u | 2 * ⁢ ( s ) - 2 ⁢ u | y | s + a ⁢ ( x ) ⁢ u in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , $\left\{\begin{aligned} &amp;\displaystyle{-}\Delta u=\mu\frac{|u|^{2^{*}(t)-2}u}{|% y|^{t}}+\frac{|u|^{2^{*}(s)-2}u}{|y|^{s}}+a(x)u&amp;&amp;\displaystyle\text{in }\Omega% ,\\ &amp;\displaystyle u=0&amp;&amp;\displaystyle\text{on }\partial\Omega,\end{aligned}\right.$ where μ ≥ 0 ${\mu\geq 0}$ , a ⁢ ( x ) &gt; 0 ${a(x)&gt;0}$ , 2 * ⁢ ( t ) = 2 ⁢ ( N - t ) N - 2 ${2^{*}(t)=\frac{2(N-t)}{N-2}}$ , 2 * ⁢ ( s ) = 2 ⁢ ( N - s ) N - 2 ${2^{*}(s)=\frac{2(N-s)}{N-2}}$ , 0 ≤ t &lt; s &lt; 2 ${0\leq t&lt;s&lt;2}$ , x = ( y , z ) ∈ ℝ k × ℝ N - k ${x=(y,z)\in\mathbb{R}^{k}\times\mathbb{R}^{N-k}}$ , 2 ≤ k &lt; N , ( 0 , z * ) ∈ Ω ¯ $2\leq k&lt;N,(0,z^{*})\in\bar{\Omega}$ and Ω is an open bounded domain in ℝ N ${\mathbb{R}^{N}}$ . By applying an abstract theorem presented in [42], we prove that if N &gt; 6 + t ${N&gt;6+t}$ when μ &gt; 0 , ${\mu&gt;0,}$ and N &gt; 6 + s ${N&gt;6+s}$ when μ = 0 , ${\mu=0,}$ and Ω satisfies some geometric conditions, then the above problem has infinitely many sign-changing solutions. The main tool is to estimate the Morse indices of these nodal solutions.

Nonexistence of p-Laplace equations with multiple critical Sobolev–Hardy terms

In this paper, we study the nonexistence of solutions for p -Laplace equations with critical Sobolev–Hardy terms and singular terms by using the Pohozaev identity. And the results can be generalized to the case of p -Laplace involving multiple critical Sobolev–Hardy terms and singular terms.

Solutions of the Quasilinear Elliptic Systems with Combined Critical Sobolev–Hardy Terms

We study the existence of multiple solutions for a quasilinear elliptic system. Based on the Ambrosetti–Rabinowitz mountain-pass theorem and the Rabinowitz symmetric mountain-pass theorem, we establish several existence and multiplicity results for the solutions and G-symmetric solutions under certain suitable conditions.

Quasilinear Elliptic Systems Involving Critical Hardy–Sobolev and Sobolev Exponents

In this paper, a system of quasilinear elliptic equations is investigated, which involves critical exponents and multiple Hardy-type terms. By variational methods and analytic techniques, the existence of positive solutions to the system is established. The conclusions are new even when the Hardy-type terms disappear.

A nonlinear p-Laplace equation with critical Sobolev-Hardy exponents and Robin boundary conditions

In this paper, we are concerned with a nonlinear p-Laplace equation with critical Sobolev-Hardy exponents and Robin boundary conditions. Through a compactness analysis of the functional corresponding to the problem, we obtain the existence of positive solutions for this problem under different assumptions.

Symmetric solutions for singular quasilinear elliptic systems involving multiple critical Hardy-Sobolev exponents

This paper deals with the existence and multiplicity of symmetric solutions for a class of singular quasilinear elliptic systems involving multiple critical Hardy-Sobolev exponents in a bounded symmetric domain. Based upon the symmetric criticality principle of Palais and variational methods, we establish several existence and multiplicity results of G-symmetric solutions under certain appropriate hypotheses on the weighted functions and the parameters.

Sign-changing solutions to elliptic problems with two critical Sobolev-Hardy exponents

We study the following elliptic equation with two Sobolev-Hardy critical exponents \begin{eqnarray} & -\Delta u=\mu\frac{|u|^{2^{*}(s_1)-2}u}{|x|^{s_1}}+\frac{|u|^{2^{*}(s_2)-2}u}{|x|^{s_2}} \quad x\in \Omega, \\ & u=0, \quad x\in \partial\Omega, \end{eqnarray} where $\Omega\subset R^N (N\geq3)$ is a bounded smooth domain, $0\in\partial\Omega$, $0\leq s_2 < s_1 \leq 2$ and $2^*(s):=\frac{2(N-s)}{N-2}$. In this paper, by means of variational methods, we obtain the existence of sign-changing solutions if $H(0)<0$, where $H(0)$ denote the mean curvature of $\partial\Omega$ at $0$.

On elliptic problems with two critical Hardy–Sobolev exponents at the same pole

In this paper we study an elliptic problem involving two different critical Hardy–Sobolev exponents at the same pole. By variational methods and concentration compactness principle, we obtain the existence of positive solution to the considered problem.

On positive G-symmetric solutions of a weighted quasilinear elliptic equation with critical hardy-sobolev exponent

In this paper, we are concerned with a weighted quasilinear elliptic equation involving critical Hardy–Sobolev exponent in a bounded G-symmetric domain. By using the symmetric criticality principle of Palais and variational method, we establish several existence and multiplicity results of positive G-symmetric solutions under certain appropriate hypotheses on the potential and the nonlinearity.

Perturbation methods in semilinear elliptic problems involving critical hardy-sobolev exponent

In this article, we deal with a class of semilinear elliptic equations which are perturbations of the problems with the critical Hardy-Sobolev exponent. Some existence results are given via an abstract perturbation method in critical point theory.

Hardy–Sobolev equations on compact Riemannian manifolds

Let (M,g) be a compact Riemannian Manifold of dimension n≥3, x0∈M, and s∈(0,2). We let 2⋆(s)≔2(n−s)n−2 be the critical Hardy–Sobolev exponent. We investigate the existence of positive distributional solutions u∈C0(M) to the critical equation Δgu+a(x)u=u2⋆(s)−1dg(x,x0)sin M where Δg≔−divg(∇) is the Laplace–Beltrami operator, and dg is the Riemannian distance on (M,g). Via a minimization method in the spirit of Aubin, we prove existence in dimension n≥4 when the potential a is sufficiently below the scalar curvature at x0. In dimension n=3, we use a global argument and we prove existence when the mass of the linear operator Δg+a is positive at x0. As a byproduct of our analysis, we compute the best first constant for the related Riemannian Hardy–Sobolev inequality.