Critical Sobolev Growth Research Articles

On a Schrödinger equation involving fractional [formula omitted]-Laplacian with critical growth and Trudinger–Moser nonlinearity

A nonlinear Schrödinger equation of fractional (N/s1,q)-Laplacian is considered with the Rabinowitz potential, critical Sobolev growth and Trudinger–Moser nonlinearity in RN−ΔN/s1s1u+−Δqs2u+V(ɛx)(uNs1−2u+uq−2u)=λfu+uqs2∗−2u.We establish the global existence of nonnegative ground-state solution for suitable parameter values primarily through variational analysis, fractional Trudinger–Moser inequality and mountain pass approach. It is a crucial ingredient to handle three aspects concerning the limiting setting s1p=N, the critical Sobolev growth and Trudinger–Moser nonlinearity.

Normalized solutions for a Schrödinger system with critical Sobolev growth in R3$\mathbb {R}^3$

AbstractWe study the following critical Schrödinger system in : where , , , and . Any solving such system (for some ) is called the normalized solution in the literature, where the normalization is settled in . We show that this system has a positive ground state for in the case of . For the case of and , we obtain the non‐existence results.

The quasi-linear Brezis-Nirenberg problem in low dimensions

We discuss existence results for a quasi-linear elliptic equation of critical Sobolev growth [3,14] in the low-dimensional case, where the problem has a global character which is encoded in sign properties of the “regular” part for the corresponding Green's function as in [9,11].

Infinitely many positive energy solutions for an elliptic equation involving critical Sobolev growth, Hardy potential and concave–convex nonlinearity

Infinitely many positive energy solutions for an elliptic equation involving critical Sobolev growth, Hardy potential and concave–convex nonlinearity

The weak solution of quasilinear System concerning Sobolev Critical conditions in RN

This work treats a typical type of quasilinear system concerning critical Sobolev growth. In the light of the Lions principle and Mountain pass lemma, the mountain pass-type solutions can be derived.

Radial symmetry for a quasilinear elliptic equation with a critical Sobolev growth and Hardy potential

We consider weak positive solutions to the critical p-Laplace equation with Hardy potential in RN−Δpu−γ|x|pup−1=up⁎−1 where 1<p<N, 0⩽γ<(N−pp)p and p⁎=NpN−p.The main result is to show that all the solutions in D1,p(RN) are radial and radially decreasing about the origin.

Existence of Solutions to Fractional p-Laplacian Systems with Homogeneous Nonlinearities of Critical Sobolev Growth

Abstract In this paper, we investigate the existence of nontrivial solutions to the following fractional p-Laplacian system with homogeneous nonlinearities of critical Sobolev growth: { ( - Δ p ) s ⁢ u = Q u ⁢ ( u , v ) + H u ⁢ ( u , v ) in ⁢ Ω , ( - Δ p ) s ⁢ v = Q v ⁢ ( u , v ) + H v ⁢ ( u , v ) in ⁢ Ω , u = v = 0 in ⁢ ℝ N ∖ Ω , u , v ≥ 0 , u , v ≠ 0 in ⁢ Ω , \left\{\begin{aligned} \displaystyle{}(-\Delta_{p})^{s}u&amp;\displaystyle=Q_{u}(u% ,v)+H_{u}(u,v)&amp;&amp;\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle(-\Delta_{p})^{s}v&amp;\displaystyle=Q_{v}(u,v)+H_{v}(u,v)&amp;&amp;% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u=v&amp;\displaystyle=0&amp;&amp;\displaystyle\phantom{}\text{in }\mathbb{R}% ^{N}\setminus\Omega,\\ \displaystyle u,v&amp;\displaystyle\geq 0,\quad u,v\neq 0&amp;&amp;\displaystyle\phantom{}% \text{in }\Omega,\end{aligned}\right. where ( - Δ p ) s {(-\Delta_{p})^{s}} denotes the fractional p-Laplacian operator, p &gt; 1 {p&gt;1} , s ∈ ( 0 , 1 ) {s\in(0,1)} , p ⁢ s &lt; N {ps&lt;N} , p s * = N ⁢ p N - p ⁢ s {p_{s}^{*}=\frac{Np}{N-ps}} is the critical Sobolev exponent, Ω is a bounded domain in ℝ N {\mathbb{R}^{N}} with Lipschitz boundary, and Q and H are homogeneous functions of degrees p and q with p &lt; q ≤ p s ∗ {p&lt;q\leq p^{\ast}_{s}} and Q u {Q_{u}} and Q v {Q_{v}} are the partial derivatives with respect to u and v, respectively. To establish our existence result, we need to prove a concentration-compactness principle associated with the fractional p-Laplacian system for the fractional order Sobolev spaces in bounded domains which is significantly more difficult to prove than in the case of single fractional p-Laplacian equation and is of its independent interest (see Lemma 5.1). Our existence results can be regarded as an extension and improvement of those corresponding ones both for the nonlinear system of classical p-Laplacian operators (i.e., s = 1 {s=1} ) and for the single fractional p-Laplacian operator in the literature. Even a special case of our main results on systems of fractional Laplacian ( - Δ ) s {(-\Delta)^{s}} (i.e., p = 2 {p=2} and 0 &lt; s &lt; 1 {0&lt;s&lt;1} ) has not been studied in the literature before.

On the singular second-order elliptic equation

Let (M,g) be a smooth and compact Riemannian manifold with dimensions n≥3. This paper deals with the existence of nontrivial solutions for a nonlinear second-order elliptic equation with a singular term and critical Sobolev growth. We provide a geometric application of this singular equation.

Positive solutions of Kirchhoff-Hénon type elliptic equations with critical Sobolev growth

We investigate the following Kirchhoff-Hénon type equation involving the critical Sobolev exponent with Dirichlet boundary condition: \[ - \bigg( a + b \bigg( \int_\Omega \lvert Du \rvert^2 dx \bigg)^{(p-2)/2} \bigg) \Delta u = \Psi u^{q-1} + \lvert x \rvert^\alpha u^{2^* - 1} \] in $\Omega$ included in a unit ball under several conditions. Here, $a, b \geq 0$, $a + b &gt; 0$, $2 &lt; p &lt; q &lt; 2^*$ and $\Psi \in L^\infty(\Omega) \setminus \{ 0 \}$ is a given non-negative function with several conditions. We show that, if either $N = 3$ with $4 &lt; q &lt; 2^* = 6$ or $N \geq 4$, there exists a positive solution for small $\alpha \geq 0$. Our methods includes the mountain pass theorem and the Talenti function.

On a nonlocal problem with critical Sobolev growth

We obtain positive solutions for a nonlinear fractional Laplacian problem with critical Sobolev growth via localization principle and variational methods.

Nondegeneracy of positive solutions to a Kirchhoff problem with critical Sobolev growth

In this paper, we prove uniqueness and nondegeneracy of positive solutions to the following Kirchhoff equations with critical growth −a+b∫R3|∇u|2Δu=u5,u>0inR3,where a,b>0 are positive constants. This result has potential applications in singular perturbation problems concerning Kirchhoff equations.

Nodal solutions for fourth order elliptic equations with critical exponent on compact manifolds

Using a variational method, we prove the existence of nodal solutions to prescribed scalar Q- curvature type equations on compact Riemannian manifolds with boundary. These equations are fourth-order elliptic equations with critical Sobolev growth.

Gradient estimates for solutions to quasilinear elliptic equations with critical sobolev growth and hardy potential

This note is a continuation of the work [17]. We study the following quasilinear elliptic equations -Δpu-μ|x|p|u|p-2u=Q(x)|u|NpN-p-2u, x∈ℝN, where 1 < p < N,0 ≤ μ < ((N-p)/p)p and Q ɛ L∞ ℝN. Optimal asymptotic estimates on the gradient of solutions are obtained both at the origin and at the infinity.

On a variational problem involving critical Sobolev growth in dimension three

In this paper we consider the following nonlinear problem: \({{-\Delta u=Ku^{5}}}\), u > 0 in \({{\Omega}}\), u = 0 on \({{\partial \Omega}}\), where K > 0 in \({{\Omega}}\), K = 0 on \({{\partial \Omega}}\) and \({{\Omega}}\) is a bounded domain of \({{\mathbb{R}^{3}}}\). We prove a version of a Morse lemma at infinity for this problem, which allows us to describe the critical points at infinity of the associated variational functional. Using a topological argument, we prove an existence result.

Some properties of the Paneitz operator and nodal solutions to elliptic equations

Given (M, g) a smooth compact Einstein manifold of dimension , we study the attainability of the second Paneitz operator by generalized metrics in the sense of B. Ammann and E. Humbert. As a consequence, we obtain the existence of nodal solutions to fourth-order elliptic equations involving the operator and with critical Sobolev growth.

A priori estimates and application to the symmetry of solutions for critical p-Laplace equations

We establish pointwise a priori estimates for solutions in D1,p(Rn) of equations of type −Δpu=f(x,u), where p∈(1,n), Δp:=div(|∇u|p−2∇u) is the p-Laplace operator, and f is a Caratheodory function with critical Sobolev growth. In the case of positive solutions, our estimates allow us to extend previous radial symmetry results. In particular, by combining our results and a result of Damascelli–Ramaswamy [6], we are able to extend a recent result of Damascelli–Merchán–Montoro–Sciunzi [7] on the symmetry of positive solutions in D1,p(Rn) of the equation −Δpu=up⁎−1, where p⁎:=np/(n−p).

Decay estimates and a vanishing phenomenon for the solutions of critical anisotropic equations

We investigate the asymptotic behavior of solutions of anisotropic equations of the form −∑i=1n∂xi(|∂xiu|pi−2∂xiu)=f(x,u) in Rn, where pi>1 for all i=1,…,n and f is a Caratheodory function with critical Sobolev growth. This problem arises in particular from the study of extremal functions for a class of anisotropic Sobolev inequalities. We establish decay estimates for the solutions and their derivatives, and we bring to light a vanishing phenomenon which occurs when the maximum value of the pi exceeds a critical value.

Asymptotic behaviors of solutions to quasilinear elliptic equations with critical Sobolev growth and Hardy potential

Optimal estimates on the asymptotic behaviors of weak solutions both at the origin and at the infinity are obtained to the following quasilinear elliptic equations−Δpu−μ|x|p|u|p−2u=Q(x)|u|NpN−p−2u,x∈RN, where 1<p<N, 0≤μ<((N−p)/p)p and Q∈L∞(RN).

Multiple solutions to singular fourth order elliptic equations on compact manifolds

Using the method of Nehari manifolds, we prove the existence of at least two distinct weak solutions to elliptic equation of four order with singularities and with critical Sobolev growth.

Concentrating Bound States for Kirchhoff Type Problems in ℝ3 Involving Critical Sobolev Exponents

Abstract We study the concentration and multiplicity of weak solutions to the Kirchhoff type equation with critical Sobolev growth, where ε is a small positive parameter and a, b &gt; 0 are constants, f ∈ C1(ℝ+,ℝ) is subcritical, V : ℝ3 → ℝ is a locally Hölder continuous function. We first prove that for ε0 &gt; 0 sufficiently small, the above problem has a weak solution uε with exponential decay at infinity. Moreover, uε concentrates around a local minimum point of V in Λ as ε → 0. With minimax theorems and Ljusternik-Schnirelmann theory, we also obtain multiple solutions by employing the topological construction of the set where the potential V(z) attains its minimum.