The Yamabe equation in a non-local setting

The aim of this paper is to deal with the non-local fractional counterpart of the Laplace equation involving critical non-linearities studied in the famous paper of Brezis and Nirenberg (1983). Namely, our model is the equation \[ { ( − Δ ) s u − λ u = | u | 2 ∗ − 2 u a m p ; in Ω , u = 0 a m p ; in R n ∖ Ω , \left \{ \begin {array}{ll} (-\Delta )^s u-\lambda u=|u|^{2^*-2}u & {\mbox { in }} \Omega ,\\ u=0 & {\mbox { in }} \mathbb {R}^n\setminus \Omega \,, \end {array} \right . \] where ( − Δ ) s (-\Delta )^s is the fractional Laplace operator, s ∈ ( 0 , 1 ) s\in (0,1) , Ω \Omega is an open bounded set of R n \mathbb {R}^n , n > 2 s n>2s , with Lipschitz boundary, λ > 0 \lambda >0 is a real parameter and 2 ∗ = 2 n / ( n − 2 s ) 2^*=2n/(n-2s) is a fractional critical Sobolev exponent. In this paper we first study the problem in a general framework; indeed we consider the equation \[ { L K u + λ u + | u | 2 ∗ − 2 u + f ( x , u ) = 0 a m p ; in Ω , u = 0 a m p ; in R n ∖ Ω , \left \{ \begin {array}{ll} \mathcal L_K u+\lambda u+|u|^{2^*-2}u+f(x, u)=0 & \mbox {in } \Omega ,\\ u=0 & \mbox {in } \mathbb {R}^n\setminus \Omega \,, \end {array}\right . \] where L K \mathcal L_K is a general non-local integrodifferential operator of order s s and f f is a lower order perturbation of the critical power | u | 2 ∗ − 2 u |u|^{2^*-2}u . In this setting we prove an existence result through variational techniques. Then, as a concrete example, we derive a Brezis-Nirenberg type result for our model equation; that is, we show that if λ 1 , s \lambda _{1,s} is the first eigenvalue of the non-local operator ( − Δ ) s (-\Delta )^s with homogeneous Dirichlet boundary datum, then for any λ ∈ ( 0 , λ 1 , s ) \lambda \in (0, \lambda _{1,s}) there exists a non-trivial solution of the above model equation, provided n ⩾ 4 s n\geqslant 4s . In this sense the present work may be seen as the extension of the classical Brezis-Nirenberg result to the case of non-local fractional operators.

In this paper we are concerned with the multiplicity of solutions for the following fractional Laplace problem $$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}u= \mu |u|^{q-2}u + |u|^{2^*_s-2}u &{}\quad \text{ in } \Omega u=0 &{}\quad \text{ in } {\mathbb {R}}^n{\setminus } \Omega , \end{array}\right. \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^n$$ is an open bounded set with continuous boundary, $$n>2s$$ with $$s\in (0,1),(-\Delta )^{s}$$ is the fractional Laplacian operator, $$\mu $$ is a positive real parameter, $$q\in [2, 2^*_s)$$ and $$2^*_s=2n/(n-2s)$$ is the fractional critical Sobolev exponent. Using the Lusternik–Schnirelman theory, we relate the number of nontrivial solutions of the problem under consideration with the topology of $$\Omega $$ . Precisely, we show that the problem has at least $$cat_{\Omega }(\Omega )$$ nontrivial solutions, provided that $$q=2$$ and $$n\geqslant 4s$$ or $$q\in (2, 2^*_s)$$ and $$n>2s(q+2)/q$$ , extending the validity of well-known results for the classical Laplace equation to the fractional nonlocal setting.

In this paper, we study the following diffusion model of Kirchhoff-type driven by a nonlocal integro-differential operator where [u]s is the Gagliardo seminorm of u, is a bounded domain with Lipschitz boundary, , is a nonlocal integro-differential operator defined in (), which generalizes the fractional Laplace operator , is the initial function, and is a continuous function and there exist two constants m0 > 0 and such that As is well-known, the nonlocal Kirchhoff problem was first introduced and motivated in Fiscella and Valdinoci (2014 Nonlinear Anal. 94 156–70) and the above problem was studied by Xiang et al (2018 Nonlinearity 31 3228–50), the main results of Xiang et al (2018 Nonlinearity 31 3228–50) are as follows: The local existence of nontrivial, nonnegative weak solution for , where . The blow-up conditions for nontrivial, nonnegative weak solution when J(u0) < 0, where J(u0) denotes the initial energy.The main purpose of this paper is to extend the above results and we get: The global existence of nontrivial, nonnegative weak solution for any . The global existence and blow-up conditions for nontrivial, nonnegative weak solution when for the case , where d is a positive constant given in ().

We are concerned with the following elliptic equation with a general nonlocal integrodifferential operator $${\mathcal {L}}_K$$ $$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} -{\mathcal {L}}_Ku=\lambda u+f(x,u), &{}\quad \text {in}\quad \Omega ,\\ u=0, &{} \quad \text {in}\quad {\mathbb {R}}^n{\setminus }\Omega , \end{array}\right. \end{aligned} \end{aligned}$$ where $$\Omega $$ be an open-bounded set of $${\mathbb {R}}^n$$ with continuous boundary, $$\lambda \in {\mathbb {R}}$$ is a real parameter, and f is a nonlinear term with subcritical growth. We show the existence of a ground state and infinitely many pairs of solutions. The proof is based on the method of Nehari manifold for the equation with $$\lambda <\lambda _1$$ , where $$\lambda _1$$ is the first eigenvalue of the nonlocal operator $$-{\mathcal {L}}_K$$ with homogeneous Dirichlet boundary condition, and the method of generalized Nehari manifold for the equation with $$\lambda \ge \lambda _1$$ . As a concrete example, we derive the existence and multiplicity of solutions for the equation driven by fractional Laplacian $$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^\alpha u=\lambda u+f(x,u),&{}\quad \text {in}\quad \Omega ,\\ u=0, &{}\quad \text {in}\quad {\mathbb {R}}^n{\setminus }\Omega , \end{array}\right. \end{aligned} \end{aligned}$$ where $$0<\alpha <1$$ . The results presented here may be viewed as the extension of some classical results for the Laplacian to nonlocal fractional setting.

This paper deals with multiplicity and bifurcation results for nonlinear problems driven by the fractional Laplace operator $$(-\Delta )^s$$ and involving a critical Sobolev term. In particular, we consider $$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^su=\gamma \left| u\right| ^{2^*-2}u+f(x,u) &{} \text{ in } \Omega u=0 &{} \text{ in } \mathbb {R}^n{\setminus } \Omega , \end{array}\right. \end{aligned}$$ where $$\Omega \subset \mathbb {R}^n$$ is an open bounded set with continuous boundary, $$n>2s$$ with $$s\in (0,1)$$ , $$\gamma $$ is a positive real parameter, $$2^*=2n/(n-2s)$$ is the fractional critical Sobolev exponent and f is a Caratheodory function satisfying different subcritical conditions. For this problem we prove two different results of multiple solutions in the case when f is an odd function. When f has not any symmetry it is still possible to get a multiplicity result: we show that the problem under consideration admits at least two solutions of different sign.

In this paper we consider a resonance problem driven by a non-local integrodifferential operator \mathcal L_K with homogeneous Dirichlet boundary conditions. This problem has a variational structure and we find a solution for it using the Saddle Point Theorem. We prove this result for a general integrodifferential operator of fractional type and from this, as a particular case, we derive an existence theorem for the following fractional Laplacian equation \left\{ \begin{alignedat}{2} (-\Delta)^s u&amp;=\lambda a(x)u+f(x,u)&amp; \quad&amp;{\text{in }} \Omega\\ u&amp;=0&amp; &amp;{\text{in }} \mathbb{R}^n\setminus \Omega, \end{alignedat} \right. when \lambda is an eigenvalue of the related non-homogenous linear problem with homogeneous Dirichlet boundary data. Here the parameter s\in (0,1) is fixed, \Omega is an open bounded set of \mathbb R^n , n&gt;2s , with Lipschitz boundary, a is a Lipschitz continuous function, while f is a sufficiently smooth function. This existence theorem extends to the non-local setting some results, already known in the literature in the case of the Laplace operator -\Delta .

In this paper, we study the existence and multiplicity of solutions for a class of quasi-linear elliptic problems driven by a nonlocal integro-differential operator with homogeneous Dirichlet boundary conditions. As a particular case, we study the following problem: \begin{equation*} \left\{ \begin{array}{l} (-\Delta)_p^s u= f(x,u) \quad \hfill \textrm{in} \ \Omega,\\ \quad u=0 \ \hfill \textrm{in} \ R^N \setminus \Omega, \end{array} \right.\\ \end{equation*} where $(-\Delta)_p^s$ is the fractional p-Laplacian operator, $\Omega$ is an open bounded subset of $R^N$ with Lipschitz boundary and $f:\Omega \times R \to R$ is a generic Carath\'eodory function satisfying either a $p-$sublinear or a $p-$superlinear growth condition.

In this paper, we study the following fractional Kirchhoff-type equation $$\begin{aligned}{\left\{ \begin{array}{ll} -(a+ b\int _{{\mathbb {R}}^{N}}\int _{{\mathbb {R}}^{N}}|u(x)-u(y)|^{2}K(x-y)dxdy){\mathcal {L}}_{K}u=|u|^{2_{\alpha }^{*}-2}u+\mu f(u), ~\ x\in \Omega ,\\ u=0, ~\ x\in {\mathbb {R}}^{N}\backslash \Omega , \end{array}\right. } \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^{N}$$ is a bounded domain with a smooth boundary, $$\alpha \in (0,1)$$ , $$2\alpha<N<4\alpha $$ , $$2_{\alpha }^{*}$$ is the fractional critical Sobolev exponent and $$\mu , a, b>0$$ ; $${\mathcal {L}}_{K}$$ is non-local integrodifferential operator. Under suitable conditions on f, for $$\mu $$ large enough, by using constraint variational method and the quantitative deformation lemma, we obtain a ground state sign-changing (or nodal) solution to this problem, and its energy is strictly larger than twice that of the ground state solutions.

In this paper, we are concerned with the problem driven by a non-local integro-differential operator with homogeneous Dirichlet boundary conditions. As a particular case, we study multiple solutions for the following non-local fractional Laplace equations:where is fixed parameter, is an open bounded subset of with smooth boundary () and is the fractional Laplace operator. By a variant version of the Mountain Pass Theorem, a multiplicity result is obtained for the above-mentioned superlinear problem without Ambrosetti–Rabinowitz condition. Consequently, the result may be looked as a complete extension of the previous work of Wang and Tang to the non-local fractional setting.

ABSTRACTIn this paper, we study the existence of sign-changing solution for a non-local problem, involving the fractional Laplacian operator and critical growth nonlinearities, namely where Ω is a bounded smooth domain of , , is the fractional critical Sobolev exponent and λ is a positive parameter. Under certain assumptions on f, we show that the problem has a least-energy sign-changing solution for λ large. The proof is based on constrained minimization in a subset of Nehari manifold, containing all the possible solutions which change sign of this equation.

In this paper we complete the study of the following non-local fractional equation involving critical nonlinearities $$ \cases (-\Delta)^s u-\lambda u=|u|^{2^*-2}u & {\text{in }} \Omega,\\ u=0 & {\text{in }} \mathbb{R}^n\setminus \Omega, \endcases $$ started in the recent papers \cite{13}, \cite{17}-\cite{19}. Here $s\in (0,1)$ is a fixed parameter, $(-\Delta )^s$ is the fractional Laplace operator, $\lambda$ is a positive constant, $2^*=2n/(n-2s)$ is the fractional critical Sobolev exponent and $\Omega$ is an open bounded subset of $\RR^n$, $n> 2s$, with Lipschitz boundary. Aim of this paper is to study this critical problem in the special case when $n\not=4s$ and $\lambda$ is an eigenvalue of the operator $(-\Delta)^s$ with homogeneous Dirichlet boundary datum. In this setting we prove that this problem admits a non-trivial solution, so that with the results obtained in \cite{13}, \cite{17}-\cite{19}, we are able to show that this critical problem admits a nontrivial solution provided \roster \item"$\bullet$" $n> 4s$ and $\lambda> 0$, \item"$\bullet$" $n=4s$ and $\lambda> 0$ is different from the eigenvalues of $(-\Delta)^s$, \item"$\bullet$" $2s< n< 4s$ and $\lambda> 0$ is sufficiently large. \endroster In this way we extend completely the famous result of Brezis and Nirenberg (see \cite{4}, \cite{5}, \cite{9}, \cite{23}) for the critical Laplace equation to the non-local setting of the fractional Laplace equation.

<abstract> In this paper, we investigate the existence of the least energy sign-changing solutions for nonlinear elliptic equations driven by nonlocal integro-differential operators with critical nonlinearity. By using constrained minimization method and topological degree theory, we obtain a least energy sign-changing solution for them under much weaker conditions. As a particular case, we drive an existence theorem of sign-changing solutions for the fractional Laplacian equations with critical growth. </abstract>

This paper is devoted to the family of optimal functional inequalities on the n-dimensional sphere , namely∥F∥Lq⁢(𝕊n)2-∥F∥L2⁢(𝕊n)2q-2≤𝖢q,s⁢∫𝕊nF⁢ℒs⁢F⁢𝑑μ for all ⁢F∈Hs/2⁢(𝕊n),where denotes a fractional Laplace operator of order , , is a critical exponent, and is the uniform probability measure on . These inequalities are established with optimal constants using spectral properties of fractional operators. Their consequences for fractional heat flows are considered. If , these inequalities interpolate between fractional Sobolev and subcritical fractional logarithmic Sobolev inequalities, which correspond to the limit case as . For , the inequalities interpolate between fractional logarithmic Sobolev and fractional Poincaré inequalities. In the subcritical range , the method also provides us with remainder terms which can be considered as an improved version of the optimal inequalities. The case is also considered. Finally, weighted inequalities involving the fractional Laplacian are obtained in the Euclidean space, by using the stereographic projection.

This paper investigates a non-local system involving the fractional p-Laplacian with a critical Sobolev-Hardy exponent. The system is characterized by coupled nonlinear equations with fractional diffusion terms, Hardy-type singular potentials, and critical nonlinearities. Specifically, we consider 0\\ {\\rm in }\\ \\Omega, \\quad u = v = 0 \\ {\\rm in } \\ \\mathbb{R}^n \\setminus\\Omega. \\end{array} \\right. \\] ]]> ( S ) { ( − Δ p ) s u − μ | u | p − 2 u | x | sp = λa ( x ) | u | q − 2 u | x | θ + c ( x ) pα α + β | u | α − 2 u | v | β | x | t in Ω , ( − Δ p ) s v − μ | v | p − 2 v | x | sp = γb ( x ) | v | q − 2 v | x | θ + c ( x ) pβ α + β | u | α | v | β − 2 v | x | t in Ω , u , v > 0 in Ω , u = v = 0 in R n ∖ Ω . where Ω ⊂ R N is a smooth bounded domain containing the origin ( 0 ∈ Ω ), N>sp, s ∈ ( 0 , 1 ) , and θ , t ∈ [ 0 , sp ) . The parameters satisfy 1<q<p, 1 $ ]]> α , β > 1 with α + β = p s ∗ ( t ) , where p s ∗ ( t ) = p ( N − t ) N − sp is the critical Sobolev–Hardy exponent. The parameters λ and γ are positive, and 0 ≤ μ < μ H , where μ H is the optimal constant in the fractional Hardy–Sobolev inequality. The functions a ( x ) , b ( x ) , and c ( x ) are non-negative, continuous, and have compact support in Ω, satisfying additional technical conditions. The fractional p-Laplacian ( − Δ p ) s is defined for smooth functions as: ( − Δ p ) s u ( x ) = 2 lim ϵ ↘ 0 ⁡ ∫ R N ∖ B ϵ | u ( x ) − u ( y ) | p − 2 ( u ( x ) − u ( y ) ) | x − y | N + sp d y , x ∈ R N . where B ϵ is a ball of radius ϵ centered at x. We use the Nehari manifold method combined with the fibering maps, in order to show the existence of at least two positive solutions for the system under suitable conditions on the parameters ( λ , γ ) . The analysis involves variational techniques, careful estimates to handle the critical exponent and singular terms, and compactness arguments to overcome the lack of compactness in the problem. This work extends previous results on fractional p-Laplacian systems by incorporating Hardy-type singularities and critical nonlinearities, providing new insights into the multiplicity of solutions in non-local settings.

High perturbations of critical fractional Kirchhoff equations with logarithmic nonlinearity