For a bounded open set $Ω\subset\mathbb R^3$ we consider the minimization problem $$ S(a+εV) = \inf_{0\not\equiv u\in H^1_0(Ω)} \frac{\int_Ω(|\nabla u|^2+ (a+εV) |u|^2)\,dx}{(\int_Ωu^6\,dx)^{1/3}} $$ involving the critical Sobolev exponent. The function $a$ is assumed to be critical in the sense of Hebey and Vaugon. Under certain assumptions on $a$ and $V$ we compute the asymptotics of $S(a+εV)-S$ as $ε\to 0+$, where $S$ is the Sobolev constant. (Almost) minimizers concentrate at a point in the zero set of the Robin function corresponding to $a$ and we determine the location of the concentration point within that set. We also show that our assumptions are almost necessary to have $S(a+εV)<S$ for all sufficiently small $ε>0$.
In this paper, we study the decomposition of Nehari manifold for the Brézis-Nirenberg problem with nonhomogeneous Dirichlet boundary conditions. By using this result, the Lusternik-Schnirelman category and the minimax principle, we establish a multiple result (four solutions) for the Brézis{Nirenberg problem with nonhomogeneous Dirichlet boundary conditions.
In the present paper, we study the effect of an elliptic operator in divergence form L defined in a bounded open set on the existence and nonexistence of positive solutions of the associated nonlocal Brezis-Nirenberg problem involving powers of L.
In this article we will study the existence and nonexistence of sign changing solutions for the Brezis-Nirenberg type problem in the Hyperbolic space. We will also establish sharp asymptotic estimates for the solutions and the compactness properties of solutions.
In this paper, we study the Brezis-Nirenberg problem on bounded smooth domains of R3. Using the algebraic topological argument of Bahri-Coron[2] as implemented in [6] combined with the Brendle[4]- Schoen[8]'s bubble construction, we solve the problem for non-contractible domains.
We show that the classical Brezis-Nirenberg problem $$ -Δu=u|u| + λu\ \hbox{in}\ Ω, u=0\ \hbox{on}\ \partialΩ, $$ when $Ω$ is a bounded domain in $\mathbb R^6$ has a sign-changing solution which blows-up at a point in $Ω$ as $λ$ approaches a suitable value $λ_0>0.$
We consider the classical Brezis-Nirenberg problem in the unit ball of $\mathbb{R}^N$, $N\geq 3$ and analyze the asymptotic behavior of nodal radial solutions in the low dimensions $N=3,4,5,6$ as the parameter converges to some limit value which naturally arises from the study of the associated ordinary differential equation.
We develop a new method for estimating the region of the spectral parameter of a generalized Brezis--Nirenberg problem for which there are no, non trivial, smooth solutions. This new method combines the standard Rellich--Pohozaev argument with a Hardy type inequality for bounded domains. The estimates we get are better than the usual estimates for low dimensions.
We prove some existence and nonexistence results for a class of critical $(p,q)$-Laplacian problems in a bounded domain. Our results extend and complement those in the literature for model cases.
In this paper, we prove that the Brezis-Nirenberg problem -Δu = |u|^{p-1}u+εu in Ω; u=0 on \partial Ωwhere Ωis a symmetric bounded smooth domain in R^N, N\geq 7 and p = (N+2)/(N-2), has a solution with the shape of a tower of two bubbles with alternate signs, centered at the center of symmetry of the domain, for all ε> 0 sufficiently small.
By means of variational methods we investigate existence, non-existence as well as regularity of weak solutions for a system of nonlocal equations involving the fractional laplacian operator and with nonlinearity reaching the critical growth and interacting, in a suitable sense, with the spectrum of the operator.
We establish some existence results for the Brezis-Nirenberg type problem of the nonlinear Choquard equation $$-Δu =\left(\int_Ω\frac{|u|^{2_μ^{\ast}}}{|x-y|^μ}dy\right)|u|^{2_μ^{\ast}-2}u+λu\4.14mm\mbox{in}\1.14mm Ω, $$ where $Ω$ is a bounded domain of $\mathbb{R}^N$, with Lipschitz boundary, $λ$ is a real parameter, $N\geq3$, $2_μ^{\ast}=(2N-μ)/(N-2)$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.
We obtain existence, multiplicity, and bifurcation results for the Brezis-Nirenberg problem for the fractional $p$\nobreakdash-Laplacian operator, involving critical Hardy-Sobolev exponents. Our results are mainly extend results in the literature for $α=0$. In the absence of an explicit formula for a minimizer in the fractional Hardy-Sobolev inequality $α\not= 0$, we get around this difficulty by working with certain asymptotic estimates for minimizers recently obtained in \cite{MM}.
This paper is devoted to study a class of integral type Brezis-Nirenbreg problem on the Heisenberg group. It is a class of new nonlinear integral equations on the bounded domains of Heisenberg group and related to the CR Yamabe problems on the CR manifold. Based on the sharp Hardy-Littlewood-Sobolev inequalities, the nonexistence and existence results are obtained by Pohozaev type identity, variational method and blow-up analysis, respectively.
In this paper we extend the well-known concentration -- compactness principle for the Fractional Laplacian operator in unbounded domains. As an application we show sufficient conditions for the existence of solutions to some critical equations involving the fractional $p-$laplacian in the whole ${\mathbb R}^n$.
In this paper, we are concerned with the following type of fractional problems: $$ \begin{cases}\dis (-Δ)^{s} u-μ\frac{u}{|x|^{2s}}-λu=|u|^{2^*_{s}-2}u+f(x,u), &\text{in} Ω,\ \ \, u=0\,&\text{in} \R^N\backslashΩ \end{cases} \eqno {(*)} $$ where $s\in (0,1)$, $2^*_{s}=2N/(N-2s)$ is the critical Sobolev exponent, $f(x,u)$ is a lower order perturbation of critical Sobolev nonlinearity. We obtain the existence of the solution for (*) through variational methods. In particular we derive a Brézis-Nirenberg type result when $f(x,u)=0$.
We consider the problem $Δu + λu +u^5 = 0$, $u>0$, in a smooth bounded domain $Ω$ in ${\mathbb R}^3$, under zero Dirichlet boundary conditions. We obtain solutions to this problem exhibiting multiple bubbling behavior at $k$ different points of the domain as $λ$ tends to a special positive value $λ_0$, which we characterize in terms of the Green function of $-Δ- λ$.
This work is devoted to the Dirichlet problem for the equation (-Δu = λu + |x|^α|u|^{2^*-2} u) in the unit ball of $\mathbb{R}^N$. We assume that $λ$ is bigger than the first eigenvalues of the laplacian, and we prove that there exists a solution provided $α$ is small enough. This solution has a variational characterization as a ground state.
In this paper, we consider the existence and non-existence of non-trivial solution to a Brezis-Nirenberg type problem with singular weights. First, we obtain a compact imbedding theorem which is an extension of the classical Rellich-Kondrachov compact imbedding theorem, and consider the corresponding eigenvalue problem. Secondly, we deduce a Pohozaev type identity and obtained a non-existence result. Thirdly, based on a generalized concentration compactness principle, we will give some abstract conditions when the functional satisfies the (PS)$_c$ condition. Finally, based on the explicit form…
We consider the Brezis--Nirenberg problem for the Laplacian with a singular drift for a (geodesic) ball in both $\mathbb{R}^{n}$ and $\mathbb{S}^n$, $3 \le n \le 5$. The singular drift we consider derives from a potential which is symmetric around the center of the (geodesic) ball. Here the potential is given by a parameter ($δ$ say) times the logarithm of the distance to the center of the ball. In both cases we determine the exact region in the parameter space for which positive smooth solutions of this problem exist and the exact region for which there are no solutions. The parameter space i…
The present paper studies the non-local fractional analogue of the famous paper of Brezis and Nirenberg in [4]. Namely, we focus on the following model, $$\begin{align*}\left(\mathcal{P}\right) \begin{cases} \left(-Δ\right)^s u-λu &= α|u|^{p-2}u + β|u|^{2^*-2}u \quad\mbox{in}\quad Ω,\\ u&=0\quad\mbox{in}\quad\mathbb{R}^N\setminusΩ, \end{cases} \end{align*}$$ where $(-Δ)^s$ is the fractional Laplace operator, $s \in (0,1)$, with $N \geq 3s$, $2<p<2^*$, $β>0, λ, α\in \mathbb{R}$ and establish the existence of nontrivial solutions and sign-changing solutions for the problem $(\mathcal{P})$.
We obtain nontrivial solutions to the Brezis-Nirenberg problem for the fractional $p$-Laplacian operator, extending some results in the literature for the fractional Laplacian. The quasilinear case presents two serious new difficulties. First an explicit formula for a minimizer in the fractional Sobolev inequality is not available when $p \ne 2$. We get around this difficulty by working with certain asymptotic estimates for minimizers recently obtained by Brasco, Mosconi and Squassina. The second difficulty is the lack of a direct sum decomposition suitable for applying the classical linking t…
We consider the Brezis-Nirenberg problem: $$-Δu =λu + |u|^{p-1}u\qquad \mbox{in}\,\, Ω,\quad u=0\,\, \mbox{on}\,\,\ \partialΩ,$$ where $Ω$ is a smooth bounded domain in $\mathbb R^N$, $N\geq 3$, $p=\frac{N+2}{N-2}$ and $λ>0$. In this paper we prove that, if $Ω$ is symmetric and $N=4,5$, there exists a sign-changing solution whose positive part concentrates and blows-up at the center of symmetry of the domain, while the negative part vanishes, as $λ\rightarrow λ_1$, where $λ_1=λ_1(Ω)$ denotes the first eigenvalue of $-Δ$ on $Ω$, with zero Dirichlet boundary condition.
The present paper deals with a parametrized Kirchhoff type problem involving a critical nonlinearity in high dimension. Existence, non existence and multiplicity of solutions are obtained under the effect of a subcritical perturbation by combining variational properties with a careful analysis of the fiber maps of the energy functional associated to the problem. The particular case of a pure power perturbation is also addressed. Through the study of the Nehari manifolds we extend the general case to a wider range of the parameters.
For dimensions $N \geq 4$, we consider the Brézis-Nirenberg variational problem of finding \[ S(εV) := \inf_{0\not\equiv u\in H^1_0(Ω)} \frac{\int_Ω|\nabla u|^2 \, dx +ε\int_ΩV\, |u|^2 \, dx}{\left(\int_Ω|u|^q \, dx \right)^{2/q}}, \] where $q=\frac{2N}{N-2}$ is the critical Sobolev exponent and $Ω\subset \mathbb{R}^N$ is a bounded open set. We compute the asymptotics of $S(0) - S(εV)$ to leading order as $ε\to 0+$. We give a precise description of the blow-up profile of (almost) minimizing sequences and, in particular, we characterize the concentration points as being extrema of a quotient in…
Let $Ω$ be a bounded smooth domain in $\mathbb{R}^N$ with $N\geq 3$, $1<α$, $2^{\ast}=\frac{2N}{N-2}$ and $\{u_α\}\subset H_{0}^{1,2α}(Ω)$ be a critical point of the functional \begin{equation*} I_{α,λ}(u)=\frac{1}{2α}\int\limits_Ω [(1+|\nabla u|^2)^α-1 ]dx-\fracλ{2}\int\limits_Ωu^2dx-\frac{1}{2^{\ast}}\int\limits_Ω|u|^{2^{\ast}}dx. \end{equation*} In this paper, we obtain the limit behaviour of $u_α$ ( $α\rightarrow 1$), energy identity, Pohozaev identity, some integral estimates, etc. And using these results, we prove infinitely many solutions for the following Brezis-Nirenberg problem for $…
In this paper, we study the higher order Brezis-Nirenberg problem under the Navier boundary condition \be\label{eq} \begin{cases} (-Δ)^m u=\varepsilon u+u^{p} & \text { in }\, Ω, \\ u>0 & \text { in }\, Ω, \\ u=-Δu=\cdots=(-Δ)^{m-1} u=0 & \text { on }\, \partial Ω, \end{cases} \ee where $Ω$ is a strictly convex smooth bounded domain in $\mathbb{R}^n$ with $n \geq 4m$, $m \in \mathbb{N}_{+}$, $\varepsilon\in (0,λ_{1})$, $λ_{1}$ is the first Navier eigenvalue for $(-Δ)^{m}$ in $Ω$, and $p=\frac{n+2m}{n-2m}$. We prove that the solutions of \eqref{eq} are unique if either $\varepsilon$ close to $λ…
In this paper we consider nodal radial solutions of the problem $$ \begin{cases} -Δu=|u|^{2^*-2}u+λu&\text{ in }B,\\ u=0&\text{ on }\partial B \end{cases} $$ where $2^*=\frac{2N}{N-2}$ with $3\le N\le6$ and $B$ is the unit ball of $\R^N$. We compute the asymptotics of the solution $u$ as well as $||u||_\infty$, its first zero and other relevant quantities as $ł$ goes to a critical value $\barł$. Also the sign of $ł-\barł$ is established in all cases. This completes an analogous analysis for $N\ge7$ given in [EGPV].
In this work, we develop a study involving some nonlinear partial differential equations on spheres and hemispheres, with the zero Neumann boundary condition, which are so-called Brezis-Nirenberg type problems, and we give conditions on which such equations have only constant solutions. We also extend these results for some nonlinear partial differential systems.
In this work we study the existence of solutions to the critical Brezis-Nirenberg problem when one deals with the spectral fractional Laplace operator and mixed Dirichlet-Neumann boundary conditions, i.e., $$ \left\{\begin{array}{rcl} (-Δ)^su & = & λu+u^{2_s^*-1},\quad u>0\quad\mbox{in}\quad Ω,\\ u & = & 0\quad\mbox{on}\quad Σ_{\mathcal{D}},\\ \displaystyle\frac{\partial u}{\partial ν} & = & 0\quad\mbox{on}\quad Σ_{\mathcal{N}}, \end{array}\right. $$ where $Ω\subset\mathbb{R}^N$ is a regular bounded domain, $\frac{1}{2}<s<1$, $2_s^*$ is the critical fractional Sobolev exponent, $0\leλ\in \math…
This paper is devoted to the existence of positive solutions for a problem related to a fourth-order differential equation involving a nonlinear term depending on a second order differential operator, $$(-Δ)^2 u=λu+ (-Δ)|u|^{p-1}u,$$ in a bounded domain $Ω\subset\mathbb{R}^N$, $N\geq 7$, and assuming homogeneous Navier boundary conditions. In particular, we study a second order equation involving a nonlocal term of the form, $$-Δu=λ(-Δ)^{-1} u+|u|^{p-1}u,$$ under Dirichlet boundary conditions and we prove the existence of positive solutions depending on the positive real parameter $λ>0$, up to…
We consider the Brezis-Nirenberg problem: \begin{equation*} \begin{cases} -Δu = λu + |u|^{2^* -2}u & \hbox{in}\ Ω\\ u=0 & \hbox{on}\ \partial Ω, \end{cases} \end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^N$, $N\geq 3$, $2^{*}=\frac{2N}{N-2}$ is the critical Sobolev exponent and $λ>0$ a positive parameter. The main result of the paper shows that if $N=4,5,6$ and $λ$ is close to zero there are no sign-changing solutions of the form $$u_λ=PU_{δ_1,ξ}-PU_{δ_2,ξ}+w_λ, $$ where $PU_{δ_i}$ is the projection on $H_0^1(Ω)$ of the regular positive solution of the critical problem in …
In this paper we are concerned with the well-known Brezis-Nirenberg problem \begin{equation*} \begin{cases} -Δu= u^{\frac{N+2}{N-2}}+\varepsilon u, &{\text{in}~Ω},\\ u>0, &{\text{in}~Ω},\\ u=0, &{\text{on}~\partial Ω}. \end{cases} \end{equation*} The existence of multi-peak solutions to the above problem for small $\varepsilon>0$ was obtained by Musso and Pistoia. However, the uniqueness or the exact number of positive solutions to the above problem is still unknown. Here we focus on the local uniqueness of multi-peak solutions and the exact number of positive solutions to the above problem fo…
Let $(M,g)$ be a closed locally conformally flat Riemannian manifold of dimension $n \ge 7$ and of positive Yamabe type. If $ξ_0$ denotes a non-degenerate critical point of the mass function we prove the existence, for any $ k \ge 1$ and $\varepsilon >0$, of a positive blowing-up solution $u_{\varepsilon}$ of $$\triangle_g u_{\varepsilon} +\big( c_n S_g +\varepsilon h\big) u_{\varepsilon} = u_{\varepsilon}^{2^*-1},$$ that blows up like the superposition of $k$ positive bubbles concentrating at different speeds at $ξ_0$. The method of proof combines a finite-dimensional reduction with the sharp…
We study the asymptotic behavior, as $λ\rightarrow 0$, of least energy radial sign-changing solutions $u_λ$, of the Brezis-Nirenberg problem \begin{equation*} \begin{cases} -Δu = λu + |u|^{2^* -2}u & \hbox{in}\ B_1\\ u=0 & \hbox{on}\ \partial B_1, \end{cases} \end{equation*} where $λ>0$, $2^*=\frac{2n}{n-2}$ and $B_1$ is the unit ball of $\R^n$, $n\geq 7$. We prove that both the positive and negative part $u_λ^+$ and $u_λ^-$ concentrate at the same point (which is the center) of the ball with different concentration speeds. Moreover we show that suitable rescalings of $u_λ^+$ and $u_λ^-$ conve…
Using a Rellich-Pohozaev argument and Hardy's inequality, we derive an improved bound on the nonlinear eigenvalue for the non existence of radial solutions of a Brezis-Nirenberg problem, with Dirichlet boundary conditions, on a geodesic ball of Hn, for 2<n<4.
In this paper we study the asymptotic and qualitative properties of least energy radial sign-changing solutions of the fractional Brezis--Nirenberg problem ruled by the s-laplacian, in a ball of $\mathbb{R}^n$, when $s \in (0,1)$ and $n > 6s$. As usual, $λ$ is the (positive) parameter in the linear part in $u$, and we consider $λ$ close to zero. We prove that if such solutions vanish at the center of the ball then they vanish everywhere, we establish a bound on the number of sign-changes and, when $s$ is close to $1$, for a suitable value of the parameter $λ$ such solutions change sign exactly…
We consider the positive solutions of the nonlinear eigenvalue problem $-Δ_{\mathbb{H}^n} u = λu + u^p, $ with $p=\frac{n+2}{n-2}$ and $u \in H_0^1(Ω),$ where $Ω$ is a geodesic ball of radius $θ_1$ on $\mathbb{H}^n.$ For radial solutions, this equation can be written as an ODE having $n$ as a parameter. In this setting, the problem can be extended to consider real values of $n.$ We show that if $2<n<4$ this problem has a unique positive solution if and only if $λ\in \left(n(n-2)/4 +L^*\,,\, λ_1\right).$ Here $L^*$ is the first positive value of $L = -\ell(\ell+1)$ for which a suitably defined …
