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Register a new userOn the Brezis-Nirenberg problem for a Kirchhoff type equation in high dimension
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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.
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For a bounded open set $Omegasubsetmathbb R^3$ we consider the minimization problem $$ S(a+epsilon V) = inf_{0 otequiv uin H^1_0(Omega)} frac{int_Omega (| abla u|^2+ (a+epsilon V) |u|^2),dx}{(int_Omega u^6,dx)^{1/3}} $$ 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 $S(a+epsilon V)-S$ as $epsilonto 0+$, 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 $S(a+epsilon V)<S$ for all sufficiently small $epsilon>0$.
For dimensions $N geq 4$, we consider the Brezis-Nirenberg variational problem of finding [ S(epsilon V) := inf_{0 otequiv uin H^1_0(Omega)} frac{int_Omega | abla u|^2 , dx +epsilon int_Omega V, |u|^2 , dx}{left(int_Omega |u|^q , dx right)^{2/q}}, ] where $q=frac{2N}{N-2}$ is the critical Sobolev exponent and $Omega subset mathbb{R}^N$ is a bounded open set. We compute the asymptotics of $S(0) - S(epsilon V)$ to leading order as $epsilon to 0+$. 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 involving the Robin function. This complements the results from our recent paper in the case $N = 3$.
The main purpose of this paper is to establish the existence, nonexistence and symmetry of nontrivial solutions to the higher order Brezis-Nirenberg problems associated with the GJMS operators $P_k$ on bounded domains in the hyperbolic space $mathbb{H}^n$ and as well as on the entire hyperbolic space $mathbb{H}^n$. Among other techniques, one of our main novelties is to use crucially the Helgason-Fourier analysis on hyperbolic spaces and the higher order Hardy-Sobolev-Mazya inequalities and careful study of delicate properties of Greens functions of $P_k-lambda$ on hyperbolic spaces which are of independent interests in dealing with such problems. Such Greens functions allow us to obtain the integral representations of solutions and thus to avoid using the maximum principle to establish the symmetry of solutions.
We establish theorems on the existence and compactness of solutions to the $sigma_2$-Nirenberg problem on the standard sphere $mathbb S^2$. A first significant ingredient, a Liouville type theorem for the associated fully nonlinear Mobius invariant elliptic equations, was established in an earlier paper of ours. Our proof of the existence and compactness results requires a number of additional crucial ingredients which we prove in this paper: A Liouville type theorem for the associated fully nonlinear Mobius invariant degenerate elliptic equations, a priori estimates of first and second order derivatives of solutions to the $sigma_2$-Nirenberg problem, and a B^ocher type theorem for the associated fully nonlinear Mobius invariant elliptic equations. Given these results, we are able to complete a fine analysis of a sequence of blow-up solutions to the $sigma_2$-Nirenberg problem. In particular, we prove that there can be at most one blow-up point for such a blow-up sequence of solutions. This, together with a Kazdan-Warner type identity, allows us to prove $L^infty$ a priori estimates for solutions of the $sigma_2$-Nirenberg problem under some simple generic hypothesis. The higher derivative estimates then follow from classical estimates of Nirenberg and Schauder. In turn, the existence of solutions to the $sigma_2$-Nirenberg problem is obtained by an application of the by now standard degree theory for second order fully nonlinear elliptic operators.
We study the problem of prescribing $sigma_k$-curvature for a conformal metric on the standard sphere $mathbb{S}^n$ with $2 leq k < n/2$ and $n geq 5$ in axisymmetry. Compactness, non-compactness, existence and non-existence results are proved in terms of the behaviors of the prescribed curvature function $K$ near the north and the south poles. For example, consider the case when the north and the south poles are local maximum points of $K$ of flatness order $beta in [2,n)$. We prove among other things the following statements. (1) When $beta>n-2k$, the solution set is compact, has a nonzero total degree counting and is therefore non-empty. (2) When $ beta = n-2k$, there is an explicit positive constant $C(K)$ associated with $K$. If $C(K)>1$, the solution set is compact with a nonzero total degree counting and is therefore non-empty. If $C(K)<1$, the solution set is compact but the total degree counting is $0$, and the solution set is sometimes empty and sometimes non-empty. (3) When $frac{2}{n-2k}le beta < n-2k$, the solution set is compact, but the total degree counting is zero, and the solution set is sometimes empty and sometimes non-empty. (4) When $beta < frac{n-2k}{2}$, there exists $K$ for which there exists a blow-up sequence of solutions with unbounded energy. In this same range of $beta$, there exists also some $K$ for which the solution set is empty.
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