Non-existence and uniqueness results for supercritical semilinear elliptic equations

This paper is devoted to non-existence and uniqueness results for various supercritical semilinear elliptic equations depending on a bifurcation parameter, in a star-shaped domain in $\mathbb{R}^{d}$. We shall distinguish the multiplicative case when the equation can be written as

and the additive case for which the equation is

We shall also distinguish two sub-cases for each equation: the local case when $\lambda$ and $\mu$ are the bifurcation parameters, and the non-local case when $\lambda$ and $\mu$ are determined by a non-local condition, respectively

and

In the multiplicative non-local case, the equation is

In many applications, the term $f(u)/\int_{\Omega}f(u)\,dx$ is interpreted as a probability measure and $\kappa$ is a coupling parameter. Such a parameter arises from physical constants after a proper adimensionalization. In the additive non-local case, the problem to solve is

The parameter $M$ is typically a mass and, in a variational setting, $\mu$ can be interpreted as a Lagrange multiplier associated with the mass constraint, that is, a chemical potential from the point of view of physics. We shall consider the four problems, (1), (2), (3) and (4), and prove that if the domain $\Omega$ is star-shaped, with boundary $\partial\Omega$ in $C^{2,\gamma}$, $\gamma\in(0,1)$, and if $f$ is a non-decreasing nonlinearity with supercritical growth at infinity, such that $f(0)>0$ in the case of (1) or (3), or such that $f>0$ on $(\bar{\mu},\infty)$ and $\lim_{\mu\to\bar{\mu}}f(\mu)=0$ for some $\bar{\mu}\in[-\infty,\infty)$ in the case of (2) or (4), then solutions are unique in $L^{\infty}\cap H^{1}_{0}(\Omega)$ in a certain range of the parameters $\lambda$, $\mu$, $\kappa$ or $M$, while no solution exists for large enough values of the same parameters. Typical nonlinearities are the exponential function $f(u)=e^{u}$ and the power law nonlinearity $f(u)=(1+u)^{p}$, for some $p>(d+2)/(d-2)$, $d\geq 3$. In the exponential case, (1) is the well known Gelfand equation, cf. [36].

Our approach is based on Pohožaev’s estimate, see [55], which is obtained by multiplying the equations by $(x\cdot\nabla u)$, integrating over $\Omega$ and then integrating by parts. Also see [63] for an earlier result based on the local dilation invariance in a linear setting. In this paper, we shall only consider solutions in $L^{\infty}\cap H^{1}_{0}(\Omega)$, which are therefore classical solutions, so that multiplying the equation by $u$ or by $(x\cdot\nabla u)$ is allowed. Some results can be extended to the $H^{1}_{0}(\Omega)$ framework, but some care is then required.

This paper is organized as follows. In Section 2, we consider the multiplicative local and non-local bifurcation problems, respectively (1) and (3). In Section 3, we study the additive local and non-local bifurcation problems, respectively (2) and (4). In all cases, we establish non-existence and uniqueness results, and give some indications on how to construct the branches of solutions, although this is not our main purpose.

Before giving the details of our results, let us give a brief review of the literature. Concerning (1), we primarily refer to the contributions of K. Schmitt in [65] and R. Schaaf in [64], which cover even more general cases than ours and will be discussed more thoroughly later in this section.

The parameter $\lambda$ in (1) can be seen as a bifurcation parameter. Equation (1) is sometimes called a nonlinear eigenvalue problem. It is well known that for certain values of $\lambda$, multiplicity of solutions can occur, see for instance [40]. In some cases there are infinitely many positive solutions, even in the radial case, when $\Omega$ is a ball. Radial solutions have been intensively studied. We refer for instance to [19] for a review of problems with positone structure, i.e. for which $f(0)<0$ and $f$ changes sign once on $\mathbb{R}^{+}$. A detailed analysis of bifurcation diagrams can be found in [52, 53]. Also see [43] for earlier and more qualitative results. Positive bounded solutions of such a nonlinear scalar field equation are often called ground states and can be characterized in many problems as minimizers of a semi-bounded coercive energy functional. They are relevant in many cases of practical interest in physics, chemistry, mathematical biology, etc.

When $\Omega$ is a ball, all bounded positive solutions are radial under rather weak conditions on the nonlinearity $f$, according to [37] and subsequent papers. Lots of efforts have been devoted to uniqueness issues for the solutions of the corresponding ODE and slightly more general problems like quasilinear elliptic ones, see, e.g., [32]. Several other results also cover the case $\Omega=\mathbb{R}^{d}$, see [67]. There are also numerous papers in case of more general nonlinearities, including, for instance, functions of $x$, $u$, and $\nabla u$, or more general bifurcation problems than the ones considered in this paper. It is out of the scope of this introduction to review all of them. In a ball, the set of bounded solutions can often be parametrized. The corresponding bifurcation diagrams have the following properties. For nonlinearities with subcritical growth, for instance for $f(u)=(1+u)^{p}$, $p<(d+2)/(d-2)$, $d\geq 3$, multiple positive solutions may exist when $\lambda$ is positive, small, while for supercritical growths, for example $f(u)=(1+u)^{p}$ with $p>(d+2)/(d-2)$, $d\geq 3$, or $f(u)=e^{u}$ and $d=3$, there is one branch of positive solutions which oscillates around some positive, limiting value of $\lambda$ and solutions are unique only for $\lambda$ positive, small. See [4, 27, 29, 43, 52, 53, 75] for more details.

Another well known fact is that, at least for star-shaped domains, Pohožaev’s method allows to discriminate between super- and subcritical regimes. This approach has been used mostly to prove the non-existence of non-trivial solutions, see [14, 57, 59], and [55, 63] for historical references. Such a method is for instance at the basis of the result of [14] on the Brezis-Nirenberg problem. Also see [4] and references therein for more details. The identity in Pohožaev’s method amounts to consider the effect of a dilation on an energy associated to the solution and therefore carries some important information on the problem, see, e.g., [28]. In this context, stereographic projection and connections between euclidean spaces and spheres are natural, as was already noted in [2] by C. Bandle and R. Benguria.

In this paper we are going to study first the regime corresponding to $\lambda$ small and show that Pohožaev’s method provides a uniqueness result also in cases for which a non-trivial solution exists. The existence of a branch of positive solutions of (1) is a widely studied issue, see for instance [25, 58]. Also see [65] for a review, and references therein. As already said, our two basic examples are based on the power law case, $f(u)=(1+u)^{p}$, and the exponential nonlinearity, $f(u)=e^{u}$, for which useful informations and additional references can be found in [27, 40, 50, 66, 77]. We shall also consider a third example, with a nonlinearity corresponding to the case of Fermi-Dirac statistics, which behaves like a power law for large, positive values of $u$, and like an exponential function for large, negative values of $u$.

The functional framework of bounded solutions and a bootstrap argument imply that we work with classical solutions. Apart from the condition that the domain is star-shaped and satisfies the Poincaré inequality, e.g., is bounded in one direction, we will assume no other geometrical condition. In the local multiplicative case, several uniqueness results are known for small $\lambda>0$, including in the case of Gelfand’s equation, see [46, 64, 65]. One should note that in the framework of the larger space $H_{0}^{1}(\Omega)$, if the boundedness assumption is relaxed, it is not even known if all solutions are radial when $\Omega$ is a ball. The results of [37] and subsequent papers almost always rely on the assumption that the solutions are continuous or at least bounded on $\overline{\Omega}$. Notice that, according to [44, 62], even for a ball, it is possible to prescribe a given isolated singularity which is not centered. In [62], the case of our two basic examples, $f(u)=e^{u}$ and $f(u)=(1+u)^{p}$, with $\frac{d+2}{d-2}<p<\frac{d+1}{d-3}$, $d>3$, has been studied and then generalized to several singularities in [61]. Also see [54] for an earlier result. These singularities are in $H^{1}_{0}(\Omega)$ and, for a given value of a parameter $\lambda$ set apart from zero, they are located at an a priori given set of points. Similar problems on manifolds were considered in [6].

We refer to [3, 35] for bounds on the solutions to Gelfand’s problem, which have been established earlier than uniqueness results but are actually a key tool. Also see [48] for a more recent contribution. Concerning the uniqueness of the solutions to Gelfand’s problem for $d\geq 3$ and $\lambda>0$, small, we refer to [46, 64, 65]. In the case of a ball, this is even known since the paper of D.D. Joseph and T.S. Lundgren, [40], when combined with the symmetry result of [37].

The local multiplicative case corresponding to Problem (1) is the subject of Section 2.1. The literature on such semilinear elliptic problems and associated biffurcation problems is huge. The results of non-existence of non-trivial solutions are well known, see [26, 57, 64] and references therein. Also see [49] for extension to systems. Concerning the uniqueness result on non-trivial solutions, the method was apparently discovered independently by several people including F. Mignot and J.-P. Puel, and X. Cabré and P. Majer, [15], but it seems that the first published reference on uniqueness results by Rellich-Pohožaev type estimates is due to K. Schmitt [65] and later, to R. Schaaf [64]. A more general result for the multiplicative case has been obtained in [13] to the price of more intricate reasoning. Numerous papers have been devoted to the understanding of the role of the geometry and they extend the standard results, mostly the non-existence results, to the case of non strictly star-shaped domains: see for instance [26, 57, 64] and several papers of J. McGough et al., see [46, 47, 48], which are, as far as we know, the most up-to-date results on such issues.

As already mentioned above, Problem (1) has been studied by K. Schmitt in [65] and R. Schaaf in [64]. In [65, Theorem 2.6.7], it is proved that if one replaces $f(u)$ in (1) by a more general function $f(x,u)$ in $C^{2}(\overline{\Omega}\times\mathbb{R}^{+})$ satisfying

then uniqueness holds for a star-shaped domain $\Omega$. A survey on the existence and continuation results for linear and superlinear (sub- and supercritical) growth of the nonlinear term $f$ in (1) can also be found in [65], as well as a study of the influence of the geometry, topology and dimension of the domain, which is of interest for our purpose.

In [64], R. Schaaf studies uniqueness results for the semilinear elliptic problem (1) under the asymptotic condition $\limsup_{u\rightarrow\infty}\tfrac{F(u)}{u\,f(u)}<\tfrac{1}{2}-M(\Omega)$ where $M(\Omega)=1/d$ for star-shaped domains. In general $M(\Omega)$ is some number in the interval $(0,1/d]$. In the autonomous case, the above asymptotic condition is equivalent to the assumption (ii) made by K. Schmitt in [65] or to our assumption (8), to be found below. Our contribution to the question of the uniqueness for (1) relies on a simplification of the proof in [64, 65].

Imposing a non-local constraint dramatically changes the picture. For instance, in case of Maxwell-Boltzmann statistics, $f(u)=e^{u}$, in a ball of $\mathbb{R}^{2}$, the solution of (1) has two solutions for any $\lambda\in(0,\lambda_{*})$ and no solution for $\lambda>\lambda_{*}$, while uniqueness holds in (3) in terms of $M$, for any $M$ for which a solution exists, see [7, 40]. Non-local constraints are motivated by considerations arising from physics. In the case of the exponential nonlinearity with a mass normalization constraint, a considerable effort has been done in the two-dimensional case for understanding the statistical properties of the so-called Onsager solutions of the Euler equation, see [16, 17, 51]. The same model, but rather in dimension $d=3$, is relevant in astrophysical models for systems of gravitating particles, see [13]. Other standard examples are the polytropic distributions, with $f(u)=u^{p}$, and Bose-Einstein or Fermi-Dirac distributions which result in nonlinearities involving special functions. Existence and non-existence results were obtained for instance in [7] and [71, 72], respectively for Maxwell-Boltzmann and Fermi-Dirac statistics.

An evolution model compatible with Fermi-Dirac statistics and the convergence of its solutions towards steady states has been thoroughly examined in [9], while the steady state problem was considered by R. Stańczy in [71, 72, 74]. See [22] and references therein for a model improved with respect to thermodynamics, [72] and references therein for more elaborate models, and [23] for a derivation of an evolution equation involving a mean field term, which also provides a relevant, stationary model studied in [13, 73]. Also see [21, 30] for an alternative, phenomenological derivation of drift-diffusion equations and their stationary counterparts, and [74] for the existence of radial solutions by fixed point methods in weighted function spaces, under nonlocal constraints. The case of a decoupled, external potential goes back to the work of Smoluchowski, see [20, 68]. For this reason, the evolution model is often referred to as the Smoluchowski–Poisson equation.

Our purpose is not to study the above mentioned evolution equations, but only to emphasize that for the corresponding steady states, non-local constraints are very natural, since they correspond to quantities which are conserved along evolution. Hence, to identify the asymptotic state of the solutions to the evolution equation, we have to solve a semilinear elliptic equation with a non-local constraint, which corresponds, for instance, to mass conservation.