On stable entire solutions of semi-linear elliptic equations with weights

We are interested in the existence versus non-existence of non-trivial stable sub- and super-solutions of {equation} \label{pop} -div(\omega_1 \nabla u) = \omega_2 f(u) \qquad \text{in}\ \ \IR^N, {equation} with positive smooth weights $ \omega_1(x),\omega_2(x)$. We consider the cases $ f(u) = e^u, u^p$ where $p>1$ and $ -u^{-p}$ where $ p>0$. We obtain various non-existence results which depend on the dimension $N$ and also on $ p$ and the behaviour of $ \omega_1,\omega_2$ near infinity. Also the monotonicity of $ \omega_1$ is involved in some results. Our methods here are the methods developed by Farina, \cite{f2}. We examine a specific class of weights $ \omega_1(x) = (|x|^2 +1)^\frac{\alpha}{2}$ and $ \omega_2(x) = (|x|^2+1)^\frac{\beta}{2} g(x)$ where $ g(x)$ is a positive function with a finite limit at $ \infty$. For this class of weights non-existence results are optimal. To show the optimality we use various generalized Hardy inequalities.


Introduction and main results
In this note we are interested in the existence versus non-existence of stable suband super-solutions of equations of the form where f (u) is one of the following non-linearities: e u , u p where p > 1 and −u −p where p > 0. We assume that ω 1 (x) and ω 2 (x), which we call weights, are smooth positive functions (we allow ω 2 to be zero at say a point) and which satisfy various growth conditions at ∞. Recall that we say that a solution u of −∆u = f (u) in R N is stable provided where C 2 c is the set of C 2 functions defined on R N with compact support. Note that the stability of u is just saying that the second variation at u of the energy associated with the equation is non-negative. In our setting this becomes: We say a C 2 sub/super-solution u of (1.1) is stable provided One should note that (1.1) can be re-written as where γ = − log(ω 1 ) and on occasion we shall take this point of view.
Remark 1. Note that if ω 1 has enough integrability then it is immediate that if u is a stable solution of (1.1) we have ω 2 f ′ (u) = 0 (provided f is increasing). To see this let 0 ≤ ψ ≤ 1 be supported in a ball of radius 2R centered at the origin (B 2R ) with ψ = 1 on B R and such that |∇ψ| and so if the right hand side goes to zero as R → ∞ we have the desired result.
The existence versus non-existence of stable solutions of −∆u = f (u) in R N or −∆u = g(x)f (u) in R N is now quite well understood, see [3,4,7,8,9,10,11,1,5,6]. We remark that some of these results are examining the case where ∆ is replaced with ∆ p (the p-Laplacian) and also in many cases the authors are interested in finite Morse index solutions or solutions which are stable outside a compact set. Much of the interest in these Liouville type theorems stems from the fact that the non-existence of a stable solution is related to the existence of a priori estimates for stable solutions of a related equation on a bounded domain.
In [12] equations similar to −∆u = |x| α u p where examined on the unit ball in R N with zero Dirichlet boundary conditions. There it was shown that for α > 0 that one can obtain positive solutions for p supercritical with respect to Sobolev embedding and so one can view that the term |x| α is restoring some compactness. A similar feature happens for equations of the form the value of α can vastly alter the existence versus non-existence of a stable solution, see [5,1,6,8,7]. We now come to our main results and for this we need to define a few quantities: The three equations we examine are and where we restrict (L) to the case p > 1 and (M ) to p > 0. By solution we always mean a C 2 solution. We now come to our main results in terms of abstract ω 1 and ω 2 . We remark that our approach to non-existence of stable solutions is the approach due to Farina, see [9,10,4].
If we assume that ω 1 has some monotonicity we can do better. We will assume that the monotonicity conditions is satisfied for big x but really all ones needs is for it to be satisfied on a suitable sequence of annuli.
There is no positive stable super-solution of (M ) provided I M → 0 as R → ∞ for some 0 < t < p + p(p + 1).
(2) There is no positive stable sub-solution of (L) if If one takes ω 1 = ω 2 = 1 in the above corollary, the results obtained for (G) and (L), and for some values of p in (M ), are optimal, see [9,10,8].
We now drop all monotonicity conditions on ω 1 .
(2) There is no positive stable sub-solution of (L) if Some of the conditions on ω i in Corollary 2 seem somewhat artificial. If we shift over to the advection equation (and we take ω 1 = ω 2 for simplicity) the conditions on γ become: γ is bounded from below and has a bounded gradient.
In what follows we examine the case where ω 1 (x) = (|x| 2 + 1) is positive except at say a point, smooth and where lim |x|→∞ g(x) = C ∈ (0, ∞). For this class of weights we can essentially obtain optimal results. Theorem 1.3. Take ω 1 and ω 2 as above.
(1) If N + α − 2 < 0 then there is no stable sub-solution for (G), (L) (here we require it to be positive) and in the case of (M ) there is no positive stable super-solution. This case is the trivial case, see Remark 1.
Assumption: For the remaining cases we assume that N + α − 2 > 0. In showing that an explicit solution is stable we will need the weighted Hardy inequality given in [2].
for all φ ∈ C ∞ c (R N ) and τ ∈ R. By picking an appropriate function E this gives, Corollary 3. For all φ ∈ C ∞ c and t, α ∈ R. We have

Proof of main results
Proof of Theorem 1.1. (1). Suppose u is a stable sub-solution of (G) with I G , J G → 0 as R → ∞ and let 0 ≤ φ ≤ 1 denote a smooth compactly supported function. Put ψ := e tu φ into (1.2), where 0 < t < 2, to arrive at Now multiply (G) by e 2tu φ 2 and integrate by parts to arrive at 2t ω 1 e 2tu |∇u| 2 φ 2 ≤ ω 2 e (2t+1)u φ 2 − 2 ω 1 e 2tu φ∇u · ∇φ, and now if one equates like terms they arrive at Now substitute φ m into this inequality for φ where m is a big integer to obtain where C m and D m are positive constants just depending on m. We now estimate the terms on the right but we mention that when ones assume the appropriate monotonicity on ω 1 it is the last integral on the right which one is able to drop.
. Now, for fixed 0 < t < 2 we can take m big enough so (2m − 2) (2t+1) 2t ≥ 2m and since 0 ≤ φ ≤ 1 this allows us to replace the power on φ in the first term on the right with 2m and hence we obtain (2.3) .
We now take the test functions φ to be such that 0 ≤ φ ≤ 1 with φ supported in the ball B 2R with φ = 1 on B R and |∇φ| ≤ C R where C > 0 is independent of R. Putting this choice of φ we obtain One similarly shows that So, combining the results we obtain We now estimate this last term. A similar argument using Hölder's inequality shows that Combining the results gives that and now we send R → ∞ and use the fact that I G , J G → 0 as R → ∞ to see that ω 2 e (2t+1)u = 0, which is clearly a contradiction. Hence there is no stable sub-solution of (G).
(2). Suppose that u > 0 is a stable sub-solution (super-solution) of (L). Then a similar calculation as in (1) shows that for p − p(p − 1) < t < p + p(p − 1), (0 < t < 1 2 ) one has One now applies Hölder's argument as in (1) but the terms I L and J L will appear on the right hand side of the resulting equation. This shift from a sub-solution to a super-solution depending on whether t > 1 2 or t < 1 2 is a result from the sign change of 2t − 1 at t = 1 2 . We leave the details for the reader. (3). This case is also similar to (1) and (2).
Proof of Theorem 1.2. (1). Again we suppose there is a stable sub-solution u of (G). Our starting point is (2.2) and we wish to be able to drop the term −D m e 2tu φ 2m−1 ∇ω 1 · ∇φ, from (2.2). We can choose φ as in the proof of Theorem 1.1 but also such that ∇φ(x) = −C(x)x where C(x) ≥ 0. So if we assume that ∇ω 1 · x ≤ 0 for big x then we see that this last term is non-positive and hence we can drop the term. The the proof is as before but now we only require that lim R→∞ I G = 0.
(2). Suppose that u > 0 is a stable sub-solution of (L) and so (2.7) holds for all p − p(p − 1) < t < p + p(p − 1). Now we wish to use monotonicity to drop the term from (2.7) involving the term ∇ω 1 · ∇φ. φ is chosen the same as in (1) but here one notes that the co-efficient for this term changes sign at t = 1 and hence by restriction t to the appropriate side of 1 (along with the above condition on t and ω 1 ) we can drop the last term depending on which monotonicity we have and hence to obtain a contraction we only require that lim R→∞ I L = 0. The result for the non-existence of a stable super-solution is similar be here one restricts 0 < t < 1 2 . (3). The proof here is similar to (1) and (2) and we omit the details.
(1). Since ∇ω 1 ·x ≤ 0 for big x we can apply Theorem 1.2 to show the non-existence of a stable solution to (G). Note that with the above assumptions on ω i we have that For N ≤ 9 we can take 0 < t < 2 but close enough to 2 so the right hand side goes to zero as R → ∞.
Both (2) and (3) also follow directly from applying Theorem 1.2. Note that one can say more about (2) by taking the multiple cases as listed in Theorem 1.2 but we have choice to leave this to the reader.
Proof of Corollary 2. Since we have no monotonicity conditions now we will need both I and J to go to zero to show the non-existence of a stable solution. Again the results are obtained immediately by applying Theorem 1.1 and we prefer to omit the details.
Proof of Theorem 1.3. (1). If N + α − 2 < 0 then using Remark 1 one easily sees there is no stable sub-solution of (G) and (L) (positive for (L)) or a positive stable super-solution of (M ). So we now assume that N + α − 2 > 0. Note that the monotonicity of ω 1 changes when α changes sign and hence one would think that we need to consider separate cases if we hope to utilize the monotonicity results. But a computation shows that in fact I and J are just multiples of each other in all three cases so it suffices to show, say, that lim R→∞ I = 0.
• (β − α + 2 < 0) Here we take u(x) = 0 in the case of (G) and u = 1 in the case of (L) and (M ). In addition we take g(x) = ε. It is clear that in all cases u is the appropriate sub or super-solution. The only thing one needs to check is the stability. In all cases this reduces to trying to show that we have for all φ ∈ C ∞ c where σ is some small positive constant; its either ε or pε depending on which equation were are examining. To show this we use the result from Corollary 3 and we drop a few positive terms to arrive at which holds for all φ ∈ C ∞ c and t, α ∈ R. Now, since N + α − 2 > 0, we can choose t such that − α 2 < t < n−2 2 . So, the integrand function in the right hand side is positive and since for small enough σ we have σ ≤ (t + α 2 )(N − 2(t + 1) |x| 2 1 + |x| 2 ) for all x ∈ R N we get stability.
With a simple calculation one sees we need just to have If one takes t = N −2 2 in the case where N = 2 and t close to zero in the case for N = 2 one easily sees the above inequalities both hold, after considering all the constraints on α, β and N .