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Uniqueness of ground states for a class of quasi-linear elliptic equations

We prove the uniqueness of positive radial solutions for a class of quasi-linear elliptic problems containing, in particular, the quasi-linear Schrodinger equation.

The goal of this manuscript is to get, for a suitable class of functions a ∈ C 2 (R), the uniqueness of radial standing waves ψ(x, t) = e imt u(x), u > 0, of (1.2), yielding to the quasi-linear elliptic problem in R N (1.3) − div(a(u)Du) + 1 2 a ′ (u)|Du| 2 + mu = u p , in R N .
Physically speaking, the value of the coefficient m > 0 of the linear term has to be interpreted as a frequency. It is readily seen by Sobolev embedding that, if the function a behaves like the power |s| k at infinity, then 1 < p < ((k + 1)N + 2)/(N − 2) is the optimal range for existence of nontrivial solutions of (1.3) in For the existence of ground state solutions to (1.3) in the polynomial quadratic case, we refer to [7,14].
The main results of the paper are the following Theorem 1.1. Let N ≥ 3, a 1 > 0 and ψ ∈ C 2 (R). Assume that there exist 2 < k < 2p such that and a(s) = a 1 |s| k + ψ(s), with Furthermore, assume that Then there exists m 0 > 0 such that the problem u is radially symmetric, admits a unique solution u ∈ X ∩ C 2 (R N ), up to translations, for all m ≥ m 0 .
Of course, in the particular but physically relevant case when k = 2, a 1 = 2 and ψ(s) ≡ 1, one gets uniqueness of ground states of the quasi-linear Schrödinger equation for m large. Essentially, thinking to more general situations where a nonlinearity q(u) is considered in place of u p , what we conjecture that should play a rǒle is the fact that a(s)/q(s) goes to zero for s → +∞, namely the source q is dominant upon the quasi-linear diffusion a(s). In the study of high-frequency standing wave solutions ψ(x, t) = e imt u(x) to (1.2) the value of m > 0 can be taken large, so that the condition in the statement of Theorems According to the achievements established in the recent papers [5,7], for the Schrödinger equation (1.2), the orbital stability range for the standing waves solutions associated with the problem (1.9) min is 1 < p < k +1+4/N, which is contained in the optimal range of the statements of Theorems 1.1-1.2. We point out that the uniqueness, up to translations, of the solutions to problem (1.7) does not easily yield uniqueness of the solutions to the minimization problem (1.9). This further conclusion, which is currently unavailable, would of course be rather useful for both analytical and numerical purposes.

Dual semi-linear problem
Problem (1.7) is formally associated with the functional Let a ∈ C 2 (R) be such that a(s) ≥ ν for any s ∈ R, for some ν > 0. Let g : R → R denote the (strictly increasing) solution to the Cauchy problem The solution is global due to the coercivity of a.

From (2.1) g is invertible and (2.3) follows.
Using g, we can related, as in [1,6], problem (1.7) to the following semi-linear equation and its associated functional I : a(g(t)) .
Consider now the following condition, namely there exists a 1 > 0 such that Lemma 2.2. Let g be as defined in (2.1) and assume condition (2.6). Then, we have Furthermore, it holds Proof. From the monotonicity of g, and from problem (2.1) we have that g(s) → +∞ as s → +∞ and g ′ (s) → 0 as s → +∞. Then, using (2.6), we have lim s→+∞ g(s) In particular, the limit on the left hand side cannot be equal to 0 or +∞. In turn, we get and the claim follows. Moreover, we have completing the proof of (2.8). Finally, observing that For the sake of completeness, we recall the following two propositions.
. Then u is a positive classical solution of (1.7).
Proof. If v ∈ H 1 (R N ), v > 0, is a critical point for I, then v is a positive weak solution of problem (2.4). From regularity theory (see e.g. [2, Lemma 1]) it follows that v ∈ C 2 (R N ). Hence, from regularity and monotonicity of g we have that u ∈ C 2 (R N ) and so that u is a classical solution of (1.7).
Next we prove that uniqueness in H 1 (R N ) for (2.4) yields uniqueness in X for the original problem, where X is as defined in (1.4).

General computations
Motivated by the conclusion of Lemma 2.5, in order to get the desired uniqueness result, in this section we study the uniqueness of positive radial solution of (2.4). To this end, we let and h(s) > 0 for all s > s 0 . Since a ∈ C 2 (R) it follows that g ∈ C 3 (R) and h ∈ C 2 (0, +∞). Then, Observe that it holds Then we only need to evaluate the quantity We start by making some calculations. Using (2.1) and the fact that g(s) > 0 for s > 0, we have that Then, we have Now, using g ′′ = − 1 2 a ′ (g) (g ′ ) 4 and g ′′′ = − 1 2 a ′′ (g) (g ′ ) 5 + (a ′ (g)) 2 (g ′ ) 7 , we have As Notice that, since g ′ > 0, we obtain In particular, in order for K to be asymptotically negative, it is sufficient to find s 1 > 0 and s 2 > 0 depending on the data of the problem, except on the value of m > 0, such that where g 0 is the function satisfying the Cauchy problem , g 0 (0) = 0.
Let us now set, for each s > 0 large,

4.2.
Sign of the term H 1 . Concerning the term H 1 , we have where we have set Then and, since G 0 (s) ≥ 0 for all s ≥ 0 by Lemma 4.1, for every s > 0 we get .
We shall now distinguish two cases.
Case I: (k > 2). In this case, exploiting assumptions (1.6), taking into account that (cf. Lemma 2.2), , as s → +∞, and recalling that G 0 is nondecreasing by virtue of Lemma 4.1, we have Whence, since p > k/2, given ε ∈ (0, 1) there exists s 1 > 0 depending upon a and k (but independent upon the value of m), such that for every s ≥ s 1 .
Recalling Lemma 4.1 (G 0 (s) > 0 eventually for s large), since p > k 2 , it follows that for every s ≥ s 1 .
In conclusion, in any case, H 1 becomes negative for values of s > 0 sufficiently large.
. Then, in light of Lemma 4.1, we get for s > 0 Taking into account assumption (1.6) (for k > 2) and (1.8) (for 0 < k ≤ 2), and observing again that (4.3) holds, it is readily verified that it always holds Hence, since p > 1, there exists s 2 > 0, depending only upon a and k, such that H 2 (s) < 0 for s ≥ s 2 . Then, recalling that s 0 := g −1 (m 1/(p−1) ) (see formula (3.1)), it is sufficient to take m 0 sufficiently large that s 0 ≥s for every m ≥ m 0 , so that (see formula (3.5)) K(s) ≤ 0 for every s ≥ s 0 , yielding (condition (3.2) if fulfilled) the desired uniqueness for problem (2.4), and in turn for the original problem (1.7) through Lemma 2.5.