On symmetry results for elliptic equations with convex nonlinearities

We investigate partial symmetry of solutions to semi-linear and quasi-linear elliptic problems with convex nonlinearities, in domains that are either axially symmetric or radially symmetric.


Introduction
Let Ω be a smooth bounded domain in R N , N ≥ 2. The goals of this paper are twofold. On the one hand, we extend some symmetry results in axially symmetric domains developed in [7,8] for the semi-linear elliptic equation with a convex nonlinearity (1.1) −∆u = f (x, u) in Ω, to a framework where the energy functional naturally associated with (1.1) is of class C 1 but not of class C 2 , that is to say when f is continuous but not differentiable in the second argument. We give sufficient conditions for symmetry in terms of the local minimality of zero for certain related functionals (see the precise statements in Proposition 2.1 and Corollary 2.6). In addition, we shall provide a further application to constrained minimization problems with convex nonlinearities in Theorem 2.7. In the framework of Morse theory, problems with the same level of regularity were investigated in [1] exploiting suitable tools of nonsmooth analysis. As pointed out in [1], the extension to the nondifferentiable case is worthwhile for certain problems in mathematical ecology where one has to deal with jumping type nonlinearities. It is well-known that, under stronger assumptions on Ω and a monotonicity condition on the mapping |x| → f (|x|, s), symmetry results can be achieved by the celebrated moving plane method (see, e.g., [4,9]). Other partial symmetry results in the framework of symmetrization and polarization theory were obtained in [2,10]. On the other hand, assuming now that f is smooth enough and it grows at infinity sufficiently fast, we obtain some symmetry results for the quasi-linear elliptic problem (1.2) −div(a(u)Du) + a (u) 2 |Du| 2 = f (x, u) in Ω, where a : R → R is smooth, positive and bounded away from zero. To this aim, we use a suitable change of variable procedure, namely, we transform the quasi-linear problem into an associated semi-linear problem −∆v = h(x, v), whose nonlinearity h depends both on a and f . By investigating the convexity or strict convexity properties of the mapping s → h(x, s), we can then apply the symmetry results obtained in [7,8] for the semi-linear case, and finally return to symmetry properties for the original problem (see Theorems 3.2, 3.3, 3.4 and 3.6 for the precise statements). A similar method has been employed in a recent paper of the second author jointly with F.
Gladiali [6], that deals with boundary blow-up solutions. These kinds of quasi-linear problems have been studied since 1995 in the framework of non-smooth critical point theory, being formally associated with (merely) continuous or lower semi-continuous functionals J : H 1 0 (Ω) → R ∪ {+∞}. Some recent applications involving (1.2) have arisen in the study of the so called quasi-linear Schrödinger equation (see [3] and the references therein). Some other applications can be traced back to differential geometry on manifolds with a general metric depending upon the solution itself. We refer the interested reader to the monograph [11] of the second author and to the references therein for further details.

Symmetry for semi-linear problems
Let Ω be a bounded domain in R N , N ≥ 2, that contains the origin and is symmetric with respect to the hyperplane and let u 0 ∈ C 2 (Ω) ∩ C(Ω) be a classical solution of the problem where f is a Carathéodory function on Ω × R that is even in x 1 . In this section we study the symmetry properties of u 0 with respect to x 1 when f is convex in the second variable.
We assume that f satisfies the growth condition where C > 0, r > 1, and where we have used the fact that u 0 solves (2.1) to write Ω ∇u 0 · ∇u = Ω f (x, u 0 ) u. Set , and note that u = 0 is also a critical point of Ψ ± . We will prove that u 0 is even in x 1 under assumptions that involve the convexity of f in t and the type of critical point that Ψ ± or Ψ has at u = 0. Let x := (−x 1 , x 2 , . . . , x N ) be the reflection of x on T and let . In particular, if u 0 ≥ 0 (resp. ≤ 0), it suffices to assume that f (x, ·) is convex on [0, max u 0 ] (resp. [min u 0 , 0]) for a.a. x ∈ Ω.
Proposition 2.1. Assume (2.2) and (C 1 ). If u = 0 is a strict local minimizer of Ψ ± , then u 0 is even in x 1 . If we have strict convexity in (C 1 ), then it suffices to assume that u = 0 is a local minimizer of Ψ ± . This proposition is immediate from the lemma below, which implies that u − ± = 0. Lemma 2.2. If (2.2) and (C 1 ) hold, then If we have strict convexity in (C 1 ) and u − ± = 0, then the strict inequality holds for t ∈ (0, 1). Proof. Since u 0 solves (2.1), u ± solve and testing with u − ± and using u ± (x) gives . Now we assume that for each M > 0, there is a constant C M > 0 such that and strengthen (C 1 ) to (C 2 ) for a.a.
Proof of Proposition 2.3. Since ∂u 0 /∂x 1 = 0 at a critical point of u 0 on T ∩ Ω, u − ± = 0 by Lemma 2.4. By Lemmas 2.5 and 2.2, If we have strict convexity in (C 1 ), then the second inequality in (2.6) is strict for t ∈ (0, 1) and hence the inequality in We now specialize to the case where Ω is either a ball or an annulus centered at the origin O of R N , and f (·, t) is radial for all t ∈ R. If u 0 = 0, then it has a critical point at some P ∈ Ω, and we may apply Proposition 2.3 to any hyperplane containing O and P to get the following Corollary 2.6. Assume (2.2), (2.5), and that f (|x|, ·) is convex for a.a. x ∈ Ω. If P = O and u 0 is not axially symmetric with respect to OP , or if P = O and u 0 is not radially symmetric, then there is a 2-dimensional subspace V ⊂ H 1 0 (Ω) containing sign-definite functions such that u 0 is a local maximizer of Φ| u 0 +V . If f (|x|, ·) is strictly convex for a.a. x ∈ Ω, then u 0 is a strict local maximizer of Φ| u 0 +V . If u 0 ≥ 0, the convexity assumptions are needed only on [0, ∞).
As an application of Corollary 2.6, consider the problem of minimizing the functional Φ defined in (2.3) on the closed set where G(x, t) = t 0 g(x, s) ds for some Carathéodory function g on Ω × R satisfying (2.2) and (2.5) with g in place of f , such that g(·, t) is radial for all t ∈ R. Let u 0 ∈ C 2 (Ω) ∩ C(Ω) be a minimizer, and assume that g(·, u 0 (·)) = 0. Then there is a neighborhood of u 0 in M that is a C 1 -submanifold of H 1 0 (Ω) of codimension 1, and u 0 solves for some λ ∈ R by the Lagrange-multiplier rule.
Proof. Suppose u 0 is not axially symmetric, and set Then there is a 2-dimensional subspace V ⊂ H 1 0 (Ω) containing sign-definite functions such that u 0 is a strict local maximizer of Φ| u 0 +V by Corollary 2.6. For u ∈ M, since u 0 minimizes Φ| M . Thus, to obtain a contradiction, it suffices to show that every neighborhood of u 0 in u 0 + V intersects M at a point different from u 0 . The tangent space to M at u 0 consists of vectors u such that Ω g(x, u 0 ) u = 0, which then have to change sign since g(·, u 0 (·)) is either positive a.e. or negative a.e. Since V contains sign-definite functions, it follows that V is not tangent to M at u 0 . The desired conclusion then follows since dim V > codim M.
For example, consider the eigenvalue problem where g(|x|, ·) is strictly convex for a.a. x ∈ Ω. If Ω g(x, u 0 ) u 0 ≥ 0, we have λ ≥ 0 and then Theorem 2.7 applies. The existence of at least one minimizer with foliated Schwarz symmetry can by obtained (without any convexity requirements) by applying the symmetric constrained version of Ekeland's variational principle proved by the second author in [12, Section 2.4].

Symmetry for quasi-linear problems
In this section we shall consider the quasi-linear elliptic problem (1.2) described in the introduction.
In order to give a precise characterization of the symmetry of the solutions to (1.2) in symmetric domains, we shall convert the (quasi-linear) problem into a corresponding semi-linear problem through a change of variable procedure involving the globally defined Cauchy problem Assuming that a is bounded away from zero from below, (3.1) admits a unique globally defined strictly increasing solution g ∈ C m+1 (R) provided that a ∈ C m (R), for m ∈ N. Furthermore, g is odd whenever a is an even function. A simple direct computation shows that u is a C 2 smooth solution to (1.2) if and only if v = g −1 (u) is a C 2 smooth solution to the semi-linear problem where we have set h(x, s) := f (x, g(s))a −1/2 (g(s)) for x ∈ Ω and s ∈ R. Formally, problem (1.2) is associated with the non-smooth functional J defined by setting while (3.2) is associated with the smoother functional I : where K(x, s) := F (x, g(s)) for all x ∈ Ω and s ∈ R. When F (x, u) ∈ L 1 (Ω) for a given u ∈ H 1 0 (Ω), then one can associate to (1. |f (x, s)| |s| p < ∞, k > 1, 1 < p < (k + 1)N + 2 N − 2 uniformly with respect to x, for every ε > 0, there exists C ε > 0 such that for all x ∈ Ω and all s ∈ R, which implies that I ∈ C 1 (H 1 0 (Ω)). If, furthermore, s → h(x, s) is C 1 , under (3.4) and similar one for a and f , arguing as in [5,Proposition 2.3], it follows that for any ε > 0, there exists C ε > 0 such that |h (x, s)| ≤ C ε + ε|s| 4/ (N −2) , for all x ∈ Ω and all s ∈ R. In turn, for v ∈ H 1 0 (Ω), ) is a continuous function on Ω, v being a solution to (3.2) with v ∈ C 2 (Ω) ∩ C(Ω). After the above connection between problems (1.2) and (3.2) is established, of course one could provide some symmetry results in symmetric domains by using the results that we have obtained in Section 2. On the other hand, we prefer to add stronger regularity assumptions and provide more concrete statements, by applying directly the results of [7,8] by investigating the convexity properties of the maps s → h(x, s) and s → h (x, s). In order to state the main results of this section, the following definition is in order.  Defining m(u, J) directly in a reasonable way seems difficult due to the lack of regularity of J.
We are now ready to state our results. First, we have the following Theorem 3.2. Let Ω be a domain in R N , N ≥ 2, which contains the origin and is symmetric with respect the hyperplane {x 1 = 0} and convex in the x 1 -direction. Let a(s) = 1 + |s| k with k > 1 and let ψ : R N → R + be continuous, even in the x 1 -variable and increasing in the x 1 -variable in {x ∈ Ω : x 1 < 0}. Then, if p > k + 1, any (smooth) solution u to the problem in Ω, is symmetric with respect to x 1 , that is u(−x 1 , x 2 , . . . , x N ) = u(x 1 , x 2 , . . . , x N ).
Secondly, we formulate the following result in radial domains.

Theorem 3.3.
Let Ω be a ball or an annulus in R N , N ≥ 2, a(s) = 1 + |s| k with k > 1, p > k + 1 and ψ : R → R + continuous. Consider a (smooth) index-one solution u to the problem in Ω, Let P ∈ Ω be a maximum point of u and denote by r p the axis passing through the origin and P . Then the following facts hold: (1) u is axially symmetric with respect to r p ; (2) if Ω is a ball and P is the origin, then u is radially symmetric; (3) if u is not radially symmetric, it is never symmetric with respect to any (N −1)-dimensional hyperplane passing through the origin and not passing through the axis r p ; (4) if u is not radially symmetric, all its critical points belong to the symmetry axis r p .
In radial domains, we also have the following partial symmetry results. We recall that a function u is said to be foliated Schwarz symmetric if there exists a unit vector ξ ∈ R N and a function η : R + × R → R such that u(x) = η(|x|, x · ξ) and η(r, ·) is nondecreasing, for all r ≥ 0.

Theorem 3.4.
Let Ω be a ball or an annulus in R N , N ≥ 2, a(s) = 1 + |s| k with k > 1 and let ψ : R → R + be a continuous function. Then there exists p k > 2 such that for every p ≥ p k , any (smooth) solution u to    Then nod(u) ≤ 1 + m(u,J) N +1 . 3.1. Some convexity results. Assume now that, for each fixed x ∈ Ω, the functions s → a(s) and s → f (x, s) are twice differentiable. Observe that, by direct computation, we obtain (3.9) h (x, s) = 2f (x, g(s))a(g(s)) − f (x, g(s))a (g(s)) 2a 2 (g(s)) , for every s ∈ R.
Proof. Assume that k ≥ 2 and p > k + 1. In particular, the functions a and f (x, ·) are of class C 2 on R. Therefore, taking into account formula (3.10), we need to prove that Hence, on account of (3.11), this inequality is fulfilled on R − , and on R + it reads as This can be rearranged as where we have set Then, by assumption, concluding the proof.
Taking into account that Π 1 (p) = O(p 3 ) and Π j (p) = o(p 3 ) as p → ∞ for all j = 2, . . . , 6, in turn there exists p k > 2 such that for every p ≥ p k it holds Then Q p (s) > 0 for all s > 0, yielding the positivity of h (s) for s > 0 and, hence, the strict convexity of h on [0, +∞). This concludes the proof.
Concerning the usual power nonlinearity f (x, s) = ψ(x)|s| p−1 s, we have the following Corollary 3.10. Let p > max{2, k + 1}, f (x, s) = ψ(x)|s| p−1 s for all s ∈ R, where ψ : R N → R + is continuous, and let a(s) = 1 + |s| k with k > 1. Then s → h (x, s) is strictly convex on R for every x ∈ Ω for any p > 2 sufficiently large, depending on the value of k.
Proof. The assertion follows from Proposition 3.9 after observing that, since f is odd and a is even (and hence g is odd), the function h is odd and, in turn, h is even with h (0) = 0. Hence h is strictly convex both on (−∞, 0) and on (0, +∞) and hence on R since h is increasing on (0, +∞) as p > k + 1 in light of Proposition 3.7.
Remark 3.11. Explicit conditions on the magnitude of p with respect to k that guarantees the validity of the assertion of Proposition 3.9 can either obtained by solving directly the inequalities in (3.13) or searching for the absolute minimum point s > 0 of Q p on (0, +∞) which satisfy the quadratic equation for the unknown Ξ = Ξ(p, k) := s k > 0 3(Π 1 (p) + Π 2 (p) + Π 3 (p) + Π 4 (p))Ξ 2 + (6Π 1 (p) + 4Π 2 (p) + 2Π 3 (p))Ξ + (3Π 1 (p) + Π 2 (p)) = 0, and finally imposing Q p (s ) > 0. In the semi-linear (corresponing to the case where a is a constant), it follows that h(x, s) = ψ(x)|s| p−1 s so that the requirement p > 2 is necessary for s → h (x, s) to be strictly convex on R. Figures 2 and 3 show how h is pushed from negative to positive values provided that the value of p is large enough in terms of k (k = 2 and p = 3.2, 4, 5, 7 respectively). For instance, if the dimension N is equal to 3, the values of p such that h > 0 are below the threshold 3k + 5 = ((k + 1)N + 2)/(N − 2) appearing in (3.4) for the growth of f which makes the problem (3.2) subcritical and, thus, nice for the existence theory via variational methods.
3.2. Proofs of Theorems 3.2, 3.3 and 3.4. We are now ready to prove the previously stated symmetry results for the quasi-linear problem.
ij v(x) for all x ∈ Ω and any i, j = 1, . . . , N . Since g > 0, x 0 is a critical point of v if and only if x 0 is a critical point of u, in which case H u (x 0 ) = g (v(x 0 ))H v (x 0 ), where H z (y) denotes the Hessian matrix of z at y. In fact, P is a maximum point for v also, since v(ξ) = g −1 (u(ξ)) ≤ g −1 (u(P )) = v(P ) for all ξ ∈ Ω, g −1 being strictly increasing. On account of Proposition 3.7, the proofs of assertions (1)-(3) follow as in the proof of Theorem 3.2 by applying [7, Theorem 3.1 (i), (ii) and (iii)]. Concerning assertion (4), assume that u is not radially symmetric. Hence, v = g −1 (u) is a nonradial (smooth) solution to −∆v = h(x, v). Whence, by [7, Theorem 3.1(4)], all its critical points belong to the symmetry axis r p , that is to say D j v(ξ) = 0 implies ξ ∈ r P . Since D j u(ξ) = g (v(ξ))D j v(ξ) for all j and g > 0, D j u(ξ) = 0 implies D j v(ξ) = 0. Hence ξ ∈ r P and the proof is complete. with Morse index m(v, I) ≤ N . In light of Corollary 3.10, the function s → h(|x|, s) has a (strictly) convex derivative on R provided that p is sufficiently large, depending on k. Then, by virtue of [8,Theorem 1.1], it follows that v is foliated Schwarz symmetric, namely, there exists a unit vector ξ ∈ R N such that v(x) = η(|x|, ξ · x) for some function η : R + × R → R such that η(r, ·) is nondecreasing for any r ≥ 0. Then u = (g • η)(|x|, ξ · x) and D s (g • η)(|x|, s) = g (η(|x|, s))D s η(|x|, s) ≥ 0 since g > 0 on R. This concludes the proof of the first assertion. The second assertion follows by arguing analogously using [8, Theorem 1.2].