A p-Laplacian supercritical Neumann problem

For $p>2$, we consider the quasilinear equation $-\Delta_p u+|u|^{p-2}u=g(u)$ in the unit ball $B$ of $\mathbb R^N$, with homogeneous Neumann boundary conditions. The assumptions on $g$ are very mild and allow the nonlinearity to be possibly supercritical in the sense of Sobolev embeddings. We prove the existence of a nonconstant, positive, radially nondecreasing solution via variational methods. In the case $g(u)=|u|^{q-2}u$, we detect the asymptotic behavior of these solutions as $q\to\infty$.


Introduction
For p > 2, we consider the following Neumann problem (1.1) Here B is the unit ball of R N , N ≥ 1, and ν is the outer unit normal of ∂B. We aim to investigate the existence of nonconstant solutions of (1.1) under very mild assumptions on the nonlinearity g, allowing in particular for Sobolev-supercritical growth.
Quasilinear equations with Neumann boundary conditions and subcritical nonlinearities in the sense of Sobolev embeddings have been studied in several papers, among which we refer to [1,2,6,14,18,19,24,26,30] and the references therein.
When g has supercritical growth, a major difficulty in analyzing the existence of solutions of (1.1) is that, due to the absence of Sobolev embeddings, the energy functional associated to the equation is not well defined in W 1,p (B), and so, a priori, it is not possible to apply variational methods. Nonetheless, the problem (1.1) with the prototype nonlinearity g(u) = u q−1 admits the constant solution u ≡ 1 for every q ∈ (1, ∞). This marks a difference with respect to the analogous problem under homogeneous Dirichlet boundary conditions, in which the Pohožaev identity is an insurmountable obstruction to the existence of non-zero solutions when q ≥ p * (see [25,Section 2,). Thus it is a natural question to ask whether (1.1) also admits nonconstant solutions.
This question has been tackled in the case p = 2 and a positive answer has been given in [28,12,11,21]. Multiplicity results have been obtained in [9,8]. The strategy used in [28,11] to obtain existence is that of establishing a priori estimates in some special classes of solutions of (1.1). This, in turn, allows to provide a variational characterization of the problem in the Sobolev space. On the other hand, in [12,21], existence is proved by a perturbative method. In [9,8] the authors apply both a priori estimates and perturbative methods to have multiplicity results. Topological methods have been used in [13] for a related problem.
The case p = 2 with a supercritical nonlinearity has been treated by S. Secchi in [27], where the right-hand side of the equation in (1.1) is of the type a(x)g(u), with a(x) nonconstant. Our paper aims to extend the results in [27] to the case a constant and p > 2. We remark that our method differs from the one in [27]: whereas S. Secchi adapts to the case p = 2 the techniques introduced in [28], we take inspiration from the techniques developed in [11].
We also mention that the existence and multiplicity of solutions to supercritical p-Laplacian problems under homogeneous Dirichlet boundary conditions have been studied in several papers, see for instance [5,15,22] and the references therein.
Our starting point to prove Theorem 1.1 is to work in the cone of nonnegative, radial, nondecreasing functions C := {u ∈ W 1,p rad (B) : u ≥ 0, u(r) ≤ u(s) for all 0 < r ≤ s ≤ 1}, (1.2) introduced by Serra and Tilli in [28], where with abuse of notation we write u(|x|) := u(x). The main advantage of working in this set is the fact that all solutions of (1.1) belonging to C are a priori bounded in W 1,p (B) and in L ∞ (B). Two strategies are available in literature. The first one, see [28,27], consists in defining the energy functional I : C → R associated to the equation and to find a critical point u of I, that is to say u ∈ C such that I (u)[ϕ] = 0 for all ϕ ∈ C. This does not imply that u is a weak solution of the problem. Under additional hypotheses on the nonlinearity g, the authors prove that it actually is.
In order to weaken the assumptions on the nonlinearity g, we follow a different strategy, see [11]. Thanks to the a priori estimates on the solutions of (1.1) belonging to C, we are allowed to truncate the nonlinearity g. Thus, we deal with a new problem involving a Sobolev-subcritical nonlinearity, with the property that all solutions of the new problem belonging to C solve also the original problem (1.1). In this way, the energy functional I associated to the truncated problem is well defined in the whole of W 1,p (B). To get a solution of (1.1), we prove that a mountain pass type theorem holds inside the cone C. The main difficulty here is the construction of a descending flow that preserves C.
Once the mountain pass solution is found, we need to prove that it is nonconstant. We further restrict our cone, working in a subset of C in which the only constant solution of (1.1) is the positive constant u 0 defined in (g 3 ). In this set, we build an admissible curve on which the energy is lower than the energy of the constant u 0 , which gives immediately that the mountain pass solution is not identically equal to u 0 . We remark that this part of the proof heavily relies on the fact that I is of class C 2 , thus it cannot be generalized to the case 1 < p < 2.
In the case in which there is more than one constant u 0 satisfying condition (g 3 ), we work in a restricted cone in order to localize the mountain pass solution. This allows us to prove the multiplicity result stated in Theorem 1.1.
We remark that in the setting p = 2 our hypoteses are slightly more general than the ones in [11]. More precisely, in [11] it is required, for p = 2, Hence our proof also provides the following generalization of [11,Theorem 1.3].
is the second radial eigenvalue of −∆ + I in B with Neumann boundary conditions. Then there exists an increasing radial solution of (1.1).
For p = 2 and g(u) = u q−1 , the result in [9] provides multiple solutions which oscillate around the constant solution u ≡ u 0 = 1. Similar oscillating solutions can be found via a bifurcation technique, as in [10]. There is a branch bifurcating in correspondence to any power q − 1 = λ rad i with i ≥ 2, where λ rad i is the i-th eigenvalue of −∆ + I under homogeneous Neumann boundary conditions in the unit ball. We note in passing that this relation between q and λ rad i seems to be in the same spirit as condition (g 3 ). It would be interesting to understand whether the bifurcation occurs also when p > 2. Theorem 1.1 ensures in particular the existence of a nonconstant, nondecreasing, radial solution of (1.1) in the case g(u) = u q−1 , for every q > p. Denoting by u q such solution, we detect its asymptotic behavior as q → ∞, in the spirit of [20] (see also [21], [9], and [8]). Theorem 1.3. Let p > 2 and g(u) = u q−1 , with q > p. Denote by u q the corresponding positive, nonconstant, radially nondecreasing solution found in Theorem 1.1. Then, as q → ∞, for any ν ∈ (0, 1), where G is the unique solution of In the proof of this theorem, we use the fact that the solutions u q are nondecreasing and that the nonlinearity g is a pure power to get an estimate on the C 1 -norm of u q , which is uniform in q. This ensures the existence of a limit profile function G which is nonnegative and radially nondecreasing. We note that it is delicate to prove that G solves the equation in (1.4) near the boundary ∂B. Heuristically, this comes from the fact that u q (1) > 1 for all q, and so lim q→∞ u q (1) q−1 may be an indeterminate form. In order to prove that G solves actually (1.4) in the whole ball B, we show that the mountain pass levels c q 's tend to a value c ∞ which is a critical level for the energy associated to (1.4). This latter result requires in turn the preliminary proof of the fact that any mountain pass level c q coincides with the minimum of the energy functional on a Nehari-type set already introduced in [28] (that is to say, the Nehari manifold intersected with the cone C). We remark here that the Neumann boundary condition is not preserved in the limit, being ∂ ν G > 0 on ∂B, by Hopf's Lemma (see for instance [17,Theorem 3.3]). Hence, the The paper is organized as follows. In Section 2, we prove a priori estimates for nonnegative, radially nondecreasing solutions of (1.1). In Section 3, we show the existence of a nonnegative, radially nondecreasing solution of (1.1) via a mountain pass type argument. Furthermore, in Section 4 we conclude the proof of Theorem 1.1, by proving the nonconstancy of the solution found in Section 3 and the multiplicity result. A sketch of the proof of Theorem 1.2 is also given in the same section. The asymptotic behavior as q → ∞ of the mountain pass solution of (1.1) in the pure power case is then studied in Section 5. Finally in Appendix A, we collect some partial results valid in the case 1 < p < 2.
As a consequence of the previous lemma, from now on in the paper we consider the equivalent problem where f ∈ C 1 ([0, ∞)) is nonnegative, nondecreasing, and satisfies (f 1 )-(f 3 ). We endow the space W 1,p (B) with the equivalent norm · : W 1,p (B) → R + defined by We look for solutions to (2.1) in W 1,p rad (B), that is to say the space of radial functions in W 1,p (B). Since p > 1, we can assume that W 1,p rad (B)-functions are continuous in (0, 1] and define the cone of nonnegative radially nondecreasing functions as in (1.2). We note that, if u ∈ C, we can set u(0) := lim r→0 + u(r) by monotonicity, and consider u ∈ C(B). Moreover, being nondecreasing, every u ∈ C is differentiable a.e. and u (r) ≥ 0 where it is defined. It is easy to prove that C is a closed convex cone in W 1,p (B), that is to say, the following properties hold for all u, v ∈ C and λ ≥ 0 (iii) if also −u ∈ C, then u ≡ 0; (iv) C is closed for the topology of W 1,p .
The cone C was first introduced in [28] in the case p = 2. It is a useful set when working with Sobolev-supercritical problems because of the following a priori estimates.
Proof. Since u ∈ C is nonnegative and nondecreasing, we get and by the radial symmetry of u ∈ C, for some C > 0 depending only on the dimension N . Moreover, being B bounded, for every q ∈ [1, ∞) there exists a constant C > 0 depending only on N and q, such that u W 1,1 (B) ≤ C u W 1,q (B) for all u ∈ W 1,1 (B). Proof. If N < p the conclusion follows at once by the Rellich-Kondrachov theorem.
In the complementary case, we take into account the fact that C-functions are bounded. More precisely, if we have (u n ) ⊂ C bounded in the W 1,p -norm, there exists u ∈ C such that up to a subsequence u n u in W 1,p (B) and so u n → u in L 1 (B). Therefore, by Lemma 2.2 we get that for every q < ∞, (2.5) The existence of δ, M > 0 follows by (f 2 ) in Lemma 2.1. We introduce the following set of functions We remark that F depends on f only through δ and M . In the remaining of this section, we shall derive some a priori estimates which are uniform in F and hence depend only on δ and M and not on the specific nonlinearity f belonging to F.
Proof. By integrating the equation in (2.7) and using the fact that ϕ ∈ F, we have Thus, where |B| is the volume of the unitary ball of R N , and so for every solution u ∈ C of (2.7) and every ϕ ∈ F.
Proof. Let u ∈ C be a solution of (2.7). We recall that the p-Laplacian of a radial function is given by Hence, since u ≥ 0 a.e., we can write Then, by integrating the equation over the interval (0, r) and using the fact that ϕ is nonnegative, we have where |∂B| is the (N − 1)-dimensional measure of the unitary sphere in R N . Together with (2.8), this gives The first estimate then follows by Lemma 2.2 (by taking q = p − 1), with K ∞ := (1 + m) 1/(p−1) K p−1 C(N, p − 1). Finally, for the last estimate, we multiply the equation of (2.7) by u, we integrate over B, and we obtain which concludes the proof.

Existence of a mountain pass radial solution
In this section we prove the existence of a radial solution of (2.1) via the Mountain Pass Theorem. Since the nonlinearity f is possibly supercritical in the sense of Sobolev spaces, we need to truncate and to replace it by a subcritical function which coincides with f in [0, K ∞ ], K ∞ being defined in Lemma 2.5. Then, we take advantage of the a priori estimates proved in the previous section to guarantee that the mountain pass solution found with the truncated function is indeed a solution of the original problem (2.1).
We define the critical Sobolev exponent

1)
and with the property that if u ∈ C solves Then, we definef (s) as in the previous case, with f replaced by f mod and s 0 by s 0 + ε.
As a consequence of the proof of the previous lemma, there exists C > 0 for whichf 3) From now on in the paper, we setf = 0 in (−∞, 0). We define the energy functional I : W 1,p (B) → R associated to the problem (3.2) by whereF (u) := u 0f (s)ds. Because of (3.1) and the Sobolev embedding, the functional I is well defined and of class C 2 , being p > 2.
We define the operator T : We observe that the definition is well posed because, for all w ∈ (W 1,p (B)) , the problem (P w ) admits a unique weak solution v ∈ W 1,p (B). To prove the existence one can apply the direct method of Calculus of Variations, while uniqueness is a consequence of the strict convexity of the map u → u p . Furthermore, by [7, Lemma 2.1] we know that T ∈ C((W 1,p (B)) ; W 1,p (B)). (3.6) We introduce also the operator with T given in (3.5).
The operatorT is compact, i.e. it maps bounded subsets of W 1,p (B) into precompact subsets of W 1,p (B). Furthermore, there exist two positive constants a, b such that for all u ∈ W 1,p (B) the following properties hold . Once the claim is proved, the first part of the statement follows by using the continuity (3.6) of the operator T . We pick any subsequence, still denoted by (u n ), and we know that, up to another subsequence, u n → u a.e. in B and that there exists h ∈ L (B) such that |u n | ≤ h a.e. in B for all n. By the continuity off we get that |f . By the arbitrariness of the subsequence picked, we have that the same convergence result holds for the whole sequence (u n ). The claim follows at once from the embedding L (B) → (W 1,p (B)) .
admits a convergent subsequence.
Proof. Let (u n ) ⊂ W 1,p (B) be a (PS)-sequence for I as in the statement. By (3.1) and L'Hôpital's rule, we get and so, there exist µ ∈ (p, ] and R 0 > 0 such thatf (s)s ≥ µF (s) for all s ≥ R 0 . Now, we estimate and, being (u n ) a (PS)-sequence, for some C > 0, where we have denoted by · * the norm of the dual space Therefore, (u n ) is bounded in W 1,p (B) and there exists u ∈ W 1,p (B) such that u n u in W 1,p (B). Hence, Proposition 3.2 guarantees that, up to a subsequence, On the other hand, by the first inequality of (3.8) we obtain Together with (3.9), we conclude that u n →T (u) = u in W 1,p (B).
We define (3.10) We point out that u + = +∞ is possible. Next, we define the set Clearly, C * is closed and convex.
Proof. We first note that u ∈ C * impliesf (u) ∈ C, by the properties off . Now, let u ∈ C * and v :=T (u). By standard regularity theory (see e.g. [23, Theorem 2]), v ∈ C 1,α (B) for some α ∈ (0, 1). Therefore, by [16, Theorem 1.1], we know that v ≥ 0 in B. Furthermore, due to uniqueness, v is radial. Now we prove that v is nondecreasing. It is enough to show that for every r ∈ (0, 1) one of the following cases occurs: which violates both (a) and (b). Now, we fix r ∈ (0, 1). Iff (u(r)) ≤ mv(r) p−1 , we consider the test function Hence, that is ϕ ≡ 0, i.e., the case (a) occurs. Analogously, iff (u(r)) > mv(r) p−1 , we consider the test function There exists a locally Lipschitz continuous operator K : W → W 1,p (B) satisfying the following properties: where a > 0 is the constant given in Proposition 3.2.
Proof. We follow the arguments in the proofs of [ We define the continuous functions δ 1 , δ 2 : W → R as where a, b are the constants introduced in Proposition 3.2. First we claim that for every u ∈ W , we can find a radius (u) > 0 such that for every v, w ∈ N (u) : Indeed, suppose by contradiction that for every n ∈ N we can find v n , w n ∈ N n (u) : (3.14) Since v n , w n ∈ N n (u) for every n, and by the continuy ofT , we get Hence, passing to the limit in (3.14), we obtain by the second inequality in (3.8) where we have used Remark 3.3 and the fact that u ∈ W implies that u ≡ 0. This yields a contradiction. Proceeding analogously, one proves that the claim holds. Now, let U be a locally finite open refinement of {N (u) : u ∈ W } and let {π U : U ∈ U} be the standard partition of unity subordinated to U, i.e.
Clearly, U ∈U π U (u) = 1 for any u ∈ W , π U is Lipschitz continuous, it satisfies supp(π U ) ⊆ U and 0 ≤ π U ≤ 1 for any U ∈ U. Furthermore, since U is a refinement of {N (u) : u ∈ W }, given any U ∈ U, (3.13) holds in particular for any v, w ∈ U . For every U ∈ U we choose an element a U ∈ U with the property that, if Therefore, K is locally Lipschitz continuous due to the Lipschitz continuity of any π U and to the local finiteness of the refinement U.
Moreover, (i) holds thanks to the facts thatT preserves the cone C * (see Lemma 3.5), that K is a convex combination of pointsT (a U ), and that C * is convex.
In view of the proof of (ii), by using the properties of the functions π U and (3.13), we estimate This gives immediately which imply the two inequalities of (ii). By using the definition of δ 2 and (3.8), we can finally prove (iii). Indeed, for every u ∈ W and the proof is concluded.
Without loss of generality, we will take from now on where is the subcritical growth off defined in Lemma 3.1. This further condition is needed in the following two lemmas.
Proof. Reasoning as in the first part of the proof of Lemma 3.4, we easily get, by the fact that I(u) ≤ c, that for some positive constant C. Then, by the second inequality of (3.8) and by Young's inequality with exponents (p , p), where C > 0 may change from line to line, and ε > 0 is sufficiently small. Hence, Young's inequality with exponents (p/p , (p − 1)/(p − 2)) then gives Now, consider the equation satisfied by v :=T (u) By testing it with v and using (3.3), Hölder's inequality, and the Sobolev embedding, we get By applying Young's inequality with exponents −1 for some C > 0, ε > 0 small. Together with (3.19), this implies the thesis.
Lemma 3.8. Let c ∈ R be such that I (u) = 0 for all u ∈ C * with I(u) = c. Then, there exist two positive constantsε andδ such that the following inequalities hold (i) I (u) * ≥δ for all u ∈ C * with |I(u) − c| ≤ 2ε; (ii) u − K(u) ≥δ for all u ∈ C * with |I(u) − c| ≤ 2ε.
Proof. (i) The proof follows by Lemma 3.4. Indeed, suppose by contradiction that (i) does not hold, then we can find a sequence (u n ) ⊂ C * such that I (u n ) * < 1 n and c − 1 n ≤ I(u n ) ≤ c + 1 n for all n. Hence, (u n ) is a Palais-Smale sequence, and since I satisfies the Palais-Smale condition at level c, up to a subsequence, u n → u in W 1,p (B). Since (u n ) ⊂ C * and C * is closed, u ∈ C * . The fact that I is of class C 1 then gives I(u n ) → c = I(u) and I (u n ) → 0 = I (u), which contradicts the hypothesis.
(ii) Let I c+2ε c−2ε := {u ∈ C * : |I(u) − c| ≤ 2ε}. By the part (i), I c+2ε c−2ε ⊂ W , where W is defined in Lemma 3.6. Hence, for all u ∈ I c+2ε c−2ε , u − K(u) ≥ 1 2 u −T (u) by Lemma 3.6-(ii). By the second inequality of (3.8) and by (i), we have for all u ∈ I c+2ε This implies by (3.17), that which in turn gives u −T (u) ≥ M for some positive M and for all u ∈ I c+2ε c−2ε . Indeed, if by contradiction we had inf u −T (u) = 0 over all u ∈ I c+2ε c−2ε , we could find a sequence (u n ) ⊂ I c+2ε c−2ε such that u n −T (u n ) → 0, and so by passing to the limit as n → ∞ in we would have the contradiction 0 ≥δ/(bC p−2 ) > 0, being β > 0 thanks to the choice of in (3.16). Therefore, for all u ∈ I c+2ε c−2ε , u − K(u) ≥ M 2 ≥ min{δ, M 2 }, still denoted byδ, and the proof is concluded. Lemma 3.9 (Descending flow argument). Let c ∈ R be such that I (u) = 0 for all u ∈ C * , with I(u) = c. Then, there exists a function η : C * → C * satisfying the following properties: (i) η is continuous with respect to the topology of W 1,p (B); (ii) I(η(u)) ≤ I(u) for all u ∈ C * ; (iii) I(η(u)) ≤ c −ε for all u ∈ C * such that |I(u) − c| <ε; (iv) η(u) = u for all u ∈ C * such that |I(u) − c| > 2ε, whereε is the positive constant corresponding to c given in Lemma 3.8.
We claim that η n (·, u) converges to the solution η(·, u) of the Cauchy problem (3.20) in W 1,p (B). Indeed, for all i = 0, . . . , n−1, we integrate by parts the equation of (3.20) in the interval [t i , t i+1 ] and we obtain On the other hand, we define the error for every i = 0, . . . , n.
Hence, by (3.23) and Lemma 3.6-(ii) and (iii), we obtain Finally, if we define with abuse of notation it is immediate to verify that η satisfies (i)-(iv).
Lemma 3.10 (Mountain pass geometry). Let τ > 0 be such that τ < min{u 0 − u − , u + − u 0 }. Then there exists α > 0 such that Proof. Suppose by contradiction that there exists a sequence (w n ) n ⊂ C * such that we get we conclude that ∇w n L p (B) → 0 and that |∇w n | → 0 a.e. in B up to a subsequence. Together with (3.24), this ensures that (w n ) is bounded in W 1,p (B) and so, up to a subsequence, it is weakly convergent to some w ∈ W 1,p (B). In particular, lim n→∞ B |∇(w n − w)| p−2 ∇(w n − w) · ∇wdx = 0.
By the Dominated Convergence Theorem, we now get that ∇w = 0 a.e. in B and so the sequence (w n ) converges to the constant solution w ≡ τ in the W 1,p -norm. Again by the Dominated Convergence Theorem we can conclude that which contradicts (3.25). Hence there exists α 1 > 0 such that (i) holds.
In a similar way, now using the fact that ms p−1 −f (s) < 0 for s ∈ (u 0 , u + ), we find α 2 > 0 such that (ii) holds if u + < ∞. The claim then follows with α := min{α 1 , α 2 }. Proof. We first observe that, by Lemma 3.1, any critical point of I solves weakly (2.1) which is equivalent to (1.1). Case u + < ∞. Pick any γ ∈ Γ. We note that by the definition of U − and U + , and by the fact that τ On the other hand, Γ is not empty, since it contains at least the path t ∈ [0, 1] → (1 − t)u − + tu + , hence c < +∞. Therefore, c is a finite number. Now, assume by contradiction that there does not exist a critical point u ∈ C * for which I(u) = c. Then, there exists a deformation η : C * → C * satisfying (i)-(iv) of Lemma 3.9, withε =ε(c) > 0 given by Lemma 3.8. Without loss of generality, we assume that 4ε < α. By the definition (3.27) of c, there exists a curve γ ∈ Γ such that max
Case u + = ∞. The existence of t − follows as in the previous case, while the existence of t + is a consequence of the facts that tu 0 −u − > τ for all t > (u − +τ )/u 0 and I(tu 0 ) → −∞ as t → +∞, see (3.32). The rest of the proof is analogous to case above, with the only change in the definition of U + .
• Proof of Theorem 1.1. By Proposition 3.11, there exists a mountain pass solution u ∈ C * \ {u − , u + } of (1.1) such that I(u) = c. Furthermore, u > 0 by [29,Theorem 5]. It only remains to prove that u ≡ u 0 . To this aim, let γs be the curve given in Lemma 4.3 and defineγ(t) := γs(t(t + − t − ) + t − ) for all t ∈ [0, 1]. Clearly, γ ∈ Γ and c ≤ max t∈[0,1] I(γ(t)) < I(u 0 ) by the previous lemma. Hence, the mountain pass solution u is different from the constant u 0 . Since u ∈ C * , and the only constant solutions of (1.1) in C * are u − , u + , and u 0 , this implies in particular that u is nonconstant.
The second part of the statement is proved by reasoning in the same way for each u 0,i , with i = 1, . . . , n. We define u (i) ± and the cone of nonnegative, radial, nondecreasing functions C (i) * , corresponding to each u 0,i . In this way, for every i, we get a nonconstant positive mountain pass solution u (i) ∈ C (i) * . Hence, u Assume without loss of generality that u 0,1 < u 0,2 < · · · < u 0,n , then u − < · · · ≤ u (n) + and so the n solutions found are distinct.
• Proof of Theorem 1.2. The proof of Theorem 1.1 works also for the case p = 2, with the only exception of Lemma 4.1. In order to prove this lemma, we need the stronger assumption (g 3 ) instead of (g 3 ), and we can proceed as in [11,Lemma 4.9].

Asymptotic behavior in the pure power case
Let q > p > 2. In this section we study the problem (1.1) for g(u) = u q−1 , namely (5.1) By Theorem 1.1 there exists a radial nondecreasing solution of (5.1) for every q > p. We remark that, concerning the notation in Sections 2-4, in this specific case, f = g, m = 1, u 0 = 1, u − = 0, u + = ∞ and C * = C. In this section we aim to find the asymptotic behavior of this solution of (5.1) as q → ∞.
For all q ≥ p + 1, the functions f q (s) := s q−1 belong to the same set F defined in (2.6), with m = 1 and δ = M − 1 for a fixed M > 1, i.e.
For our analysis we need an additional property (namely (5.2) below) on the truncated functionf introduced in Lemma 3.1; in order to ensure it, we provide here a more explicit construction off .
In order to prove thatf q ∈ F, it remains to show that We denote byF q the primitive off q and by I q the associated energy functional. We introduce the Nehari-type set Proof. Suppose by contradiction that there exist (q n ), with q n ≥ p + 1 for any n, and (u n ) ⊂ N qn such that u n L ∞ (B) → 0 as n → ∞. Then, for n sufficiently largẽ f qn (u n ) = u qn−1 n and, being q n ≥ p + 1, there exists ε > 0 such thatf qn (u n )u n = u qn n ≤ (1 − ε)u p n for every n. Therefore, since u n ∈ N qn , for n large we get which is impossible, since 0 ∈ N qn .
Let c q be the mountain pass level corresponding to q as in (3.27), that is to say with τ < min{σ, 1}, σ given in Lemma 5.2, and α q is given as in Lemma 3.10 for g(s) = s q−1 . We notice that property (5.2) is crucial for the proof of the following lemma.
is a homeomorphism.
For the Dominated Convergence Theorem we obtain By the uniqueness of the limit, this yields 1 h p−1 Bf q (hu)udx = u p , that ish = h q (u) and in particular H is continuous.
Finally, the continuous map v ∈ N q → v/ v ∈ C ∩ S 1 is the inverse of H, by the uniqueness of h q (u) and by the fact that h q (u) = 1 if and only if u ∈ N q .
The preceding lemma allows to prove that the mountain pass level in the cone coincides with a Nehari-type level in the cone. (5.14) Proof. We shall split the proof of (5.14) into three steps.
Step 1. We first prove that inf u∈C\{0} sup t≥0 I q (tu) = inf u∈Nq I q (u). From Lemma 5.3, we know that being h q (u)u ∈ N q . On the other hand, where we have used the fact that H defines a homeomorphism between C ∩ S 1 and N q .
In conclusion, we have that there existst ∈ [0, 1) for which h q (γ(t)) > 1, and that h q (γ(1)) < 1. By the continuity of h q proved in Lemma 5.3, and the continuity of γ, there exists t γ ∈ (t, 1) for which h q (γ(t γ )) = 1, that is γ(t γ ) ∈ N q . By Theorem 1.1 there exists a nonconstant, nondecreasing, radial solution u q of (5.1), which by Proposition 3.11 can be caracterized as a mountain pass solution, that is to say c q = I q (u q ) and I q (u q ) = 0. (5.15) We shall now provide some a priori bounds on u q , uniform in q.
Lemma 5.5. There exists C > 0 independent of q such that, for all q ≥ p + 1, Proof. By integrating the equation satisfied by u q , we get Since u q ≡ 1 is positive and nondecreasing, we deduce that u q (0) < 1, u q (1) > 1 for all q ≥ p + 1. (5.16) Consider the equation satisfied by u q in radial form. We multiply it by u q ≥ 0 to obtain We deduce that the function is nonincreasing in r, and hence, using (5.16), We note that L q (r) ≤ 0 is equivalent to This implies (see Figure 1) The previous a priori bounds ensure the existence of a limit profile. Lemma 5.6. There exists a function u ∞ ∈ C for which for any ν ∈ (0, 1). Furthermore, u ∞ (1) = 1.
Proof. The existence of u ∞ and the convergence are consequences of the previous lemma, together with the compactness of the embedding C 1 → C 0,ν . Since, up to a subsequence, u q → u ∞ pointwise, we deduce that u ∞ ∈ C. From (5.16) we immediately get that u ∞ (1) ≥ 1. It only remains to show that u ∞ (1) = 1. To this aim, suppose by contradiction that u ∞ (1) > 1. Then there exist s ∈ (0, 1) and δ > 0 such that u q (r) ≥ 1 + δ for every s ≤ r ≤ 1, (5.18) and for every q sufficiently large. We integrate the equation satisfied by u q in the interval (s, 1) and we replace (5.18) to obtain as q → ∞, in contradiction with Lemma 5.5.
Lemma 5.7. The quantity is achieved by the unique radial function G satisfying (1.4).
Proof. By the Direct Method of the Calculus of Variations, is uniquely achieved by G. Let us prove that c ∞ = c ∞ . Clearly, c ∞ ≤ c ∞ . On the other hand, by the comparison principle, G > 0 in B, and by the radial symmetry G (0) = 0. If we integrate the equation in (1.4) in its radial form, we get Hence, G ∈ C and so c ∞ ≤ c ∞ . This concludes the proof.
Proof. We take the function u ∞ introduced in Lemma 5.6 as test function in the definition of c ∞ and we get for some constant C > 0 independent of q, where we used (5.1), (5.15), and Lemma 5.5.
• Proof of Theorem 1.3. Let G be the unique solution of (1.4). Since G ∈ C \ {0}, by Lemma 5.3 there exists a unique h q (G) > 0 such that h q (G)G ∈ N q . Sincẽ f q (s) = s q−1 for s ≤ 1 = G L ∞ (B) , we have This implies Now, since h q (G)G ∈ N q , we can rewrite the last term as by (5.20). Then, since h q (G)G ∈ N q , Lemma 5.4 implies that The previous two equations provide c ∞ ≥ lim sup q→∞ c q . By combining this inequality with Lemma 5.8, we obtain that As a consequence, the inequalities in (5.19) are indeed equalities, so that lim q→∞ u q = G and u ∞ = G .
Hence, u ∞ achieves c ∞ and, by Lemma 5.7, u ∞ = G. Together with the W 1,p -weak convergence and the uniform convexity of W 1,p (B), this implies that u q → G in W 1,p (B). By Lemma 5.6 the convergence is also C 0,ν (B) for any ν ∈ (0, 1).
Appendix A. Some remarks in the case 1 < p < 2 In this appendix we consider the case 1 < p < 2. We prove that Proposition 3.11 holds under an additional assumption on g, that is to say, a mountain pass solution exists also in this case. Nevertheless, we do not know whether the mountain pass solution is nonconstant. In particular, we prove that Lemma 4.1-(ii) does not hold for 1 < p < 2 and g(u) = u q−1 .
We require g to be of class C 1 ((0, ∞)) ∩ C([0, ∞)), to satisfy (g 1 )-(g 3 ) and We remark that the assumption on the regularity of g is slightly weaker than in case p ≥ 2. This allows us to cover the nonlinearities which behaves like s q−1 (q > p) near the origin. The results in Section 2 hold also in this setting with exactly the same proofs. The only difference is that the function f in Lemma 2.1 is of class C 1 ((0, ∞)) ∩ C([0, ∞)) as g.
Furthermore, proceeding as in Lemma 3.1 we can build, also in this case, the subcritical nonlinearityf . and with the property that, if u ∈ C solves (3.2), then u solves (2.1).
The associated energy functional I is defined as in (3.4) and is of class C 1 . In this setting there exists a mountain pass solution of the problem, as stated in the following proposition.
The proof of Proposition A.2 relies on several preliminary results, which hold under the same assumptions. Proof. The proof is analogous to the one of Proposition 3.2.
Proof. Reasoning as in Lemma 3.4, we obtain that any (PS)-sequence (u n ) is weakly converging to some u in W 1,p (B) and that (3.9) holds. Now, by the first inequality of (A.2) we get u n −T (u n ) 2 ( u n + T (u n ) ) p−2 ≤ 1 a I (u n ) * u n −T (u n ) .
We conclude that u n →T (u) = u in W 1,p (B).
Lemma 3.5 holds for all 1 < p < ∞, hence also in this case the operatorT preserves the cone C * defined in (3.12).  Lemma A.6. Let c ∈ R be such that I (u) = 0 for all u ∈ C * with I(u) = c. Then there exist two positive constantsε andδ such that the following inequalities hold (i) I (u) * ≥δ for all u ∈ C * with |I(u) − c| ≤ 2ε; (ii) u − K(u) ≥δ for all u ∈ C * with |I(u) − c| ≤ 2ε.
Proof. The proof of part (i) is analogous to the one given in Lemma 3.8. We prove now (ii). Let I c+2ε c−2ε := {u ∈ C * : |I(u) − c| ≤ 2ε}. By the part (i), I c+2ε c−2ε ⊂ W , where W is defined in Lemma A.5. Furthermore, for all u ∈ I c+2ε c−2ε , u − K(u) ≥ 1 2 u −T (u) by Lemma A.5-(ii). Now, by the second inequality of (A.2) and by the (i) part of the present lemma, we have for all u ∈ I c+2ε Hence, u − K(u) ≥ min δ , we have proved that η satisfies (ii) and (iii). Properties (i) and (iv) are immediate. The fact that η preserves the cone can be proved as in Lemma 3.9, sinceT (C * ) ⊂ C * also for 1 < p < 2.
• Proof of Proposition A.2. The preliminary results shown in this appendix allow us to prove Proposition A.2 by proceeding as in the proof of Proposition 3.11.
In the case 1 < p < 2, we cannot conclude that the mountain pass solution found in Proposition A.2 is nonconstant. In particular, Proposition A.9 below implies that Lemma 4.1 does not hold for 1 < p < 2 and g(u) = u q−1 .
Since we are in the pure power case, we refer to the truncated nonlinearityf defined in ( Hence we see that h(s) is unique and regular in s. Therefore the proof of (i) is concluded.
In order to prove (ii), we write the Taylor expansion at the first order of h h(s) = 1 + h (0)s + o(s).
Since v is nonconstant, the statement follows.