Existence theorem for a class of semilinear totally characteristic elliptic equations involving supercritical cone sobolev exponents

In this paper, we prove the existence of bounded positive solutions for a class of semilinear degenerate elliptic equations involving supercritical cone Sobolev exponents. We also obtain the existence of multiple solutions by the Ljusternik-Schnirelman theory.

1. Introduction. In this paper, we consider the existence of positive solutions for the following Dirichlet problem where 2 < q < 2 * ≤ p, 2 * = 2N N −2 is the critical cone Sobolev exponents, N ≥ 3 and λ > 0 is a parameter. Here the domain B is [0, 1)×X for X ⊂ R N −1 compact, which is regarded as the local model near the conical points on manifolds with conical singularities and {0} × X ⊂ ∂B. Moreover, the operator ∆ B in (P λ ) is defined by (x 1 ∂ x1 ) 2 + ∂ 2 x2 + . . . + ∂ 2 x N , which is an elliptic operator with totally characteristic degeneracy on the boundary x 1 = 0 (we also call it Fuchsian type Laplace operator), and the corresponding gradient operator is denoted by ∇ B := (x 2 1 ∂ x1 , ∂ x2 , . . . , ∂ x N ). Near ∂B we will often use coordinates (x 1 , x ) = (x 1 , x 2 , . . . , x N ) for 0 ≤ x 1 < 1, x ∈ X. The main purpose of this article is to establish the existence theorem for the problem (P λ ) in the cone Sobolev space H 1, N 2 2,0 (B). The space H 1, N 2 2,0 (B) was introduced in [5] and will be explained later in Section 2 for the completeness. We always assume that P (x) and Q(x) are positive and continuous on B and there exist δ 1 > 0 and δ 2 > 0 such that where θ 1 and θ 2 are two positive constants. The analysis on manifolds with conical singularities has been studied in many references, see [11,12,17,18,19]. The foundation of this analysis have been developed through the fundamental works by Schulze [12], and subsequently further expended by him and his collaborators. Recently, H. Chen, X. Liu and Y. Wei established the corresponding Sobolev inequality and Poincaré inequality on the cone Sobolev spaces in [5]. For related nonlinear problems with totally characteristic degeneracy, these inequalities seem to be of fundamental importance to obtain the existence and multiplicity results. Consider the following semi-linear boundary value problem For f (x, u) = Q(x)|u| p−2 u in (1.1), the authors in [5] consider the following Dirichlet problem and obtained the existence result for 2 < q < 2 * = 2N N −2 with Q(x) satisfying some appropriate conditions. For f (x, u) = λu + |u| 2 * −2 u, the authors in [6] proved the Dirichlet problem −∆ B u = λu + |u| 2 * −2 u in B, u = 0 on ∂B, (1.2) possesses at least one positive solution for N ≥ 4, λ ∈ (0, λ 1 ) where λ 1 is the first eigenvalue of −∆ B and admits infinitely many solutions for N ≥ 7, λ > 0. They also studied problem (1.1) for general nonlinearities of subcritical and critical cone Sobolev exponents. Fan and Liu in [15] obtained the existence of multiple positive solutions to (1.1) with f (x, u) = f λ |u| q−2 u + g(x)|u| 2 * −2 u, where 1 < q < 2, N ≥ 3, f λ and g(x) satisfies some suitable conditions. Also the authors in [14] considered the degenerate elliptic equations with singularity and critical cone Sobolev exponents and obtained the existence of multiple positive solutions for λ > 0 and γ ∈ (0, 1). Other problems were investigated in [4,6,7,8,9,10,13,14,16] and the references therein.
To the authors' knowledge, there are very few results for problem (1.2) involving supercritical Sobolev exponent. J. Chabrowski and J. Yang in [3] studied the situation of classical Laplacian. Inspired by [3], we investigate problem (P λ ) with supercritical exponent.
In order to solve problem (P λ ) we first consider a truncated problem which involves only a subcritical cone Sobolev exponent. It turns out that weak solutions of the truncated problem are bounded. This allows us to choose a suitable parameter λ and a truncation so that a solution of truncated problem in fact satisfies problem (P λ ). We use the cone Sobolev inequality and Poincaré inequality to prove the existence theorem for the truncated problem. A solution of the truncated problem will be obtained by the mountain-pass theorem.
Our main results are as follows.
2,0 (B) and u 0 is a solution of the subcritical problem (P 0 ). This paper is organized as follows. In Section 2 we recall the cone Sobolev spaces and some necessary propositions. In Section 3, we prove the existence of positive solutions for the truncated problem corresponding to (P λ ). In Section 4, we give the proof of Theorem 1.1 by Moser iteration. We will prove Theorem 1.2 and Theorem 1.3 in Section 5.

2.
Preliminaries. In this section, we introduce a totally characteristic operator and its corresponding cone Sobolev spaces and their properties. For more details we refer to [5].
Let X ⊂ R N −1 be a closed, compact, C ∞ manifold and B = [0, 1) × X. We consider the following Riemannian metric Then the gradient operator with respect to the metric g is first order differential operator. By direct calculation, we know the corresponding gradient operator is ∇ B := (x 2 1 ∂ x1 , ∂ x2 , . . . , ∂ x N ). Moreover, the Laplace-Beltrami operator corresponding to the metric g is then of the form Moreover, the weighted L p -spaces with weight data γ ∈ R is denoted by Now we can define the weighted Sobolev space for 1 ≤ p < +∞.
where X ∧ = R + × X is the open stretched cone with the base X and W m,p 0 (int B) denotes the closure of C ∞ 0 (int B) in Sobolev spaces W m,p (X) whenX is a closed compact C ∞ manifold of dimension N that containing B as a submanifold with boundary.
where the constant c = c 1 + c 2 , and c 1 and c 2 are given in (2.2).
, and the constant c depending only on B and p. The following compactness result was first proved by H. Chen, X. Liu and Y. Wei in [5] and later corrected by themselves.
Proof. According to Definition 2.3, we can write H 1,γ2 2,0 (B) and H 0,γ1 p,0 (B) as follows, . With the help of Sobolev inequality, we know that the embedding [ and it induces an isomorphism For , and it also induces an isomorphism is also an isomorphism. Since the embedding On the other hand, if δ > 0, we set ϕ(r) = e −rδ · r s . And then all derivatives of ϕ(r) are uniformly bounded on suppw for every s > 0. Then it follows where the last embedding is compact. This completes the proof.
For the completeness, we introduce the well-known Mountain Pass Lemma (see [1,22,23]). Definition 2.4. We say that a functional I satisfies the (P S) c condition, if for any sequence {u n } ⊂ X with the properties: there exists a subsequence which is convergent, where X is the dual space of X.
: h(0) = 0 and h(1) = e , then c is a critical value of I and c ≥ α.
3. The truncated problem. In this section, we consider the truncated problem for some positive constant K and it will be determined later. The idea of truncation was used in [3,21]. Let

2) and
Either in the case 0 ≤ u ≤ K or in the case u > K, it is easy to check that We define the following functional I λ on H 1, N 2 2,0 (B), i.e., and it is easy to see that In fact, by Proposition 2.2, H 1, N 2 2,0 (B) can be regarded as the closure of C ∞ 0 (B) with respect to the norm u . Then I λ (u) is well-defined and belongs to Therefore, a critical point of the variational functional I λ is a weak solution of problem (T λ ). The Mountain Pass Lemma (Proposition 2.4) will be applied to find a critical point of (T λ ).
Therefore there exist α 0 > 0 and ρ 0 > 0 such that I λ (u) ≥ α 0 > 0 for all u ∈ for t ≥ 0. Since 2 < q < 2 * , there exists a t 0 large enough such that e 0 > ρ 0 and I λ (e) < 0 with e 0 = t 0 u 0 . The conclusion (2) holds.  for q ∈ (2, 2 * ) and N q − δ ≤ γ < N q , where δ = min{δ 1 , δ 2 }. Moreover, we obtain By the condition (H), there existC 1 ,C 2 > 0 such that Then it follows With the help of Hölder inequality, we have . Therefore, by (3.8), we have A 11,n → 0 as n → ∞. Analogously we get A 12,n → 0 as n → ∞, which means that A 1,n → 0 as n → ∞. On the other hand, by (3.2) and (3.9), we obtain Summing up, we get It shows that u n → u strongly in H 1, N 2 2,0 (B) as n → ∞. We get the assertion. Combining Lemma 3.1 and Lemma 3.2, we have the following existence result. The positivity of a solution can be ensured by replacing I λ to I + λ defined by Moreover, we may apply the cone maximum principle in order to obtain u λ > 0 in int B.

4.
Proof of Theorem 1.1. In this section, we prove that a solution u λ of the problem (T λ ) obtained in Proposition 3.1 must be bounded, namely u λ L ∞ (B) ≤ K provided that K and λ are appropriately chosen. For this purpose, we will use the Moser iteration method [20]. This implies obviously that such u λ solves problem (P λ ). Let be the best constant corresponding to the Sobolev embedding H . Then for any B, we know S(B) = S(R N + ) (see [15]). Thus we denote S := S(B) = S(R N + ) for simplicity, and S is achieved by the function We introduce the functional I * : H 1, N 2 2,0 (B) → R defined by If we denote c 0 is the mountain pass level related to I * , which is associated to the problem in [5] We conclude that I λ (u) ≤ I * (u) for each u ∈ H Therefore, by (4.1) it yields that Proof of Theorem 1.1. We first prove that there exists a constant λ 0 > 0 such that for 0 < λ ≤ λ 0 the positive solution u λ of (T λ ) satisfies , w L = uu β−1 L , where β ≥ 1 will be determined later. Taking φ as a test function in (T λ ), we get By the definition of u L , we have Therefore, we obtain by (3.2) and (4.7) where M 1 = M (1 + λK p−q ). On the other hand, we have (4.10) Hence, by (4.8)-(4.10), we obtain (4.11)

5.
Existence of multiple solutions. In this section we consider the existence of multiple solutions of (P λ ). In particular, we obtain that the number of solutions is increasing as λ gets closer to 0. In order to get multiple solutions, the Ljusternik-Schnirelman theory of critical points ([1, 2, 3]) will be applied and the function h will be extended as an odd function, i.e., Next, let us state a variant of the dual variational principle of A. Ambrosetti and P. Rabinowitz [1]. Let E be a Banach space and For I ∈ C 1 (E, R), we set E + = {u ∈ E : I(u) ≥ 0} and We introduce the following lemma (Lemma 3.1 in [2]), where the proof is exactly the same as that in [1]. In what follows, we always take E = H Since E λ ⊂ E * , we have Γ m * ⊂ Γ m λ for m = 1, 2, . . . We now define min-max levels for I λ and I * as follows, for m = 1, 2, . . . By Lemma 3.2, I λ satisfies the (P S) c condition for each c = c m λ . Similarly, it is easy to check that I * satisfies the (P S) c condition for each c = c m * . It is easy to show that I λ and I * satisfy all conditions in Lemma 5.1. Therefore, c m λ and c m * are critical values of I λ and I * respectively, m = 1, 2, . . . That implies, for fixed m ∈ N + , there exist m distinct pairs of solutions (−u j λ , u j λ ) (j = 1, · · · , m) of (T λ ) for each λ > 0 and m distinct pairs of solutions (−u j 0 , u j 0 ) (j = 1, . . . , m) of (P 0 ).
Since each c m * is independent of K and λ, following the proof of Lemma 4.1, we can obtain that its assertion continues to hold for any solution u ∈ H Choose λ m satisfying and we get that each critical point u m λ corresponding to c m λ for 0 < λ ≤ λ m satisfies u L ∞ (B) ≤ K. For any given m ∈ N + above and any 0 < λ ≤ λ m , we have that each critical point u j λ of I λ at the level c j λ (1 ≤ j ≤ m − 1) satisfies Here we apply Lemma 4.1 and the fact thatc j * = ( 2q q−2 c j * ) 1 2 ≤c m * for 1 ≤ j ≤ m − 1. Therefore, for each 0 < λ ≤ λ m , the problem (P λ ) has at least m distinct pairs of solutions (−u j , u j ) (j = 1, . . . , m). We finish the proof.