ON A P-LAPLACIAN EIGENVALUE PROBLEM WITH SUPERCRITICAL EXPONENT

. In this paper, we prove the existence of the positive and negative solutions to p-Laplacian eigenvalue problems with supercritical exponent. This extends previous results on the problems with subcritical and critical exponents.


(I')
For the case p = 2, the authors in [1,15,19] proved the existence of multiple solutions to (I') when g is odd, later the existence of three solutions was obtained in [9] and Y.Q. Li and Z.L. Liu in [10] applied the descent flow to abtain the multiple and sign-changing solutions when g is non odd. Recently, we studied the existence of positive and negative solutions to (I) for the supercritical exponent(i.e., q > 2 * ) by establishing a new framework in [21].
Using the Ljusternik-Schnirelman category, Y.Q. Li in [8] showed the existence of solutions to the following problem with g is non odd and N α has a hyperboloid structure.
For the general p in (I'), A. Lê in [11] considered linear eigenvalue problems for the p-Laplace operator subject to different kinds of boundary conditions in a bounded domain. In the whole space R N , p-Laplace eigenvalue problems were discussed in [12,5] by the Ljusternik-Schnirelman principle. Using degree theoretic arguments, S.C. Hu and N.S. Papageorgiou in [7] investigated the nonlinear eigenvalue problem for the p-Laplace operator with a nonsmooth potential(see also [4]). The nontrivial eigenvalue or principal eigenvalue was considered in [17,2]. One positive, one negative and one sign-changing solutions were obtained in [3] in the case of Ω bounded and in [6] in the case of Ω = R N . Now, we study (I) in the supercritical case(q > p * ). We first set up an appropriate space such that the functionals Φ, Ψ are well-posed; Secondly, as in [21], we construct a proper formula for the constraint functional Ψ (u) and test the Palais-Smale condition for it.
Due to the Best Approximation Theorem in a strictly convex reflexive Banach space, the following fundamental lemma holds(see also Lemma 1.1 in [21]). where ·, · denotes the dual pair between X and X * , the element e(u) ∈ X dependents on u. Furthermore, there exist two positive constants M 1 , M 2 independent on u, such that Then, let ker(Φ (u)) = {w ∈ X : Φ (u), w = 0} and ker(e(u)) = {w * ∈ X * : w * , e(u) = 0}. We give our new form for Ψ (u) with the best approximation e(u) given above(see also Theorem 1.1 in [21]).
Moveover, provided X is a Hilbert space, Ψ (u) is just a tangent vector(in the sense of isomorphism).
From (1.7), the existence of solutions for the eigenvalue problem (I) is shown in the following Theorem: This article is organized as follows: In Section 2, some preliminaries are given; In Section 3, we check the Palais-Smale condition for the formula (1.7); In Section 4, we show Theorem 1.4. Throughout the rest of the article, C, C i , c, c i , i = 1, 2, . . . denote constants and may be different in different places; " → " and " " represent strong and weak convergence in related function spaces, respectively.

2.
Preliminaries. We begin with the following lemma about the convergence property.
Since the proof is similar to that of Lemma 2.1 in [21], we omit it here. Next is to give the differential of functionals Φ(u), Ψ(u).
By the mean value theorem and the Lebesgue dominant convergence, it is easy to prove Lemma 2.2. We shall not develop this here and refer the interested reader to Lemma 2.2 in [21].
Before considering the formula of Ψ (u), we show the smoothness of the manifold N α .
3. Palais-Smale condition for Ψ (u) defined in (1.7). Recall that there is a oneto-one correspondence between critical points of Ψ and weak solutions to problem (I). Therefore, (1.7) implies that Ψ (u) = 0 if and only if u is a weak solution to problem (I) with λ = 1 Ψ (u),e(u) X * ,X , where Ψ (u), e(u) X * ,X = 0 follows from the subsequent remark. Lemma 3.2. Assume that a sequence u n ∈ N α satisfies Ψ(u n ) → c( = 0) and Ψ (u n ) → 0, then it has a convergent subsequence in X.

From (2.1), it is equivalent to showing
i.e., Ω |∇u n k | p−2 ∇u n k − |∇u| p−2 ∇u (∇u n − ∇u)dx → 0. (3.15) For this aim, we rewrite the equation (3.4) as Multiplying (3.16) with u n k and integrating on Ω(noting that {u n k } is bounded in Recall that 1 < q − 1 and q > p * . Owing to the Strauss Lemma and (2.2), the right hand side of (3.17) yields From (3.17) and (3.18), we obtain Similarly, multiplying (3.16) by u and integrating over Ω, as k → ∞, we get that Clearly, the right hand side of (3.20) leads to Using the Strauss Lemma together with (2.4), (3.20) and (3.21), we deduce that where the term lim k→∞ Ω |∇u n k | p−2 ∇u n k · ∇udx exists because {|∇u n k | p−2 ∇u n k } is bounded in L p p−1 (Ω) and by choosing a subsequence, {|∇u n k | p−2 ∇u n k } is weakly convergent in L p p−1 (Ω).
Due to u n k u in W 1,p 0 (Ω), taking the difference of (3.19) and (3.22), we derive that Here the final inequality in (3.23) follows from the Fatou's lemma with the fact that u n → u almost everywhere in Ω. Nevertheless, one has |∇u n k | p−2 ∇u n k − |∇u| p−2 ∇u ∇u n k − ∇u 0. Combining with (2.4), one has Φ (u n ) Φ (u) in X * . The proof of Lemma 3.2 is completely finished.
By applying the similar argument as Theorem 2.1 in [21], we have the convergence of sequences {e(u n k )} defined in (3.16) as u n → u in X. This plays an important role in our formula Ψ (u n k ) = Ψ (u n k ) − Ψ (u n k ), e(u n k ) Φ (u n k ). From the assumptions ( * ), ( * * ) on g, we know that J(u) < 0 if u 0 and u ≡ 0. We claim that Claim 3: If u ∈ N α , then |u| ∈ N α and J(|u|) J(u). Proof. In fact, for any u ∈ N α , then Ω can be decomposed into Ω = Ω 1 + Ω 2 + Ω 2 with Based on the Theorem 6.17, p. 152, of [13], we know that One can easily deduce that |u| ∈ N α with |u| 0 and |u| ≡ 0. Thus (4.4) follows from the assumption ( * ) on g and (4.2).
If we set g(x, t) = g(x, t), if t 0, 0, if t > 0. (4.13) By a similar argument, we obtain a negative solution (denoted by u 2 ) to problem (I), that is the conclusion of Theorem 1.4.
As an auxiliary result, the following holds: Proposition 1. Both the positive and the negative solutions are local minima ofΨ in the C 1 0 (Ω) topology. For the readers' convenience, we provide its proof below(see also [21]).

Remark 2.
Our method also works for the subcritical or critical case.