Quasilinear equations with indefinite nonlinearity

In this paper, we are concernedwith quasilinear equations with indefinite nonlinearity and explore the existence of infinitely many solutions.


Introduction
Consider the quasilinear equation with the indefinite nonlinearity where Ω ⊂ ℝ N , N ≥ 3, is a bounded smooth domain, r > 4, 4 < s < 4N N−2 , and a ± are nonnegative continuous functions in Ω. A great number of theoretical issues concerning nonlinear elliptic equations with indefinite nonlinearity have received considerable attention in the past few decades. In particular, the existence of solutions has been studied extensively. For example, the existence of positive solutions and their multiplicity was studied by variational techniques [2], and the existence of nontrivial solutions was investigated by two different approaches (one involving the Morse theory and the other using the min-max method) [1]. It was shown that the existence of positive solutions, negative solutions and sign-changing solutions could be established by means of the Morse theory [7,8]. For the results on a priori estimates and more comparable relations among various solutions etc., we refer the reader to [3,5,6,9,10,12] and the references therein. However, as far as one can see from the literature, not much has been known about the existence of solutions to quasilinear equations with indefinite nonlinearity. From the variational point of view, there are two main difficulties that arise in the study. One lies in the fact that there is no suitable space in which the corresponding functional enjoys both smoothness and compactness. Compared with quasilinear equations with the definite nonlinearity, the other one is to prove the boundedness of the associated Palais-Smale sequences. In this work, we will get over these obstacles by means of the variational techniques and the perturbation method to study the existence of infinitely many solutions of system (1.1).
Assume that (a) Ω + ̸ = 0 and Ω + ∩ Ω − = 0. We are looking for u ∈ H 1 0 (Ω) ∩ L ∞ (Ω) satisfying equation ( for φ ∈ H 1 0 (Ω) ∩ L ∞ (Ω), which is formally the variational formulation of the following functional: In view of the perturbation (regularization) approach [11], due to the lack of a suitable working space, we introduce the corresponding perturbed functionals, which are smooth functionals in the given space and satisfy the necessary compactness property. For μ ∈ (0, 1], we define the perturbed functional I μ on the Sobolev space W 1,p 0 (Ω) with p > N by Note that I μ is a C 1 -functional. We shall show that I μ satisfies the Palais-Smale condition. The critical points of I μ will be used as the approximate solution of problem (1.1). Now let us briefly summarize our main results of this paper.
(a1) There exist constants c 1 , c 2 > 0 such that uniformly in x ∈ Ω. The rest of the paper is organized as follows: The proof of Theorem 1.1 is presented in Section 2, and the proof of Theorem 1.2 is shown in Section 3. Section 4 is dedicated to the existence of infinitely many solutions to the more general quasilinear equation (1.3).
Taking ψ = uψ p as the test function in (2.1), we get In view of the Sobolev inequality By choosing q ∈ (4, s), we deduce that and thus

Lemma 2.2.
Assume that μ n > 0, μ n → 0, u n ∈ W 1,p 0 (Ω), DI μ n (u n ) = 0 and I μ n (u n ) ≤ C 0 . Then there exists a constant C > 0 independent of n such that Moreover, there holds Proof. Assume that Let v n = ρ −1 n u n . Then and have We further obtain the estimates as
Since v 2 ∈ H 1 0 (Ω 0 ), we have v 2 ≡ 0 in Ω. It follows from (2.4) and Lemma 2.1 that It is easy to see that So we further deduce that Consequently, the left-hand side of (2.5) converges to zero, which yields a contradiction.
We separate the proof into three steps.
Step 1. Moser's iteration shows that the sequence {u n } is uniformly bounded.
Note that for p > N, the convergence W 1,p 0 (Ω) → C α (Ω) holds for some α ∈ (0, 1). The function u n belongs to L ∞ (Ω). Here we prove that ‖u n ‖ L ∞ (Ω) is uniformly bounded. The term a + |u| s−2 u in equation (1.2)  Step 2. Choose a suitable test function and show that the limit function u satisfies equation (1.2).
In the following context, we call c ∈ ℝ a critical value of the functional I, provided there exists a function u ∈ H 1 0 (Ω) ∩ L ∞ (Ω) satisfying equation (1.2) and I(u) = c. Theorem 1.1 implies that if μ n → 0, c n is a critical value of I μ n and c = lim n→∞ c n , then c is a critical value of I.

Proof of Theorem 1.2
In this section, we prove Theorem 1.2. We construct a sequence of critical values of the functional I μ with μ > 0. The corresponding critical points will be used as the approximate solutions of equation (1.2).

Lemma 3.1. Suppose that {u n } is a Palais-Smale sequence of the functional I μ with μ > 0. Then u n is bounded in W
Proof. The proof is similar to the one of Lemma 2.2. Since μ > 0 is fixed and ‖DI μ (u n )‖ → 0, by Lemma 2.1 we have As in the proof of Lemma 2.2, we prove it by way of contradiction. Assume that Then it is easy to see that Let v n = ρ −1 n u n . As in the proof of Lemma 2.1, we have v n → v in L q (Ω), 1 ≤ q < 4N N−2 , v 2 n ⇀ v 2 in H 1 0 (Ω), and v = 0 in Ω + ∪ Ω − . Since p > N and μ is fixed, we find So we see v 2 n → v 2 in C α (Ω) for some α ∈ (0, 1). Given ψ ≥ 0 and ψ ∈ C ∞ 0 (Ω 0 ), we take φ n = ψu n /(1 + u 2 n ) as the test function, and thus have So one can see v 2 ≡ 0. Otherwise, we assume that By a density argument, inequality (3.2) also holds for ψ ≥ 0 and ψ ∈ W and v 2 = 0 in Ω + ∪ Ω − , we have v 2 ≡ 0 on ∂Ω 0 and C n → 0 as n → ∞. Define ψ n ∈ W 1,p 0 (Ω 0 ) by ψ n (x) = (v 2 n (x) − C n ) + , for x ∈ Ω 0 , |∇ψ n | ≤ 2|v n ∇v n | and ‖ψ n ‖ ≤ C. Taking ψ n as the test function in inequality (3.2), we have Taking n → ∞, we obtain Hence, v 2 = 0 in Ω, which leads to a contradiction by virtue of Lemma 2.2. In fact, it follows from (3.1) that So, {u n } is a Cauchy sequence of W , φ is odd and I 1 (φ(t)) < 0 for t ∈ ∂B k }, and B k is the unit ball of ℝ k . Then we have where w is defined by Dw = (1 + u 2 ) 1/2 Du and w ∈ H 1 0 (Ω), and J is a C 1 -functional defined on H 1 0 (Ω). Define the critical values of J by , φ is odd and J(φ(t)) < 0 for t ∈ ∂B k }.

More general cases
In this section, we consider the more general quasilinear equation (1.3) and prove Theorem 1.3. Equation (1.3) has a variational structure, given by the functional Again we apply the perturbation method and introduce the perturbed functional H μ with μ ∈ (0, 1]: Note that H μ is defined on the Sobolev space W 1,p 0 (Ω) with p > N. It is a C 1 -functional on W 1,p 0 (Ω), and satisfies 1,p 0 (Ω). As we have seen in the preceding section, for the quasilinear equations with indefinite nonlinearity, compared with ones with definite nonlinearity, the difficulty is to prove the boundedness of some associated sequences, either the sequence of approximate solutions (Lemmas 2.1 and 2.2) or the Palais-Smale sequence of the perturbed functional (Lemma 3.1). When we have proved the boundedness of these sequences, we can deal with the quasilinear equations as before to obtain the convergence and the existence results.
In the following, we will prove the boundedness of sequences of the approximate solutions, and the boundedness of the Palais-Smale sequences of the functional H μ . Proof. The proof is closely analogous to the one of Lemma 2.1. Choose ψ ∈ C ∞ 0 (ℝ N ) such that ψ ≥ 0, ψ(x) = 1 for x ∈ Ω − , and ψ(x) = 0 for x ∈ Ω + . Taking φ = uψ p as the test function, we have In view of assumptions (a2) and (a3), it follows from the Sobolev inequality (2.3) that Hence, we further have uD s a ij (x, u)))D i uD j u dx By virtue of (4.2) and (4.3), we arrive at (4.1).

Lemma 4.2.
Assume that μ n > 0, μ n → 0, u n ∈ W 1,p 0 (Ω), DH μ n (u n ) = 0 and H μ n (u n ) ≤ C. Then there exists a constant C independent of n such that Moreover, we have Proof. As in the proof of Lemma 2.2, we apply the indirect argument by assuming that By Lemma 4.1, we get Let v n = ρ −1 n u n . Then v n → u in L q (Ω), 1 ≤ q < 4N N−2 and v n Dv n ⇀ v∇v in L 2 (Ω), where v 2 ∈ H 1 0 (Ω 0 ) ⊂ H 1 0 (Ω). Given ψ ≥ 0 and ψ ∈ C ∞ 0 (Ω 0 ), we set φ n = ψu n /(M + u 2 n ) with M being the constant given in condition (a 3 ). From condition (a3) we have It follows from Lemma 4.3 that A ij (x)v∂ i v∂ j ψ dx as n → ∞.
That is, , by the density argument we have and v 2 ≡ 0 in Ω. The remainder is the same as that shown in Lemma 2.2, so we omit it here. Proof. For T > 0, let u T n be the truncated function of u n , that is, u T n = u n if |u n | ≤ T, and u T n = ±T if ±u n ≥ T. Taking u T n as the test function, we have 0 = ⟨DH μ n (u n ), u T n ⟩ = μ n ( ∫ In view of assumption (a 3 ), there exists a T 0 > 0 such that for |s| ≥ T > T 0 and x ∈ Ω, where ξ ∈ ℝ N . By condition (a2) and (4.5), we get Hence, we obtain the estimate The following proposition can be regarded as a counterpart of Theorem 1.1.

Proposition 4.4.
Assume that μ n > 0, μ n → 0, u n ∈ C 1 0 (Ω), DH μ n (u n ) = 0 and H μ n (u n ) ≤ C. Then the following assertions hold: Proposition 4.4 can be proved similarly to Theorem 1.1, so we omit it and refer to [11]. As we have seen in the proof of Theorem 1.1, with the help of estimate (4.6), the proof of Theorem 1.1 can also be done in the same way as that for quasilinear equations with definite nonlinearity.