Existence of positive solutions for nonlocal problems with indefinite nonlinearity

In this paper, we consider the following new nonlocal problem: {−(a−b∫Ω|∇u|2dx)Δu=λf(x)|u|p−2u,x∈Ω,u=0,x∈∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left \{ \textstyle\begin{array}{l@{\quad}l} - (a-b\int_{\varOmega} \vert \nabla u \vert ^{2}\,dx )\Delta u=\lambda f(x) \vert u \vert ^{p-2}u, & x\in\varOmega,\\u=0, & x\in\partial\varOmega, \end{array}\displaystyle \right . $$\end{document} where Ω is a smooth bounded domain in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{3}$\end{document}, a,b>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a,b>0$\end{document} are constants, 3<p<6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$3< p<6$\end{document}, and the parameter λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda>0$\end{document}. Under some assumptions on the sign-changing function f, we obtain the existence of positive solutions via variational methods.


Introduction and main resluts
In this paper, we are concerned with the existence of positive solutions for the following new nonlocal problem: -(ab Ω |∇u| 2 dx) u = λf (x)|u| p-2 u, x ∈ Ω, u = 0, x ∈ ∂Ω, (1.1) where Ω is a smooth bounded domain in R 3 , a, b > 0 are constants, 3 < p < 6, and the parameter λ > 0. f (x) is sign changing in Ω, which is the reason why we call it indefinite nonlinearity in the title. In recent years, the Kirchhoff type problems in a bounded domain have been investigated by a lot of scholars under different assumptions on f (x, u), see e.g. [1][2][3][4][5][6][7][8][9]. There are also many studies for problem (1.2) on the whole space R N (N ≥ 3). For this case, we refer the interested readers to [10][11][12][13][14][15][16][17]. In particular, Chen [10] obtained the existence and multiplicity of positive solutions to the Kirchhoff problem with indefinite nonlinearity by using mountain pass theorem and minimization argument. In problem (1.2), if we replace a + b Ω |∇u| 2 dx by ab Ω |∇u| 2 dx, it turns to be the following new nonlocal problem: So far, there are only a few works about such a nonlocal problem. Yin and Liu [18] first studied this kind of nonlocal one and obtained two nontrivial solutions for problem (1.3) when f (x, u) = |u| p-2 u with 2 < p < 2 * (2 * = 2N N-2 when N ≥ 3 and 2 * = +∞ when N = 1, 2). Lei et al. [19] proved that problem (1.3) has at least two positive solutions in the case of f (x, u) = f λ (x)|u| q-2 u, where 1 < q < 2 and f λ (x) ∈ L ∞ (Ω) changes sign. Generalization of the result of [18] and [19] was done by Duan et al. [20]. In [20], the authors proved the existence and multiplicity results for problem (1.3) Lei et al. [21] also obtained the multiplicity of nontrivial solutions for problem (1.3) with singularity. On the whole space R 4 , Wang et al. [22] Under the assumption f (x) ∈ L 4/3 (R 4 ), they showed the existence of multiple positive solutions for the considered problem.
From the works described above, we do not see any existence result of positive solution to problem (1.1) in the case that 3 < p < 6, the nonlocal term ab Ω |∇u| 2 dx, and indefinite nonlinearity. Thus, it is natural to ask what the case would be. Our purpose of this paper is to show some existence of positive solutions for problem (1.1).
In order to state our main results, we first make the following hypotheses on f : ( Our main results can be stated as follows.
Problem (1.1) is variational in nature. Note that, for any u ∈ H 1 0 (Ω), the functional is well defined, where u 2 = Ω |∇u| 2 dx. Furthermore, it belongs to C 1 (H 1 0 (Ω), R) and its critical points are precisely the weak solutions of problem (1.1). Here, we say u ∈ H 1 0 (Ω) is a weak solution to problem (1.1) if, for any v ∈ H 1 0 (Ω), it holds In the proof of Theorem 1.1, the main difficulty is to prove the boundedness of the Palais-Smale ((PS) for short) sequence, which cannot be proved directly in the case of 4 ≤ p < 6 and indefinite nonlinearity. We will prove the boundedness of (PS) sequence indirectly, which is inspired by [10]. In the proof of Theorem 1.2, since (f 2 ) without assuming |Σ 0 | = 0, we cannot get the compactness property for the corresponding energy functional as in the proof of Theorem 1.1. We overcome this difficulty by pulling the energy level down below some critical level. However, we now face the situation with superlinear and indefinite nonlinearity, it is difficult to achieve the critical energy level. To this end, we choose a suitable test function presented in [23] (see also [12]). At this point, we need to require the restrictive condition p > 3. Without this condition, we do not know if the result is still right.
Throughout this paper, we make use of the following notations. H 1 0 (Ω) and L q (Ω) are standard Sobolev spaces with the usual norm denotes an open ball centered at x with radius r > 0. → and denote strong and weak convergence, respectively. C and C i denote various positive constants whose values may vary from line to line. o(1) denotes a quantity such that o(1) → 0 as n → +∞. All limitations hold as n → ∞ unless otherwise stated. Let S be the best Sobolev constant for the embedding of H 1 0 (Ω) into L 6 (Ω), that is, The paper contains two more sections, one is devoted to the proof of Theorem 1.1, the other is dedicated to the proof of Theorem 1.2. Proof By (f 1 ), (1.4), and Hölder's inequality, one has Thanks to 4 ≤ p < 6, the above inequality implies that, for any λ > 0, there are α, δ > 0 such that I(u) ≥ α for all u = δ.
Moreover, by (f 2 ), we know that there is u ∈ H 1 0 (Ω) satisfying Ω f (x)|u| p dx > 0. Then we have that which means that there is t 0 large enough such that e > ρ and I(e) < 0, where e = t 0 u.
Thus, we can use the mountain pass theorem without (PS) condition [24] to obtain a sequence {u n } ⊂ H 1 0 (Ω) satisfying I(u n ) → c * and I (u n ) → 0 for To complete the proof of the lemma, it suffices to prove that c * < a 2 4b . By easy calculations, one has This together with the definition of c * implies c * < a 2 4b .
and for any ψ ∈ H 1 0 (Ω), First, we show that {u n } is bounded in H 1 0 (Ω). Inspired by [10], arguing by contradiction, we define t n = u n and assume that t n → +∞. Set v n = u n /t n . Then we have and We distinguish two cases. In case p = 4, multiplying (2.5) by 4 and combining with (2.6), we obtain lim n→∞ a = 0, which is impossible, since a > 0 is constant. In case 4 < p < 6, we also obtain .
Combining this with (2.6) and u n → ∞, we must have which is a contradiction to (2.4).
In conclusion, either of the two cases above will lead to a contradiction. Thus, we obtain that {u n } is bounded in H 1 0 (Ω). Then, up to a subsequence (still denoted by {u n }), we may assume that ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ u n u, in H 1 0 (Ω), u n → u, in L r (Ω), 1 ≤ r < 6, u n → u, a.e. in Ω.

(2.7)
By using Hölder's inequality, we obtain from (2.7) that It then follows from (2.1) that Next, we show that the former alternative above does not occur. If, to the contrary, namely, Let us define a functional by Consequently, By using Hölder's inequality again, we obtain Hence, by using (2.7), we obtain Moreover, we also have This with (2.9) implies that φ (u n ) → 0. Thus, we deduce from (2.10) that Then we can apply the variational method fundamental lemma [25] to obtain Combining this with assumption (f 2 ) that |Σ 0 | = 0, we obtain u = 0. By (f 1 ) and (2.7), we have and hence, This together with (2.9) yields which is a contradiction to I(u n ) → c < a 2 4b . Therefore, we obtain Ω ∇u n ∇(u nu) dx → 0, which gives u n → u . This and the weak convergence of {u n } in H 1 0 (Ω) imply that u n → u in H 1 0 (Ω). The proof of Lemma 2.2 is complete. Now, we are in a position to prove Theorem 1.1.
Proof of Theorem 1.1 In view of Lemma 2.1, there is a sequence {u n } ⊂ H 1 0 (Ω) such that I(u n ) → c * < a 2 4b and I (u n ) → 0 for any λ > 0. It follows from Lemma 2.2 that, along a subsequence, u n → u in H 1 0 (Ω), and u is a weak solution of problem (1.1). Furthermore, if we replace the functional I with the following one: It is clear that all the above calculations can be repeated word by word. Thus, there is a nontrivial critical point u ∈ H 1 0 (Ω) ofĨ. Then we have Ĩ (u), u -= 0, and hence Due to u n → u and the fact u n 2 → a b is false, we deduce u -2 = 0. In turn, we deduce u ≥ 0. By the standard elliptic regularity argument and the strong maximum principle, we get u > 0, namely, u is a positive solution of problem (1.1). This finishes the proof.

Proof of Theorem 1.2
In order to prove Theorem 1.2, we need the following four lemmas.

Lemma 3.1
Suppose that (f 1 ) holds, then the functional defined by Proof It is easy to verify that the functional χ is well defined. Assume that u n u in H 1 0 (Ω), then we have u n → u in L r (Ω), 2 ≤ r < 6. Hence, u n → u a.e. in Ω. Since {u n } is bounded in H 1 0 (Ω), we have {u n } is bounded in L 6 (Ω), and so {|u n | p } is bounded in L 6/p (Ω). Furthermore, we obtain that |u n | p |u| p in L 6/p (Ω), and consequently, we can apply (f 1 ) to deduce that Thus, the proof is complete.
(ii) The proof is similar to Lemma 2.1.
which implies that {u n } is bounded in H 1 0 (Ω), that is, there exists m > 0 such that u n ≤ m for each n ∈ N.
Set v n = u nu * and we claim that v n → 0. If not, we may suppose v n → l with l > 0. Passing the limit as n → ∞, we get that l 2 abl 2b u * 2 = 0, that is, -D 0 λ. (3.11) Combining (3.9) and (3.11), we have that, for any λ ∈ (0, λ * ), This completes the proof of Lemma 3.4. Now, we are ready to prove Theorem 1.2.
Proof of Theorem 1.2 Define the minimax level of the mountain pass theorem as in Lemma 2.1. By the definition of c * and Lemma 3.4, we know there is λ * > 0 such that c * < a 2 4b -D 0 λ for any λ ∈ (0, λ * ). According to Lemma 3.3, we further obtain the compactness property on the level c * . Then we can argue as in the proof of Theorem 1.1 to conclude that there exists u * ∈ H 1 0 (Ω) such that u * is a positive solution of problem (1.1). This completes the proof.