Multiplicity of positive solutions for Kirchhoff type problem involving critical exponent and sign-changing weight functions

This paper is devoted to the study of a class of Kirchhoff type problems with critical exponent, concave nonlinearity, and sign-changing weight functions. By means of variational methods, the multiplicity of the positive solutions to this problem is obtained.

MSC:35J20, 35J60, 47J30, 58E50.

This paper is concerned with the existence and multiplicity of positive solutions for the following problem:

where Ω is a smooth bounded domain in R^3 with and the parameters a,b,\lambda >0. The weight functions f(x), g(x) satisfy the following conditions:

(f_1) f(x)\in L^{q_0}(\mathrm{\Omega }) and f^{+}=max\{f,0\}\ne 0, where q_0=\frac{6}{5-q};

(f_2) there exist positive constants {\beta }_0 and {\delta }_0 such that B(x_0,2{\delta }_0)\subset \mathrm{\Omega } and f(x)\ge {\beta }_0 in B(x_0,2{\delta }_0);

(g_1) g(x)\in L^{\mathrm{\infty }}(\mathrm{\Omega }) and g^{+}=max\{g,0\}\ne 0;

(g_2) g(x_0)={\parallel g\parallel }_{\mathrm{\infty }} and g(x)>0 for all x\in B(x_0,2{\delta }_0);

(g_3) there exists k>3 such that g(x)=g(x_0)+o({|x-x_0|}^k) as x\to x_0.

In (1), if we replace \lambda f(x)u^q+g(x)u^5 by h(x,u), it reduces to the following Dirichlet problem of Kirchhoff type:

Problem (2) is related to the stationary analogue of the equation

proposed by Kirchhoff in [1] as an extension of the classical d’Alembert wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in the length of the string produced by transverse vibrations. It received great attention only after Lions [2] proposed an abstract framework for the problem. The solvability of the Kirchhoff type problem (2) has been paid much attention to by various authors. The positive solutions of such a problem are obtained by using variational methods [3–5]. Perera and Zhang [6] obtained a nontrivial solution of problem (2) via the Yang index and the critical group. He and Zou [7] obtained infinitely many solutions by using the local minimum methods and the fountain theorems. Recently, when h(x,u) is a continuous superlinear nonlinearity with critical growth, the existence of positive solutions of the Kirchhoff type problem has been studied [8–13]. Moreover, the paper [14] considered problem (2) with concave and convex nonlinearities by using a Nehari manifold and fibering map methods, and one obtained the existence of multiple positive solutions. In addition, the corresponding results of the Kirchhoff type problem can be found in [15–25], and the references therein.

In the present paper, we deal with problem (1) and consider the existence and multiplicity of positive solutions of problem (1). About the critical growth situation, the aforementioned papers only showed the existence of positive solutions of the Kirchhoff type problem. Moreover, involving the concave and convex nonlinearities, [14] only considered the subcritical growth case. Therefore, our purpose is to extend the result of [14] to critical growth. The main results of this paper extend the corresponding results in [11] and [14].

Before stating our results, we give some notations and assumptions. Let \parallel w\parallel ={({\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }w|}^{2}dx)}^{\frac{1}{2}}, {\parallel w\parallel }_s={({\int }_{\mathrm{\Omega }}{|w|}^{s}dx)}^{\frac{1}{s}} (), . In addition, we denote positive constants by C,C_1,C_2,\dots . The main results of this paper are as follows.

Theorem 1 Let a>0, b>0 and . Suppose that (f_1) and (g_1) hold, then there exists \mathrm{\Lambda }>0 such that problem (1) for all \lambda \in (0,\mathrm{\Lambda }) has at least one positive solution.

Theorem 2 Let a>0, b>0 and . Suppose that (f_1), (f_2), (g_1), (g_2) and (g_3) hold, then there exists {\lambda }^{\ast }>0 such that problem (1) for all \lambda \in (0,{\lambda }^{\ast }) has at least two positive solutions.

Remark 1 Our Theorem 2 extends the results for the critical case of Theorem 1.1 in [11]. Our Theorem 2 shows that we have at least two positive solutions of problem (1), but the authors of the reference only obtain at least one positive solution of problem (1). In addition, the results of Theorem 2.1 in [14] are extended to critical growth.

This paper is organized as follows. In Section 2, we give the local Palais-Smale condition. The proof of Theorems 1 and 2 is provided in Section 3.

In this section, we show that the corresponding functional of problem (1) satisfies the {(\mathit{PS})}_c condition. Let u^{\pm }=max\{\pm u,0\}, the corresponding functional of problem (1) is

It is well known that the critical points of the functional I in H_0^1(\mathrm{\Omega }) are weak solutions of problem (1). By the definition of weak solution u of problem (1), it means that u\in H_0^1(\mathrm{\Omega }) satisfies

for any v\in H_0^1(\mathrm{\Omega }).

Define the best Sobolev constant,

From [26], we know that S is attained when \mathrm{\Omega }=R^3 by functions

Definition A sequence \{u_n\}\subset H_0^1(\mathrm{\Omega }) is called a {(\mathit{PS})}_c sequence of I if I(u_n)\to c and I^{\prime }(u_n)\to 0 as n\to \mathrm{\infty }. We say that I satisfies the {(\mathit{PS})}_c condition if any {(\mathit{PS})}_c sequence \{u_n\}\subset H_0^1(\mathrm{\Omega }) of I has a convergent subsequence.

Lemma 1 Let a>0, b>0 and . Assume that (f_1) and (g_1) hold. If \{u_n\}\subset H_0^1(\mathrm{\Omega }) is a {(\mathit{PS})}_c sequence of I, then \{u_n\} is bounded in H_0^1(\mathrm{\Omega }).

Proof By the Hölder inequality and the Young inequality, it follows from (3) and (f_1) that

for any u\in H_0^1(\mathrm{\Omega }), where C(\eta )={(\frac{{\parallel f\parallel }_{q_0}}{q+1})}^{\frac{2}{1-q}}{(\eta S)}^{\frac{q+1}{q-1}}. Let \{u_n\} be a {(\mathit{PS})}_c sequence of I. It follows from (4) that

which implies

Set , we see that \{u_n\} is bounded in H_0^1(\mathrm{\Omega }). □

Lemma 2 Let a>0, b>0, and . Assume that (f_1) and (g_1) hold. If \{u_n\} is a {(\mathit{PS})}_c sequence of I, then there exists a positive constant A depending on a, q, S, and {\parallel f\parallel }_{q_0} such that

Proof Let \{u_n\} be a {(\mathit{PS})}_c sequence of I. By Lemma 1, we know that \{u_n\} is bounded. Therefore, up to a subsequence, there exists u\in H_0^1(\mathrm{\Omega }) such that u_n converges weakly in H_0^1(\mathrm{\Omega }), strongly in L^s(\mathrm{\Omega }) with and a.e. in Ω. By (f_1), (f_2), and the Dominated Convergence Theorem, we have

Thus, by using also the fact that 〈I^{\prime }(u_n),u_n-u〉\to 0, we get

from which it follows that u_n\to u in H_0^1(\mathrm{\Omega }). Since I is C^1, we obtain I^{\prime }(u)=0. In particular, we have 〈I^{\prime }(u),u〉=0, which implies that

It follows from (4) that

Set \eta =\frac{2a}{5-q}, and we have I(u)\ge -A{\lambda }^{\frac{2}{1-q}}. □

Lemma 3 Let a>0, b>0 and . Assume that (f_1) and (g_1) hold, then I satisfies the {(\mathit{PS})}_c condition with = \frac{ab}{4}{\parallel g\parallel }_{\mathrm{\infty }}^{-1}{S}^3 + \frac{b^3}{24}{\parallel g\parallel }_{\mathrm{\infty }}^{-2}{S}^6 + \frac{aS}{6}\sqrt{b^2{\parallel g\parallel }_{\mathrm{\infty }}^{-2}{S}^4+4a{\parallel g\parallel }_{\mathrm{\infty }}^{-1}S} + \frac{b^2}{24}{\parallel g\parallel }_{\mathrm{\infty }}^{-1}{S}^4\sqrt{b^2{\parallel g\parallel }_{\mathrm{\infty }}^{-2}{S}^4+4a{\parallel g\parallel }_{\mathrm{\infty }}^{-1}S} − A{\lambda }^{\frac{2}{1-q}}, where A is the positive constant given in Lemma  2.

Proof Let \{u_n\}\subset H_0^1(\mathrm{\Omega }) be a {(\mathit{PS})}_c sequence of I with . By Lemma 1, we know that \{u_n\} is bounded. Up to a subsequence, we may assume that

From Lemma 2, we have I^{\prime }(u)=0. By (f_1) and the Dominated Convergence Theorem, we obtain

Let w_n=u_n-u; by the Brezis-Lieb lemma [27], one has

and

Since I(u_n)=c+o(1), we obtain

According to I^{\prime }(u_n)=o(1) and 〈I^{\prime }(u),u〉=0, we get

Assume that \parallel w_n\parallel \to l, it follows from (6) that

From (3), we have

As n\to \mathrm{\infty }, we deduce that

It follows from (5), (6), and Lemma 2 that

which contradicts the fact that . Therefore, we have l=0, which implies that u_n\to u in H_0^1(\mathrm{\Omega }). Hence I satisfies the {(\mathit{PS})}_c condition with . □

In this section, we show the proofs of our Theorems 1 and 2. Before we come to the proof of Theorem 1, we first recall the following lemma in [28].

Lemma 4 Let r,s>1, \psi \in L^s(\mathrm{\Omega }) and {\psi }^{+}=max\{\psi ,0\}\ne 0. Then there exists w_0\in C_0^{\mathrm{\infty }}(\mathrm{\Omega }) such that {\int }_{\mathrm{\Omega }}\psi {(w_0^{+})}^{r}dx>0.

Proof of Theorem 1 Using the hypotheses (f_1) and (g_1), it follows from (3) and (4) that

Let \eta =\frac{a}{4}, we can find \rho >0 and {\mathrm{\Lambda }}_1>0 such that for all \lambda \in (0,{\mathrm{\Lambda }}_1)

where C_1={(\frac{{\parallel f\parallel }_{q_0}{\mathrm{\Lambda }}_1}{q+1})}^{\frac{2}{1-q}}{(\frac{a}{4}S)}^{\frac{q+1}{q-1}}.

From Lemma 4, we obtain the result that there exists {\phi }_0\in C_0^{\mathrm{\infty }}(\mathrm{\Omega })\subset H_0^1(\mathrm{\Omega }) such that

Therefore, one has

Fix \lambda \in (0,{\mathrm{\Lambda }}_1); noticing that , it implies from (8) that there exists k_0=k(\lambda )>0 small enough such that . Thus we deduce that

By applying the Ekeland’s variational principle in \overline{B_{\rho }(0)} [29], we obtain the result that there exists a {(\mathit{PS})}_{c_{\lambda }} sequence \{u_n\}\subset \overline{B_{\rho }(0)} of I.

By the expression of c^{\ast }, we can choose such that c^{\ast }>0 for all \lambda \in (0,\mathrm{\Lambda }). It follows from and Lemma 3 that I satisfies the {(\mathit{PS})}_{c_{\lambda }} condition. Therefore, one has a subsequence still denoted by \{u_n\} and u_{\lambda }\in H_0^1(\mathrm{\Omega }) such that u_n\to u_{\lambda } in H_0^1(\mathrm{\Omega }) and

which implies that u_{\lambda } is a solution of problem (1). After a direct calculation, we derive \parallel u_{\lambda }^{-}\parallel =〈I^{\prime }(u_{\lambda }),-u_{\lambda }^{-}〉=0, which implies u_{\lambda }\ge 0. Since , we have u_{\lambda }\ne 0. Applying the Harnack inequality [30], we see that u_{\lambda } is a positive solution of problem (1). The proof of Theorem 1 is completed. □

Lemma 5 Let a>0, b>0 and . Assume that (f_1), (f_2), (g_1), (g_2), and (g_3) hold, then there exists {\mathrm{\Lambda }}^{\ast }>0, such that for any \lambda \in (0,{\mathrm{\Lambda }}^{\ast }), we can find {\overline{u}}_{\lambda }\in H_0^1(\mathrm{\Omega }) such that .

Proof For convenience, we consider the functional J:H_0^1(\mathrm{\Omega })\to R defined by

for all u\in H_0^1(\mathrm{\Omega }). According to (g_2) and (g_3), we can choose such a cut-off function \varphi (x)\in C_0^{\mathrm{\infty }}(\mathrm{\Omega }) that \varphi (x)=1 for x\in B(x_0,{\delta }_0), \varphi (x)=0 for x\in R^3\setminus B(x_0,2{\delta }_0), 0\le \varphi (x)\le 1 and |\mathrm{\nabla }\varphi |\le C_2, where C_2>0 is a positive constant. Define

According to (g_2) and (g_3), similar to the calculation of [31], we have the following estimate (as \epsilon \to 0)

where U(x)={(1+{|x-x_0|}^2)}^{-\frac{1}{2}}. Define

for all t\ge 0. From (9), we have {lim}_{t\to +\mathrm{\infty }}h(t)=-\mathrm{\infty }. Note that h(0)=0 and h(t)>0 for t\to 0^{+}, so {sup}_{t\ge 0}h(t) is attained for some t_{\epsilon }>0. By

one has

Therefore, we deduce from (9) that

By the expression of c^{\ast }, we can choose {\mathrm{\Lambda }}_2>0 such that c^{\ast }>0 for all \lambda \in (0,{\mathrm{\Lambda }}_2). Using the definitions of I and u_{\epsilon }, from (f_2) and (g_2), we have

for all t\ge 0 and \lambda >0. It follows that there exist T\in (0,1) and {\epsilon }_1>0 such that

for all and . Moreover, using the definitions of I and u_{\epsilon }, it follows from (f_2) and (10) that

Let \epsilon ={\lambda }^{\frac{8}{1-q}}\in (0,{\delta }_0^2), it follows that

By the above two inequalities, for any , we have

Hence, we can choose {\mathrm{\Lambda }}_3>0 such that for all

Therefore, for all and \epsilon ={\lambda }^{\frac{8}{1-q}}, we have

Set {\mathrm{\Lambda }}^{\ast }=min\{{\mathrm{\Lambda }}_2,{\mathrm{\Lambda }}_3,{\epsilon }_1^{\frac{1-q}{8}}\}. Let \lambda \in (0,{\mathrm{\Lambda }}^{\ast }), \epsilon ={\lambda }^{\frac{8}{1-q}} and {\overline{u}}_{\lambda }=u_{\epsilon }, we deduce from (11) and (13) that

 □

Proof of Theorem 2 Choose {\lambda }^{\ast }=min\{\mathrm{\Lambda },{\mathrm{\Lambda }}^{\ast }\}, from the proof of Theorem 1, we have already seen that problem (1) for any \lambda \in (0,{\lambda }^{\ast }) has a positive solution u_{\lambda } with . Now we only need to find the second positive solution of problem (1). According to (f_1), we can see that (4) and (7) hold. It follows from (g_1) and Lemma 4 that there exists {\varphi }_0\in C_0^{\mathrm{\infty }}(\mathrm{\Omega }) such that

According to (4), we have

which implies that

Hence, there exists a positive number t_0 such that \parallel t_0{\varphi }_0\parallel >\rho and for any \lambda \in (0,{\lambda }^{\ast }). It implies from (7) that the functional I has the mountain pass geometry. Define

From Lemma 5, we have . Applying Lemma 3, we know that I satisfies the {(\mathit{PS})}_{{\tilde{c}}_{\lambda }} condition. By the Mountain Pass Theorem [32], we obtain the result that problem (1) has the second solution {\tilde{u}}_{\lambda } with I({\tilde{u}}_{\lambda })>0. After a direct calculation, we derive

which implies that {\tilde{u}}_{\lambda }^{-}=0. Hence we have {\tilde{u}}_{\lambda }\ge 0. Since I({\tilde{u}}_{\lambda })>0=I(0), we have {\tilde{u}}_{\lambda }\ne 0. By the Harnack inequality, we obtain the result that {\tilde{u}}_{\lambda } is the second positive solution of problem (1). The proof of Theorem 2 is completed. □

The author would like to thank the referees for valuable comments and suggestions on improving this paper. This paper was supported by Science and Education Youth culture project in Guizhou Province (Contract Number: Guizhou Provincial Institute of Zi (2012) No. 157). This paper also was supported by Science and Technology Foundation of Guizhou Province (No. J[2013]2141; No. LKM[2011]31).

Affiliations

1. College of Science, Guizhou Minzu University, Guiyang, Guizhou, 550025, People’s Republic of China

   • Chang-Mu Chu

Corresponding author

Correspondence to Chang-Mu Chu.

Competing interests

The author declares that they have no competing interests.

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