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

where Ω is a smooth bounded domain in $R^3$ with $0<q<1$ 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 $x\to x_0$.

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 $\parallel w\parallel ={({\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }w|}^{2}dx)}^{\frac{1}{2}}$, ${\parallel w\parallel }_s={({\int }_{\mathrm{\Omega }}{|w|}^{s}dx)}^{\frac{1}{s}}$ ($1<s<\mathrm{\infty }$), $B(x_0,\delta )=\{x\in R^3:|x-x_0|<\delta \}$. 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 $0<q<1$. 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 $0<q<1$. 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.

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

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 $0<q<1$. 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 $H_0^1(\mathrm{\Omega })$.

Set $\eta <\frac{2a}{5-q}$, we see that $\{u_n\}$ is bounded in $H_0^1(\mathrm{\Omega })$. □

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

which contradicts the fact that $c<c^{\ast }$. 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 $c<c^{\ast }$. □

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 ${\int }_{\mathrm{\Omega }}\psi {(w_0^{+})}^{r}dx>0$.

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

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.

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 $I(u_{\lambda })=c_{\lambda }<0=I(0)$, 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 $0<q<1$. 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 ${sup}_{t\ge 0}I(t{\overline{u}}_{\lambda })<c^{\ast }$.

 □

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. □