Existence of a positive radial solution for semilinear elliptic problem involving variable exponent

1.   Introduction and main result

In recent years, the following nonlinear elliptic equation

was studied due to the fact that it can be applied to fluid mechanics and the field of image processing (see ^[1,2]), where $\Omega\subset \mathbb{R}^{N}(N\geq 3)$ is a bounded smooth domain, $p\in C(\overline{\Omega}, \mathbb{R})$, $1 < p^- = \mathop {\min }\limits_{x\in \overline{\Omega}}p(x)\leq p(x)\leq\mathop {\max}\limits_{x\in \overline{\Omega}}p(x) = p^+ < N$, $\Delta_{p(x)}u: = div(|\nabla u|^{p(x)-2}\nabla u)$ and $f: \overline{\Omega}\times\mathbb{R} \rightarrow \mathbb{R}$.

In 2003, Fan and Zhang in ^[3] gave several sufficient conditions for the solvability of nontrivial solutions for problem (1.1). These conditions include either the sublinear growth condition

or Ambrosetti-Rabinowitz type growth condition ($(AR)$-condition, for short): there is $\theta > p^+$ such that

where $C > 0$, $F(x, t) = \int_0^tf(x, s)\, ds$, and $|f(x, t)t|\leq C(1+|t|^{p^*(x)})$ with $p^*(x) = \frac{Np(x)}{N-p(x)}$. Subsequently, Chabrowski and Fu in ^[4] discussed problem (1.1) in a more general setting than that in ^[3].

As is well known, the $(AR)$-condition ensure the boundedness of Palais-Smale sequence of the corresponding function. However, there are some papers considering the nonlinearity without $(AR)$-condition. ^[5] proveed the existence of strong solutions of problem (1.1) without the growth condition of the well-known Ambrosetti–Rabinowitz type. Subsequently, ^[6] extended the results of ^[5]. Under no Ambrosetti–Rabinowitz's superquadraticity conditions, ^[7] and ^[8] obtained the existence and multiplicity of the solution of problem (1.1) by different methods. In addition, ^[9] and ^[10] pointed out the importance of the Cerami condition. In fact, these papers still require nonlinearity to satisfy superlinear growth condition:

As far as we know, there are few results in the case $f(x, t)t = p(x) F(x, t)$ for some $x\in\Omega$ and $|t|$ large enough. In addition to the eigenvalue problem was studied in ^[11] and ^[12], we only see that ^[13] and ^[14] discussed the multiplicity of nontrivial solutions and sign-changing solutions, respectively. As described in ^[15], there are new difficulties in dealing with this situation.

Let $S_N = \mathop {\inf } \limits_{0\neq u\in H^{1}_0(B_1)}\frac{\int_{B_1}|\nabla u|^2dx}{\left(\int_{B_1}|u|^{2^*}dx\right)^{\frac{2}{2^*}}}$ be the best Sobolev constant and $B_1$ be the unit ball in $\mathbb{R}^N(N\geq 3)$, we consider the following elliptic problem

where $q(x) = q(|x|)$ is a continuous radial function satisfying $2\leq q(x) < 2^* = \frac{2N}{N-2}$ and $q(0) > 2$.

Theorem 1.1. Let $q(x)$ be a continuous radial function satisfying

Suppose that $\Omega_0 = \{x\in B_1 :q(x) = 2\}$ is not empty and the measure satisfies

Then problem (1.2) has at least a positive radial symmetric solution.

Remark 1.2. In ^[15], the authors considered the existence of a nontrivial solution of $-\Delta_{p} u+u^{p-1} = u^{q(x)-1}$, $u\in W^{1, p}_r(\mathbb{R}^N)$ and $u\geq 0$ in $\mathbb{R}^N$ for $1 < p < N$, where $\Delta_{p}u: = div(|\nabla u|^{p-2}\nabla u)$. They showed that if there exist positive constants $R_1$, $R_2$, $C_1$, $C_2$ and $0 < l_1, \ l_2 < 1$ such that $ess \inf_{x\in B_{R_1}} \{q(x)\} > p$, $q(x)\geq p+\frac{C_1}{|\log|x||^{l_1}}$ for $x\in \mathbb{R}^N\backslash B_{R_1}$ and $q(x)\leq \frac{Np}{N-p}-\frac{C_2}{|\log|x||^{l_2}}$ for $x\in B_{R_2}$, then there exists a nontrivial solution to this equation. However, our Theorem 1.1 allows $q(x) = 2$ for some $x\in B_1$.

Remark 1.3. The hypothesis of Theorem 1.1 can not ensure that problem (1.2) satisfies the Ambrosetti-Rabinowitz growth condition. Indeed, for the case $p(x)\equiv 2$ and $f(x, t) = t^{q(x)-1}$, we have $p^+ = 2$ and $F(x, t) = \frac{1}{q(x)}t^{q(x)}$ for $t\geq 0$. It follows that $f(x, t)t = p^+ F(x, t)$ for any $x\in \Omega_0$.

Remark 1.4. In our paper, the $L^{\infty}$ estimate is an essential tool that makes the solution go back to the original problem. The condition of radial symmetry plays a major role in the estimation of the solution.

2.   The auxiliary problem

According to $q(x)\geq 2$, It is not easy to determine whether the functional $I$ satisfies the Palais-Smale condition. To apply the mountain pass theorem, the first step is to modify the nonlinearity. By the continuity of $q(x)$, $2\leq q(x)\leq q^+ < 2^*$ and $q(0) > 2$, we see that there exist $\delta\in (0, \frac{1}{4})$ and $r > 0$ such that

Let $\psi(t)\in C^{\infty}_0(\mathbb{R}, [0, 1])$ be an even function satisfying $\psi(t) = 1$ for $|t|\leq 1$, $\psi(t) = 0$ for $|t|\geq 2$ and $\psi(t)$ decreases monotonically over $\mathbb{R}^+$. Define

for $\mu\in (0, 1]$. We consider the auxiliary problem

where $Q(x) = Q(|x|)\in C(B_1, [0, 1])$ satisfies $Q(x) = 1$ for $x\in B_{\delta}$ and $Q(x) = 0$ for $x\in B_1\setminus B_{2\delta}$.

Theorem 2.1. Assume that $q(x) = q(|x|)$ is a continuous radial function satisfying $2\leq q(x)\leq q^+ < 2^*$ and $q(0) > 2$, the measure of $\Omega_0 = \{x|q(x) = 2\}$ satisfies $S^{-1}_N|\Omega_{0}|^{\frac{2^*-2}{2^*}} < \frac{1}{2}$. Then problem (2.2) has at least a positive radial symmetric solution for any $\mu\in(0, 1]$.

Set $H_{0, r}^1(B_1) = \left\{u\in H_0^1(B_1)\ |\ u(x) = u(|x|)\right\}$, $\|\cdot\| = \|\nabla (\cdot)\|_{L^2(B_1)}$. $I_\mu: H^1_{0, r}(B_1)\to\mathbb{R}$ by

where $k_{\mu}(x, t) = k_{\mu}(t) = \left(\frac{t}{m_{\mu}(t)}\right)^rt^{q(x)-1}$, $K_{\mu}(x, t) = K_{\mu}(t) = \int_0^tk_{\mu}(s)\, ds$.

Lemma 2.2. $K_{\mu}(x, t)$ have the following properties:

for $t > 0$, where $C_{\mu} > 0$.

Proof. According to the monotonicity of $b_{\mu}(t)$, one has

for $t > 0$, where $\xi\in(0, t)$. Therefore, $\frac{t}{m_{\mu}(t)}$ is monotonically increasing on $\mathbb{R}^+$. Hence, $\frac{k_{\mu}(x, t)}{t^{q(x)-1}} = \left(\frac{t}{m_{\mu}(t)}\right)^r$ is also monotonically increasing on $\mathbb{R}^+$. It implies that

for $t > 0$. Obviously, $m_{\mu}(t) = \frac{A}{\mu}$ for $t\geq \frac{2}{\mu}$, where $A = 1+\int_1^2\psi(\tau)d\tau$. For $t > \frac{2}{\mu}$, one has

Combining (2.3) with (2.4), we obtain $K_{\mu}(x, t)\leq \frac{1}{q(x)+r}tk_{\mu}(x, t)+C_{\mu}$ for $t > 0$.

Lemma 2.3. Suppose that $q(x) = q(|x|)$ is a continuous radial function satifying $2\leq q(x)\leq q^+ < 2^*$ and $q(0) > 2$. Then $I_{\mu}$ satisfies the $(PS)$ condition for all $\mu\in (0, 1]$.

Proof. Let $\{u_n\}$ be a $(PS)$ sequence of $I_{\mu}$ in $H^1_{0, r}(B_1)$. There exists $C > 0$ such that

By (2.1) and Lemma 2.2, we have

which implies that $\frac{r}{2(2+r)}\|u_{n}\|^{2}\leq C+C_{\mu}+o(\|u_{n}\|)$. We obtain $\{u_{n}\}$ is bounded in $H^1_{0, r}(B_1)$. Up to a subsequence, we may assume that

It implies that

It follows from (2.5) that

It is not difficult to see that

By the Sobolev imbedding theorem and $2\leq q(x) < q(x)+r < q^++r < 2^*$, one has

and

as $i$ and $j$ tend to $+\infty$. From (2.6)–(2.8), we have $\|u_i-u_j\|\rightarrow 0$ as $i, \ j \rightarrow +\infty$, which implies that $\{u_n\}$ contains a strongly convergent subsequence in $H^1_{0, r}(B_1)$. Hence $I_{\mu}$ satisfies the $(PS)$ condition.

Lemma 2.4. $I_{\mu}$ has the following properties:

(1) there exist $m, \ \rho > 0$ such that $I_{\mu}(u) > m$ for any $u\in H^1_{0, r}(B_1)$ with $\|u\| = \rho$;

(2) there exists $w\in H^1_{0, r}(B_1)$ such that $\|w\| > \rho$ and $I_{\mu}(w) < 0$.

Proof. By definition of the function $k_{\mu}$, we have

It follows that

Therefore, there exists $C > 0$ such that

By the Sobolev imbedding theorem, it implies from $2\leq q(x) < q(x)+r < 2^*$ that

Set $\Omega_{\varepsilon} = \{x\in B_1|2\leq q(x) < 2+\varepsilon\}$. By the Sobolev imbedding theorem and the Hölder inequality, we obtain

Since $S^{-1}_N|\Omega_{0}|^{\frac{2^*-2}{2^*}} < \frac{1}{2}$, for $\varepsilon > 0$ small enough, one has $S^{-1}_N|\Omega_{\varepsilon}|^{\frac{2^*-2}{2^*}} < \frac{1}{4}+\frac{1}{2}S^{-1}_N|\Omega_{0}|^{\frac{2^*-2}{2^*}}$. From (2.9)–(2.11), we obtain

Therefore, there exist $m, \ \rho > 0$ such that $I_{\mu}(u) > m$ for any $u\in H^1_{0, r}(B_1)$ with $\|u\| = \rho$.

Fix a nonnegative radial function $v_0\in H_{0, r}^1(B_{\delta})\backslash\{0\}$. We have

for $t > 0$ sufficiently large. Choosing $w = tv_0$, we have $\|w\| > \rho$ and $I_{\mu}(w) < 0$ for $t > 0$ large enough.

Proof of Theorem 2.1. By Lemmas 2.3 and 2.4, we know that $I_{\mu}$ satisfy the $(PS)$ condition and the mountain pass geometry. Define

We obtain that problem (2.2) has a solution $u_{\mu}$ by the mountain pass theorem (see ^[16]). After a direct calculation, we derive that $\|u_{\mu}^-\|^2 = \langle I'_{\mu}(u_{\mu}), u_{\mu}^-\rangle = 0$, which implies that $u_{\mu}^- = 0$. Hence, $u_{\mu}\geq 0$. Since $I_{\mu}(u_{\mu}) > 0 = I(0)$, we have $u_{\mu}\neq 0$. One has $u_{\mu}$ is a positive solution to problem (2.2) by the Strong Maximum Principle (see ^[17]).

It follows from (2.1) that

Therefore, ${c}_{\mu}$ is uniformly bounded. In other words, we have the following results.

Remark 2.5. ${c}_{\mu}\leq D$, where $D$ is a positive constant independent of $\mu$.

3.   $L^\infty$-estimate and the proof of main results

In this section, we will show that solutions of auxiliary problem (2.2) are indeed solutions of original problem (1.2) for sufficiently small $\mu$.

Lemma 3.1. If $v$ is a positive critical point of $I_\mu$ with $I_\mu(v) = c_{\mu}$, then $\int_{B_{\frac{\delta}{2}}} (|\nabla v|^2+v^2) \, dx\leq L$, where $L$ is a positive constant independent of $\mu$.

Proof. From (2.1) and Lemma 2.2, one has

Let $\varphi\in C^{\infty}_0 (B_{\delta}, \mathbb{R})$ satisfies $|\varphi(x)|\leq 1$, $\varphi(x) = 1$ for $|x|\leq \frac{\delta}{2}$ and $|\nabla\varphi|\leq \frac{4}{\delta}$. Multiply problem (2.2) by $v\varphi^2$ and integrate to obtain

According to (3.1) and (3.2), we have

It implies from Remark 2.5 that $\int_{B_{\frac{\delta}{2}}} \left(|\nabla v|^2+v^2\right) \, dx\leq L$, where $L$ is a positive constant independent of $\mu$.

Lemma 3.2. If $v$ is a positive radial symmetric critical point of $I_\mu$ with $I_\mu(v) = c_\mu$, then $\|v\|_{L^{\infty}\left(B_{1}\right)}\leq M$, where $M$ is a positive constant independent of $\mu$.

Proof. Let $\alpha > 2$ and $\zeta\in C^{\infty}_0 (B_{\frac{\delta}{2}}, \mathbb{R})$. On the one hand, by the Young inequality, we have

On the other hand, one has

Combining (3.3) with (3.4), and noticing that $v$ is a solution to problem (2.2), we obtain

Set $\delta_k = \frac{\delta}{4}\left(1+\frac{1}{2^k}\right)$. Let $\zeta_k\in C^{\infty}_0 (B_{\delta_k}, \mathbb{R})$ satisfies the following properties: $0\leq \zeta_k\leq 1$, $\zeta_k = 1$ for $x\in B_{\delta_{k+1}}$ and $|\nabla\zeta_k|\leq\frac{1}{4(\delta_k-\delta_{k+1})} = \frac{2^{k+1}}{\delta}$. $B_{\frac{\delta}{2}}$ and $\zeta$ are taken to be $B_{\delta_k}$ and $\zeta_k$ in inequality (3.5), respectively. Using the Sobolev embedding theorem, the Hölder inequality and Lemma 3.1, we obtain

It implies that

Set $\beta_k = 2\big(\frac{2^*-q^++2}{2}\big)^{k}$ for $k = 0, 1, \cdots$. Then $\frac{ 2}{2^*-q^++2}\beta_{k+1} = \beta_{k}$. By (3.6), we have

Doing iteration yields

Since $\beta_1 > 2$, the series $\sum\limits_{j = 1}^\infty \left(\frac{2}{\beta_1}\right)^{j}$ and $\sum\limits_{j = 1}^\infty j\left(\frac{2}{\beta_1}\right)^{j}$ are convergent. Letting $k\to\infty$, we conclude that

Set $\rho = |x|$. Since $v$ is positive radially symmetric, one has

which implies that $\frac{d}{d\rho}\left(\rho^{N-1}\frac{dv}{d\rho}\right)\leq 0$. Notice that $\rho^{N-1}\frac{dv}{d\rho}|_{\rho = 0} = 0$, we have $\rho^{N-1}\frac{dv}{d\rho}\leq 0$. That is $\frac{dv}{d\rho}\leq 0$. Hence,

Proof of Theorem 1.1. By definition of the function $m_{\mu}$, we have $m_{\mu}(t) = t$ for $t\leq \frac{1}{\mu}$. It is easy to see problem (2.2) reduce to problem (1.2) for $|u|\leq \frac{1}{\mu}$. Let $\mu > \frac{1}{M}$. We see that a positive solution $u_\mu$ problem (2.2) is indeed a positive solution of problem (1.2).

Acknowledgments

Thanks for the support of National Natural Science Foundation of China (No. 11861021).

Conflict of interest

The authors declared that there was no competition of interests.