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An inhomogeneous perturbation for a class of nonlinear scalar field equations

Abstract

This paper deals with a class of nonlinear scalar field equations with an inhomogeneous perturbation. Two positive solutions were obtained using the variational methods.

Introduction and main result

In this paper, we consider the following nonlinear scalar field equations with an inhomogeneous perturbation

$$\begin{aligned} -\Delta u =&g(u)+h(x),\quad u\in H^{1}\bigl( \mathbb{R}^{N}\bigr),N\geq 3. \end{aligned}$$
(1.1)

Actually, g satisfies the Berestycki–Lions conditions:

\((g_{1})\):

\(g\in C(\mathbb{R},\mathbb{R})\);

\((g_{2})\):

\(-\infty <\liminf_{s\rightarrow 0^{+}}\frac{g(s)}{s}\leq \limsup_{s\rightarrow 0^{+}}\frac{g(s)}{s}=-m<0\);

\((g_{3})\):

\(\lim_{s\rightarrow +\infty }\frac{g(s)}{s^{2^{*}-1}}=0\), where \(2^{*}=\frac{2N}{N-2}\);

\((g_{4})\):

there exists \(\zeta >0\) such that \(G(\zeta ):=\int _{0}^{\zeta }g(\tau )\,d\tau >0\);

and h satisfies

\((h_{1})\):

there exists \(p\in [\frac{2N}{N+2},2]\) such that \(h\in L^{p}(\mathbb{R}^{N})\);

\((h_{2})\):

h is nonnegative and \(h\not \equiv 0\);

\((h_{3})\):

h is radially symmetric;

\((h_{4})\):

\((\nabla h,x)\in L^{\frac{2N}{N+2}}(\mathbb{R}^{N})\), where \((\cdot ,\cdot )\) denotes scalar product in \(\mathbb{R}^{N}\).

When \(h\equiv 0\), Eq. (1.1) reduces to the following nonlinear scalar field equations

$$\begin{aligned}& -\Delta u=g(u), \quad u\in H^{1}\bigl( \mathbb{R}^{N}\bigr), N\geq 3. \end{aligned}$$
(1.2)

Equation (1.2) possesses strong physical background as introduced in [1, 6] and has been extensively studied, for example, in [3, 5, 10, 12]. Especially in [3], Berestycki and Lions gave nearly optimal conditions known as the Berestycki–Lions conditions.

In [2, 8, 9], the authors studied Eq. (1.2) with a homogeneous perturbation and obtained a positive solution using the variational method. In this paper, we consider the effect of an inhomogeneous perturbation. In other words, we investigate the existence of positive solutions of Eq. (1.1). With regard to Eq. (1.1), there are some results, for example, [4, 14, 15]. Compared with those results, in the present paper, the nonlinearity g is almost optimal.

Set \(|\cdot |_{s}=(\int _{\mathbb{R}^{N}}|\cdot |^{s}\,dx)^{\frac{1}{s}}\). Using the variational method, we get the following

Theorem 1.1

Suppose that \((g_{1})\)\((g_{3})\) and \((h_{1})\)\((h_{3})\) hold. Then there exists \(\Lambda >0\) such that when \(|h|_{p}<\Lambda \), Eq. (1.1) has a positive solution. If we add \((g_{4})\) and \((h_{4})\), then when \(|h|_{p}<\Lambda \), Eq. (1.1) has another positive solution.

We have something to say about the perturbation h. The assumptions \((h_{1})\) and \((h_{2})\) are necessary, and \((h_{3})\) is to overcome the lack of compactness. Moreover, to prove the second positive solution, we need to use the Pohožaev identity, and then \((h_{4})\) seems appropriate.

Set \(f(s)=g(s)+ms\), then Eq. (1.1) equals to the following equation

$$\begin{aligned}& -\Delta u+mu=f(u)+h(x), \quad u\in H^{1}\bigl( \mathbb{R}^{N}\bigr), N\geq 3. \end{aligned}$$
(1.3)

where f satisfies

\((f_{1})\):

\(f\in C(\mathbb{R},\mathbb{R})\);

\((f_{2})\):

\(-\infty <\liminf_{s\rightarrow 0^{+}}\frac{f(s)}{s}\leq \limsup_{s\rightarrow 0^{+}}\frac{f(s)}{s}=0\);

\((f_{3})\):

\(\lim_{s\rightarrow +\infty }\frac{f(s)}{s^{2^{*}-1}}=0\);

\((f_{4})\):

there exists \(\zeta >0\) such that \(F(\zeta ):=\int _{0}^{\zeta }f(\tau )\,d\tau >\frac{1}{2}m\zeta ^{2}\).

We only need to prove the following

Theorem 1.2

Suppose that \((f_{1})\)\((f_{3})\) and \((h_{1})\)\((h_{3})\) hold. Then there exists \(\Lambda >0\) such that when \(|h|_{p}<\Lambda \), Eq. (1.3) has a positive solution. If we add \((f_{4})\) and \((h_{4})\), then when \(|h|_{p}<\Lambda \), Eq. (1.3) has another positive solution.

Remark 1.3

(i) f can be sign-changing. (ii) There exist some functions that satisfy \((h_{1})\)\((h_{4})\). For example,

$$ h_{1}(x)=\frac{\Lambda }{2\sqrt{\omega _{N}}(1+ \vert x \vert ^{N})},\qquad h_{2}(x)= \frac{\Lambda e^{-\frac{ \vert x \vert }{2}}}{\sqrt{N\omega _{N}[1+(N+1)!]}},$$

where \(\omega _{N}\) denotes the volume of the unit ball in \(\mathbb{R}^{N}\). By computing, we have \(h_{i}\in L^{2}(\mathbb{R}^{N})\), \((\nabla h_{i},x)\in L^{\frac{2N}{N+2}}(\mathbb{R}^{N})\) and \(|h_{i}|_{2}<\Lambda \), \(i=1,2\).

The rest of the paper is organized as follows: In Sect. 2, we introduce some preliminaries. In Sect. 3, we give the proof of the first positive solution. Section 4 is devoted to obtaining the second positive solution.

Preliminaries

From now on, \(C,C_{1},C_{2},\ldots \) , denotes various positive constant, \(u^{\pm }=\max \{\pm u,0\}\) and \((H, \|\cdot \|)\) is a Hilbert space, where

$$\begin{aligned}& H=\bigl\{ u\in H^{1}\bigl(\mathbb{R}^{N}\bigr):u(x)=u\bigl( \vert x \vert \bigr)\bigr\} ,\qquad \Vert \cdot \Vert = \biggl[ \int _{ \mathbb{R}^{N}}\bigl( \vert \nabla \cdot \vert ^{2}+m \vert \cdot \vert ^{2}\bigr)\,dx \biggr]^{ \frac{1}{2}}. \end{aligned}$$

To ensure the positivity of solutions and for simplicity, we always take \(f(s)=0\) for all \(s\leq 0\). As is well known, the solutions of Eq. (1.3) correspond to the critical points of the following energy functional

$$\begin{aligned} I(u)=\frac{1}{2} \Vert u \Vert ^{2}- \int _{\mathbb{R}^{N}}F(u)\,dx- \int _{ \mathbb{R}^{N}}h(x)u\,dx. \end{aligned}$$

By Principle of symmetric criticality [13], we know that if u is a critical point of I restricted to H, then u is a critical point of I in \(H^{1}(\mathbb{R}^{N})\). Set

$$ F_{1}(s)= \int _{0}^{s}f^{+}(t)\,dt \quad \text{and}\quad F_{2}(s)= \int _{0}^{s}f^{-}(t)\,dt,$$

then \(F_{1}(s)\geq 0\), \(F_{2}(s)\geq 0\), \(F(s)=F_{1}(s)-F_{2}(s)\) for all \(s\in \mathbb{R}\),

$$\begin{aligned} I(u)=\frac{1}{2} \Vert u \Vert ^{2}+ \int _{\mathbb{R}^{N}}F_{2}(u)\,dx- \int _{ \mathbb{R}^{N}}F_{1}(u)\,dx- \int _{\mathbb{R}^{N}}h(x)u\,dx \end{aligned}$$

and by \((f_{1})\)\((f_{3})\), we have

$$\begin{aligned} \lim_{s\rightarrow 0^{+}}\frac{f^{+}(s)}{s}=\lim _{s \rightarrow +\infty }\frac{f^{\pm }(s)}{s^{2^{*}-1}}=0. \end{aligned}$$
(2.1)

The first positive solution of Eq. (1.3)

In this section, we prove that Eq. (1.3) has a local minimal solution.

Lemma 3.1

Suppose that \((f_{1})\)\((f_{3})\) and \((h_{1})\) hold. Then there exist \(\rho >0\), \(\Lambda >0\), \(\alpha >0\) such that when \(|h|_{p}<\Lambda \), \(I(u)\geq \alpha \) for all \(\|u\|=\rho \).

Proof

From (2.1), it follows that

$$ F_{1}(s)\leq \frac{m}{4} \vert s \vert ^{2}+C_{1} \vert s \vert ^{2^{*}} \quad \text{for all } s \in \mathbb{R}.$$

Combining with Hölder’s inequality and Sobolev’s inequality, we get

$$\begin{aligned} I(u) \geq &\frac{1}{2} \Vert u \Vert ^{2}- \frac{m}{4} \int _{\mathbb{R}^{N}}u^{2}\,dx-C_{1} \int _{\mathbb{R}^{N}} \vert u \vert ^{2^{*}}\,dx- \vert h \vert _{p} \vert u \vert _{\frac{p}{p-1}} \\ \geq &\bigl(C_{2} \Vert u \Vert -C_{3} \Vert u \Vert ^{2^{*}-1}-C_{4} \vert h \vert _{p}\bigr) \Vert u \Vert . \end{aligned}$$

Define \(k(t)=C_{2}t-C_{3}t^{2^{*}-1}\) for \(t>0\), then there exists \(\rho >0\) such that \(k(t)\) is increasing in \([0,\rho ]\), \(k(t)\) is decreasing in \([\rho ,+\infty )\), and \(k(\rho )=\max_{t>0}k(t)\). Hence when \(|h|_{p}<\Lambda :=\frac{k(\rho )}{C_{4}}\), we have \(I(u)\geq \alpha :=[k(\rho )-C_{4}|h|_{p}]\rho\) for all \(\|u \|=\rho \). □

Define \(\overline{B}_{\rho }=\{u\in H:\|u\|\leq \rho \}\) and \(\mathfrak{m}=\inf_{u\in \overline{B}_{\rho }}I(u)\), then we have

Lemma 3.2

Suppose that \((f_{1})\)\((f_{3})\) and \((h_{1})\)\((h_{2})\) hold. Then \(\mathfrak{m}\in (-\infty ,0)\).

Proof

It follows from \((f_{1})\)\((f_{2})\) that there exist \(M>0\) and \(\theta >0\) such that

$$ F(s)\geq -Ms^{2}\quad \text{for all } s\in [0,\theta ).$$

By \((h_{2})\), there exist \(L\in (0,\theta )\) and \(\varphi \in H\) such that \(\int _{\mathbb{R}^{N}}h\varphi \,dx>0\) and \(0\leq \varphi (x)\leq L\) for all \(x\in \mathbb{R}^{N}\). Then we have

$$ \lim_{t\rightarrow 0^{+}}\frac{I(t\varphi )}{t}\leq \lim_{t\rightarrow 0^{+}} \biggl[\frac{t \Vert \varphi \Vert ^{2}}{2}+Mt \int _{\mathbb{R}^{N}}\varphi ^{2}\,dx- \int _{\mathbb{R}^{N}}h\varphi \,dx \biggr]=- \int _{\mathbb{R}^{N}}h\varphi \,dx< 0,$$

which implies that there exists \(t_{0}>0\) such that \(\|t_{0}\varphi \|\leq \rho \) and \(I(t_{0}\varphi )<0\). Hence \(\mathfrak{m}<0\). It is obvious that \(\mathfrak{m}>-\infty \). □

Lemma 3.3

Suppose that \((f_{1})\)\((f_{3})\) and \((h_{1})\)\((h_{3})\) hold. Then \(\mathfrak{m}\) is achieved.

Proof

By the definition of \(\mathfrak{m}\), there exists a sequence \(\{u_{n}\}\subset H\) such that \(\|u_{n}\|\leq \rho \) and \(I(u_{n})=\mathfrak{m}+o(1)\). Then there exists \(u\in H\) such that up to a subsequence,

$$ \textstyle\begin{cases} u_{n}\rightharpoonup u,& \text{in } H, \\ u_{n}\rightarrow u, & \text{in } L^{s}_{\mathrm{loc}}(\mathbb{R}^{N}), 2< s< 2^{*}, \\ u_{n}(x)\rightarrow u(x), & \text{a.e. in } \mathbb{R}^{N}. \end{cases} $$
(3.1)

The weakly lower semicontinuity of the norm infers

$$\begin{aligned} \Vert u \Vert \leq \liminf_{n\rightarrow \infty } \Vert u_{n} \Vert . \end{aligned}$$
(3.2)

Thus \(\|u\|\leq \rho \). Fatou’s lemma [11] and Strauss’s compactness lemma [3] yield

$$\begin{aligned} \int _{\mathbb{R}^{N}}F_{2}(u)\,dx\leq \liminf _{n\rightarrow \infty } \int _{\mathbb{R}^{N}}F_{2}(u_{n})\,dx \end{aligned}$$
(3.3)

and

$$\begin{aligned} \int _{\mathbb{R}^{N}}F_{1}(u_{n})\,dx= \int _{\mathbb{R}^{N}}F_{1}(u)\,dx+o(1). \end{aligned}$$
(3.4)

Since \((h_{1})\) holds,

$$\begin{aligned} \int _{\mathbb{R}^{N}}hu_{n}\,dx= \int _{\mathbb{R}^{N}}hu\,dx+o(1). \end{aligned}$$
(3.5)

By (3.2)–(3.5), we get

$$\begin{aligned} \mathfrak{m} =&\liminf_{n\rightarrow \infty }I(u_{n}) \\ \geq &\frac{1}{2} \Vert u \Vert ^{2}+ \int _{\mathbb{R}^{N}}F_{2}(u)\,dx- \int _{ \mathbb{R}^{N}}F_{1}(u)\,dx- \int _{\mathbb{R}^{N}}h(x)u\,dx \\ =&I(u) \\ \geq &\mathfrak{m}. \end{aligned}$$

Hence \(I(u)=\mathfrak{m}\). □

The proof of the first positive solution

From Lemma 3.3, there exists \(u\in H\) such that \(\|u\|\leq \rho \) and \(I(u)=\mathfrak{m}\). Lemma 3.1 infers \(\|u\|<\rho \). Thus for any \(v\in H\),

$$ \bigl\langle I'(u),v\bigr\rangle =\lim_{t\rightarrow 0^{+}} \frac{I(u+tv)-I(u)}{t}\geq 0$$

and

$$ \bigl\langle I'(u),v\bigr\rangle =\lim_{t\rightarrow 0^{-}} \frac{I(u+tv)-I(u)}{t}\leq 0,$$

which imply \(I'(u)=0\). By \(\langle I'(u),u^{-}\rangle =0\), we know \(u^{-}=0\). The strong maximum principle deduces \(u>0\) in \(\mathbb{R}^{N}\). □

The second positive solution of Eq. (1.3)

In this section, we prove that Eq. (1.3) has another positive solution. In order to obtain a bounded Palais–Smale sequence, we use the following Jeanjean’s theorem [7].

Theorem 4.1

Let X be a Banach space equipped with a norm \(\|\cdot \|_{X}\) and let \(J\subset \mathbb{R}^{+}\) be an interval. We consider a family \(\{\Phi _{\mu }\}_{\mu \in J}\) of \(C^{1}\)-functionals on X of the form

$$ \Phi _{\mu }(u)=A(u)-\mu B(u), \quad \forall \mu \in J,$$

where \(B(u)\geq 0\) for all \(u\in X\) and such that either \(A(u)\rightarrow +\infty \) or \(B(u)\rightarrow +\infty \) as \(\|u\|_{X}\rightarrow +\infty \). We assume that there are two points \(v_{1}\), \(v_{2}\) in X such that

$$ c_{\mu }=\inf_{\gamma \in \Gamma }\max_{t\in [0,1]} \Phi _{\mu }\bigl(\gamma (t)\bigr)>\max \bigl\{ \Phi _{\mu }(v_{1}), \Phi _{\mu }(v_{2})\bigr\} ,$$

where

$$ \Gamma =\bigl\{ \gamma \in C\bigl([0,1],X\bigr):\gamma (0)=v_{1},\gamma (1)=v_{2}\bigr\} .$$

Then for almost every \(\mu \in J\), there is a sequence \(\{u_{n}\}\subset X\) such that

  1. (i)

    \(\{u_{n}\}\) is bounded in X,

  2. (ii)

    \(\Phi _{\mu }(u_{n})\rightarrow c_{\mu }\) and

  3. (iii)

    \(\Phi '_{\mu }(u_{n})\rightarrow 0\) in the dual \(X^{*}\) of X.

Moreover, the map \(\mu \rightarrow c_{\mu }\) is non-increasing and continuous from the left.

From [3], we know that Eq. (1.2) has a positive ground state solution \(\omega \in H\) and

$$ \int _{\mathbb{R}^{N}}F_{1}(\omega )\,dx- \int _{\mathbb{R}^{N}}F_{2}( \omega )\,dx-\frac{m}{2} \int _{\mathbb{R}^{N}}\omega ^{2}\,dx>0.$$

Then there exists \(\delta \in (0,1)\) such that

$$\begin{aligned} \delta \int _{\mathbb{R}^{N}}F_{1}(\omega )\,dx- \int _{\mathbb{R}^{N}}F_{2}( \omega )\,dx-\frac{m}{2} \int _{\mathbb{R}^{N}}\omega ^{2}\,dx>0. \end{aligned}$$
(4.1)

In Theorem 4.1, we set

$$ X=H, \qquad \Vert \cdot \Vert _{X}= \Vert \cdot \Vert , \qquad \Phi _{\mu }=I_{\mu },\qquad J=[\delta ,1],\qquad B(u)= \int _{\mathbb{R}^{N}}F_{1}(u)\,dx$$

and

$$\begin{aligned} A(u)=\frac{1}{2} \Vert u \Vert ^{2}+ \int _{\mathbb{R}^{N}}F_{2}(u)\,dx- \int _{ \mathbb{R}^{N}}hu\,dx. \end{aligned}$$

By Hölder’s inequality and Sobolev’s inequality, we have

$$ \frac{A(u)}{ \Vert u \Vert }\geq \frac{1}{2} \Vert u \Vert - \int _{\mathbb{R}^{N}}h \frac{u}{ \Vert u \Vert }\,dx \geq \frac{1}{2} \Vert u \Vert -C \vert h \vert _{p}\rightarrow +\infty \quad \text{as } \Vert u \Vert \rightarrow +\infty ,$$

which implies \(A(u)\rightarrow +\infty \) as \(\|u\|\rightarrow +\infty \). Note that

$$\begin{aligned} I_{\mu }(u)=A(u)-\mu B(u)=\frac{1}{2} \Vert u \Vert ^{2}+ \int _{\mathbb{R}^{N}}F_{2}(u)\,dx- \mu \int _{\mathbb{R}^{N}}F_{1}(u)\,dx- \int _{\mathbb{R}^{N}}hu\,dx. \end{aligned}$$

In the following, \(I_{1}\) will always replace I. The next lemma is to verify the assumptions of Theorem 4.1.

Lemma 4.2

Suppose that \((f_{1})\)\((f_{4})\) and \((h_{1})\)\((h_{2})\) hold. Then when \(|h|_{p}<\Lambda \), there exist \(v_{1},v_{2}\in E\) such that for any \(\mu \in J\), \(c_{\mu }\geq \alpha >\max \{I_{\mu }(v_{1}),I_{\mu }(v_{2})\}\), where Λ, α are from Lemma 3.1.

Proof

From Lemma 3.1, it follows that for any \(\mu \in J\), \(I_{\mu }(u)\geq I_{1}(u)\geq \alpha \) for all \(\|u\|=\rho \). Define

$$\begin{aligned} \omega _{t}(x)= \textstyle\begin{cases} 0, &t=0, \\ \omega (t^{-1}x), &t>0, \end{cases}\displaystyle \end{aligned}$$

where ω satisfies (4.1). For any \(\mu \in J\), one has

$$\begin{aligned} I_{\mu }(\omega _{t})\leq \frac{t^{N-2}}{2} \int _{\mathbb{R}^{N}} \vert \nabla \omega \vert ^{2}\,dx- t^{N} \int _{\mathbb{R}^{N}} \biggl[\delta F_{1}( \omega )-F_{2}(\omega )-\frac{m}{2}\omega ^{2} \biggr]\,dx. \end{aligned}$$

Note that

$$ \Vert \omega _{t} \Vert ^{2}=t^{N-2} \int _{\mathbb{R}^{N}} \vert \nabla \omega \vert ^{2} \,dx+mt^{N} \int _{\mathbb{R}^{N}} \vert \omega \vert ^{2} \,dx.$$

Thus, there exists \(t_{0}>0\) such that \(\|\omega _{t_{0}}\|>\rho \) and \(I_{\mu }(\omega _{t_{0}})<0\). Set \(v_{1}=0\), \(v_{2}=\omega _{t_{0}}\). Hence for any \(\gamma \in \Gamma \), \(\max_{t\in [0,1]}I_{\mu }(\gamma (t))\geq \alpha >0\). Consequently, \(c_{\mu }\geq \alpha >0=\max \{I_{\mu }(v_{1}),I_{\mu }(v_{2})\}\). □

From Theorem 4.1, we know that for almost every \(\mu \in J\), there is a sequence \(\{u_{n}\}\subset H\) such that

$$ \begin{aligned} &\mathrm{(i)}\quad \{u_{n}\} \quad \text{is bounded in } H, \\ &\mathrm{(ii)}\quad I_{\mu }(u_{n})\rightarrow c_{\mu }, \\ &\mathrm{(iii)}\quad I'_{\mu }(u_{n})\rightarrow 0 \quad \text{in the dual } H^{*} \text{ of } H. \end{aligned} $$
(4.2)

Moreover, the map \(\mu \rightarrow c_{\mu }\) is non-increasing and continuous from the left.

Lemma 4.3

Fix \(\mu \in J\). Suppose that \((f_{1})\)\((f_{4})\) and \((h_{1})\)\((h_{3})\) hold. Assume that \(\{u_{n}\}\subset H\) satisfies (4.2). Then there exists a positive function \(u\in H\) such that \(I_{\mu }(u)=c_{\mu }\) and \(I'_{\mu }(u)=0\).

Proof

Since \(\{u_{n}\}\subset H\) satisfies (4.2), there exists \(u\in H\) such that up to a subsequence, (3.1)–(3.5) hold, and Fatou’s lemma [11] and Strauss’s compactness lemma [3] yield

$$\begin{aligned} \int _{\mathbb{R}^{N}}f^{-}(u)u\,dx\leq \liminf _{n\rightarrow \infty } \int _{\mathbb{R}^{N}}f^{-}(u_{n})u_{n} \,dx \end{aligned}$$
(4.3)

and

$$\begin{aligned} \int _{\mathbb{R}^{N}}f^{+}(u_{n})u_{n} \,dx= \int _{\mathbb{R}^{N}}f^{+}(u)u\,dx+o(1). \end{aligned}$$
(4.4)

Obviously, \(I'_{\mu }(u)=0\). That is

$$\begin{aligned} \Vert u \Vert ^{2}-\mu \int _{\mathbb{R}^{N}}f^{+}(u)u\,dx + \int _{\mathbb{R}^{N}}f^{-}(u)u\,dx- \int _{\mathbb{R}^{N}}hu\,dx=0. \end{aligned}$$
(4.5)

Note that

$$\begin{aligned} 0 =&\liminf_{n\rightarrow \infty }\bigl\langle I'_{\mu }(u_{n}),u_{n} \bigr\rangle \\ \geq &\liminf_{n\rightarrow \infty } \Vert u_{n} \Vert ^{2}-\limsup_{n\rightarrow \infty } \biggl[\mu \int _{\mathbb{R}^{N}}f^{+}(u_{n})u_{n}\,dx - \int _{\mathbb{R}^{N}}f^{-}(u_{n})u_{n} \,dx+ \int _{\mathbb{R}^{N}}hu_{n}\,dx \biggr]. \end{aligned}$$

Using (3.1)–(3.5) and (4.3)–(4.5), we obtain

$$\begin{aligned} \Vert u \Vert ^{2} \leq &\liminf_{n\rightarrow \infty } \Vert u_{n} \Vert ^{2} \\ \leq &\limsup_{n\rightarrow \infty } \biggl[\mu \int _{ \mathbb{R}^{N}}f^{+}(u_{n})u_{n}\,dx - \int _{\mathbb{R}^{N}}f^{-}(u_{n})u_{n} \,dx+ \int _{\mathbb{R}^{N}}hu_{n}\,dx \biggr] \\ \leq &\mu \int _{\mathbb{R}^{N}}f^{+}(u)u\,dx - \int _{\mathbb{R}^{N}}f^{-}(u)u\,dx+ \int _{\mathbb{R}^{N}}hu\,dx \\ =& \Vert u \Vert ^{2}. \end{aligned}$$

It is easy to know that \(\|u_{n}\|\rightarrow \|u\|\). Combining with (3.1), we get \(u_{n}\rightarrow u\) in H. Therefore, \(I_{\mu }(u)=c_{\mu }\) and \(I'_{\mu }(u)=0\). By \(\langle I'_{\mu }(u),u^{-}\rangle =0\), we know \(u^{-}=0\). The strong maximum principle deduces \(u>0\) in \(\mathbb{R}^{N}\). □

The proof of the second positive solution

We choose \(\mu _{n}\in J\) and \(\mu _{n}\nearrow 1\). Lemma 4.3 implies that there exists a positive sequence \(\{u_{\mu _{n}}\}\subset H\) such that \(I_{\mu _{n}}(u_{\mu _{n}})=c_{\mu _{n}}\) and \(I_{\mu _{n}}'(u_{\mu _{n}})=0\). Note that \((h_{4})\) holds. Then we have

$$\begin{aligned} 0 =& \bigl\langle I_{\mu _{n}}'(u_{\mu _{n}}),u_{ \mu _{n}} \bigr\rangle \\ =& \Vert u_{\mu _{n}} \Vert ^{2}+ \int _{\mathbb{R}^{N}}f^{-}(u_{ \mu _{n}})u_{\mu _{n}} \,dx -\mu _{n} \int _{\mathbb{R}^{N}}f^{+}(u_{\mu _{n}})u_{ \mu _{n}} \,dx- \int _{\mathbb{R}^{N}}h(x)u_{\mu _{n}}\,dx \end{aligned}$$
(4.6)

and the following Pohožaev identity

$$\begin{aligned} 0 =&P_{\mu _{n}}(u_{\mu _{n}}) \\ :=&\frac{N-2}{2} \int _{\mathbb{R}^{N}} \vert \nabla u_{\mu _{n}} \vert ^{2}\,dx+ \frac{Nm}{2} \int _{\mathbb{R}^{N}} u_{\mu _{n}}^{2}\,dx+N \int _{ \mathbb{R}^{N}}F_{2}(u_{\mu _{n}})\,dx \\ &{}-N\mu _{n} \int _{\mathbb{R}^{N}}F_{1}(u_{\mu _{n}})\,dx-N \int _{ \mathbb{R}^{N}}hu_{\mu _{n}}\,dx - \int _{\mathbb{R}^{N}}(\nabla h,x)u_{ \mu _{n}}\,dx. \end{aligned}$$

Combining with Hölder’s inequality and Sobolev’s inequality, we get

$$\begin{aligned} c_{\delta } \geq &c_{\mu _{n}} \\ =&I_{\mu _{n}}(u_{\mu _{n}})-\frac{1}{N}P_{\mu _{n}}(u_{\mu _{n}}) \\ =&\frac{1}{N} \int _{\mathbb{R}^{N}} \vert \nabla u_{\mu _{n}} \vert ^{2}\,dx+ \frac{1}{N} \int _{\mathbb{R}^{N}}(\nabla h,x)u_{\mu _{n}}\,dx \\ \geq &\frac{1}{N} \int _{\mathbb{R}^{N}} \vert \nabla u_{\mu _{n}} \vert ^{2}\,dx-C \bigl\vert ( \nabla h,x) \bigr\vert _{\frac{2N}{N+2}} \biggl( \int _{\mathbb{R}^{N}} \vert \nabla u_{ \mu _{n}} \vert ^{2}\,dx \biggr)^{\frac{1}{2}}. \end{aligned}$$

So

$$\begin{aligned} \int _{\mathbb{R}^{N}} \vert \nabla u_{\mu _{n}} \vert ^{2}\,dx\leq C \quad \text{for all } n\in \mathbb{N}. \end{aligned}$$
(4.7)

By (2.1), (4.6), (4.7), Hölder’s inequality, and Sobolev’s inequality, we yield

$$\begin{aligned} m \int _{\mathbb{R}^{N}}u_{\mu _{n}}^{2}\,dx \leq & \Vert u_{\mu _{n}} \Vert ^{2}+ \int _{\mathbb{R}^{N}}f^{-}(u_{\mu _{n}})u_{\mu _{n}} \,dx \\ =&\mu _{n} \int _{\mathbb{R}^{N}}f^{+}(u_{\mu _{n}})u_{\mu _{n}} \,dx+ \int _{\mathbb{R}^{N}}h(x)u_{\mu _{n}}\,dx \\ \leq &\frac{m}{2} \int _{\mathbb{R}^{N}}u_{\mu _{n}}^{2}\,dx+C_{1} \int _{ \mathbb{R}^{N}}u_{\mu _{n}}^{2^{*}}\,dx +C_{2} \vert h \vert _{p} \Vert u_{\mu _{n}} \Vert \\ \leq &\frac{m}{2} \int _{\mathbb{R}^{N}}u_{\mu _{n}}^{2}\,dx+C_{3}+C_{2} \vert h \vert _{p} \biggl(C+m \int _{\mathbb{R}^{N}}u_{\mu _{n}}^{2}\,dx \biggr)^{\frac{1}{2}}. \end{aligned}$$

So

$$ \int _{\mathbb{R}^{N}}u_{\mu _{n}}^{2}\,dx\leq C \quad \text{for all } n\in \mathbb{N}.$$

Hence \(\{u_{\mu _{n}}\}\) is bounded in H. Note that \(\mu _{n}\nearrow 1\),

$$ I_{1}(u_{\mu _{n}})=I_{\mu _{n}}(u_{\mu _{n}})+(\mu _{n}-1) \int _{ \mathbb{R}^{N}}F_{1}(u_{\mu _{n}})\,dx =c_{\mu _{n}}+(\mu _{n}-1) \int _{ \mathbb{R}^{N}}F_{1}(u_{\mu _{n}})\,dx$$

and

$$ \bigl\Vert I'_{1}(u_{\mu _{n}}) \bigr\Vert _{*}= \bigl\Vert I'_{1}(u_{\mu _{n}})-I'_{\mu _{n}}(u_{ \mu _{n}}) \bigr\Vert _{*}=\sup_{ \Vert v \Vert =1} \biggl\vert (\mu _{n}-1) \int _{ \mathbb{R}^{N}}f^{+}(u_{\mu _{n}})v\,dx \biggr\vert ,$$

where \(\|\cdot \|_{*}\) denotes the norm in \(H^{*}\). Therefore, \(I_{1}(u_{\mu _{n}})\rightarrow c_{1}\) and \(\|I'_{1}(u_{\mu _{n}})\|_{*}\rightarrow 0\). According to Lemma 4.2 and Lemma 4.3, we get that there exists a positive function \(u\in H\) such that \(I_{1}(u)=c_{1}>0\) and \(I'_{1}(u)=0\). □

By Sect. 3 and Sect. 4, we complete the proof of Theorem 1.2.

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Acknowledgements

The authors wish to thank the referees and the editor for their valuable comments and suggestions.

Funding

This research was supported by NNSFC (11861052), Natural Science Foundation of Education of Guizhou ([2019]065, KY[2020]144), Science and Technology Foundation of Guizhou ([2019]5653) and Funds of QNUN (QNYSKYTD2018012).

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Sun, X., Duan, Y. & Liu, J. An inhomogeneous perturbation for a class of nonlinear scalar field equations. Bound Value Probl 2022, 26 (2022). https://doi.org/10.1186/s13661-022-01607-z

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MSC

  • 35J20
  • 35B20
  • 35B09

Keywords

  • Nonlinear scalar field equations
  • Inhomogeneous perturbation
  • Variational methods
  • Positive solution