1 Introduction

The Schrödinger equation is central in quantum mechanics and it plays the role of Newton’s laws and conservation of energy in classical mechanics, that is, it predicts the future behaviour of a dynamical system. It is striking to point out that talking about his celebrating equation, Erwin Schrödinger said: “I don’t like it, and I’m sorry I ever had anything to do with it”. The linear Schrödinger equation is a central tool of quantum mechanics, which provides a thorough description of a particle in a non-relativistic setting. Schrödinger’s linear equation is

$$\begin{aligned} \Delta \psi +\frac{8\pi ^2m}{\hbar ^2}\,\left( E-V(x)\right) \psi =0, \end{aligned}$$

where \(\psi \) is the Schrödinger wave function, m is the mass of the particle, \(\hbar \) denotes Planck’s renormalized constant, E is the energy, and V stands for the potential energy.

Schrödinger also established the classical derivation of his equation, based upon the analogy between mechanics and optics, and closer to de Broglie’s ideas. He developed a perturbation method, inspired by the work of Lord Rayleigh in acoustics, proved the equivalence between his wave mechanics and Heisenberg’s matrix, and introduced the time dependent Schrödinger’s equation

$$\begin{aligned} i\hbar \psi _t=-\frac{\hbar ^2}{2m}\,\Delta \psi +V(x)\psi -\gamma |\psi |^{p-1}\psi \qquad x\in {\mathbb R}^N (N\ge 2), \end{aligned}$$
(1.1)

where \(p<2N/(N-2)\) if \(N\ge 3\) and \(p<+\infty \) if \(N=2\).

In physical problems, a cubic nonlinearity corresponding to \(p = 3\) in equation (1.1) is common; in this case problem (1.1) is called the Gross–Pitaevskii equation. In the study of equation (1.1), Floer and Weinstein [26] and Oh [36] supposed that the potential V is bounded and possesses a non-degenerate critical point at \(x=0\). More precisely, it is assumed that V belongs to the class (\(V_a\)) (for some real number a) introduced in Kato [31]. Taking \(\gamma >0\) and \(\hbar >0\) sufficiently small and using a Lyapunov–Schmidt type reduction, Oh [36] proved the existence of bound state solutions of problem (1.1), that is, a solution of the form

$$\begin{aligned} \psi (x,t)=e^{-iEt/\hbar }u(x). \end{aligned}$$
(1.2)

Using the Ansatz (1.2), we reduce the nonlinear Schrödinger equation (1.1) to the semilinear elliptic equation

$$\begin{aligned} -\frac{\hbar ^2}{2m}\,\Delta u+\left( V(x)-E\right) u=|u|^{p-1}u. \end{aligned}$$

The change of variable \(y=\hbar ^{-1}x\) (and replacing y by x) yields

$$\begin{aligned} -\Delta u+2m\left( V_\hbar (x)-E\right) u=|u|^{p-1}u\qquad x\in {\mathbb R}^N, \end{aligned}$$
(1.3)

where \(V_\hbar (x)=V(\hbar x)\).

Let us also recall that in his 1928 pioneering paper, Gamow [27] proved the tunneling effect, which lead to the construction of the electronic microscope and the correct study of the alpha radioactivity. The notion of “solution” used by him was not explicitly mentioned in the paper but it is coherent with the notion of weak solution introduced several years later by other authors such as Leray, Sobolev and Schwartz. Most of the study developed by Gamow was concerned with the bound states \(\psi (x,t)\) defined in (1.2), where u solves the stationary equation

$$\begin{aligned} -\Delta u+V(x)u=\lambda u\quad \text{ in }\ {\mathbb R}^N, \end{aligned}$$

for a given potential V(x). Gamow was particularly interested in the Coulomb potential but he also proposed to replace the resulting potential by a simple potential that keeps the main properties of the original one. In this way, if \(\Omega \) is a subdomain of \({\mathbb R}^N\), Gamow proposed to use the finite well potential

$$\begin{aligned} V_{q,\Omega }(x)=\left\{ \begin{array}{lll} &{}\displaystyle V(x)\quad &{}\text{ if }\ x\in \Omega \\ &{}\displaystyle q\quad &{}\text{ if }\ x\in {\mathbb R}^N{\setminus }\Omega \end{array}\right. \quad {\text{ for } \text{ some }} \, q\in {\mathbb R}. \end{aligned}$$

It seems that the first reference dealing with the limit case, the so-called infinite well potential,

$$\begin{aligned} V_{\infty }(x;R,V_0)=\left\{ \begin{array}{lll} &{}\displaystyle V_0\quad &{}\text{ if }\ x\in \Omega \\ &{}\displaystyle +\infty \quad &{}\text{ if }\ x\in {\mathbb R}^N{\setminus }\Omega \end{array}\right. \quad \text{ for } \text{ some }\quad V_0\in {\mathbb R}, \end{aligned}$$

was the book by the 1977 Nobel Prize Mott [35]. The more singular case in which \(V_0\) is the Dirac mass \(\delta _0\) is related with the so-called Quantum Dots, see Joglekar [29]. In contrast with classical mechanics, in quantum mechanics the incertitude appears (the Heisenberg principle). For instance, for a free particle (i.e. with \(V(x)\equiv 0\)), in nonrelativistic quantum mechanics, if the wave function \(\psi (\cdot ,t)\) at time \(t=0\) vanishes outside some compact region \(\overline{\Omega }\) then at an arbitrarily short time later the wave function is nonzero arbitrarily far away from the original region \(\overline{\Omega }\). Thus, the wave function instantaneously spreads to infinity and the probability of finding the particle arbitrarily far away from the initial region is nonzero for all \(t>0\). We refer to Díaz [25] for more details.

2 The main results

In this paper, we consider Schrödinger equations with sublinear nonlinearity and non-symmetric potentials, which are affected by a nonnegative perturbation. We are interested in the multiplicity of solutions and we establish several sufficient conditions for the existence of two solutions.

We point out that sublinear problems on the whole space do not have necessarily a solution. In fact, the existence of solutions is in relationship not only with the nonlinearity but also with the behaviour of a certain potential. Brezis and Kamin [18] pointed out a striking phenomenon, which asserts that a sublinear problem on the whole space has a solution if and only if a linear equation depending only on the potential has a solution. They considered the nonlinear problem

$$\begin{aligned} -\Delta u = \rho (x)\, u^p,\quad x\in \mathbb {R}^n\ (n\ge 3), \end{aligned}$$
(2.1)

with \(0<p<1\), \(\rho \in L^\infty _\mathrm{loc}(\mathbb {R}^n){\setminus }\{0\}\), \(\rho \ge 0\). Brezis and Kamin [18] proved that the nonlinear problem (2.1) has a bounded positive solution if and only if the linear equation

$$\begin{aligned} -\Delta u = \rho (x),\quad x\in \mathbb {R}^n \end{aligned}$$

has a bounded solution. Their analysis showed that such a solution exists for potentials like

$$\begin{aligned} \rho (x)=\frac{1}{1+|x|^\alpha }\quad \text{ or }\quad \rho (x)=\frac{1}{(1+|x|^2)\,|\log (2+|x|)|^\alpha }\ \ (\alpha >2), \end{aligned}$$

while no solution exists if

$$\begin{aligned} \rho (x)=\frac{1}{1+|x|^\alpha }\quad \text{ with } \alpha \le 2. \end{aligned}$$

Consider the following class of sublinear Schrödinger equations

$$\begin{aligned} -\Delta u+ V(x)u= a(x)\left| u\right| ^{q-1}u+ f(x), \quad x\in \mathbb {R}^{N}, \end{aligned}$$
(2.2)

where

figure a

Equation (2.2) arises in the study of solitary waves in nonlinear equations of the Klein-Gordon or Schrödinger type.

We look for the existence of two solutions for problem (2.2), where the potential V(x) and the weight a(x) are indefinite, that is, they are sign-changing functions in \(\mathbb {R}^{N}\). Such problems, with indefinite linear and nonlinear terms, present challenging mathematical difficulties.

Equation (2.2) can be considered as a perturbation of the homogeneous problem

$$\begin{aligned} -\Delta u+ V(x)u= a(x) \left| u\right| ^{q-1} u, \ \ \ \ \ u\in H^{1}(\mathbb {R}^{N})\cap L^{q+1}\left( \mathbb {R}^{N}\right) . \end{aligned}$$
(2.3)

Equation (2.3) has been extensively studied in the last few decades, see [6,7,8, 18, 20, 21, 38]. In 2008, using the Nehari manifold method, Chabrowski and Costa [21] proved the existence of two solutions of problem (2.3), where both a(x) and V(x) change sign in \(\mathbb {R}^{N}\). In [7], Bahrouni et al. improved this result and established the existence of infinitely many solutions of problem (2.3) when V(x) and a(x) change sign.

In the case of bounded domains and under Dirichlet boundary conditions, there is a large literature on existence and a multiplicity of solutions for (2.2), see [3, 5, 17, 19, 28, 30, 32, 34] and the references therein. In a pioneering paper, Ahmad, Lazer and Paul [3] considered the resonant problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u-\lambda _{k}u=g(x,u) \quad \text{ in } \ \ \Omega \\ u=0 \quad \quad \quad \quad \quad \quad \quad \quad \quad \text{ on } \ \ \partial \Omega , \end{array}\right. } \end{aligned}$$
(2.4)

where \(\lambda _{k}\) denotes the kth eigenvalue of the Laplace operator. They proved that a sufficient condition for the existence of a solution is

$$\begin{aligned} \lim _{\left\| u\right\| \rightarrow +\infty , u\in E_{0}} \int _{\Omega }G\left( x,u(x)\right) dx =\mp \infty , \end{aligned}$$

where \(G(x,s) = \int _{0}^{s}g(x,t) dt\) and \(E_{0}=\text{ Ker }\,\left( -\Delta -\lambda _{k}\right) \).

Brown [19] proved the existence of two solutions of problem (2.2) when \(V=f=0\) and a(x) changes sign.

In the case of unbounded domains, many authors have studied the existence of solutions of problem (2.2) with superlinear subcritical nonlinearity (\(1<q<2^{*}=\frac{2N}{N-2}\)), see [1, 2, 4, 33]. For instance, Li and Wu [33] treated problem (2.2) where \(f=\mu h, \ \ V=\lambda \) and \(a(x)\in \left( 0,1\right) \). The authors proved the existence of positive numbers \(\epsilon >0\) and \(\lambda _{0}\), \(\mu >0\) such that for any \(\lambda > \lambda _{0}\) and \(\mu \lambda ^{\frac{N}{4}-\frac{p-1}{p-2}}<\epsilon \), problem (2.2) admits multiple positive solutions.

For the sublinear case and especially for the whole space \(\mathbb {R}^{N}\), to our best knowledge, few results are known. We can for example quote the papers [5, 14,15,16]. For instance, Benrhouma [14] proved the existence of at least three solutions of problem (2.2), provided that \(a(x)< 0\) and V changes sign. As far as we know, the only existence result for problem (2.2) where both a and V change sign in \(\mathbb {R}^{N}\), is obtained by Tehrani [37]. He considered the equation

$$\begin{aligned} -\Delta u+V(x)u=a(x) g(u)+f, \end{aligned}$$

under the following assumptions:

  • \(\left( G_{1}\right) \ \ g \in C\left( \mathbb {R}, \mathbb {R}\right) , \ \ g(s)s\ge 0\) for all \(s\in \mathbb {R}\);

  • \(\left( G_{2}\right) \ \ \lim _{\left| s\right| \rightarrow + \infty } \frac{g(s)}{\left| s\right| ^{q-1}s}=1\), for some \(0<q<1\);

  • \(\left( G_{3}\right) \ \ \exists A>0, \ \ \left| g(s)\right| \le A \left| s\right| ^{q}\) for all \(s\in \mathbb {R}\);

  • \(\left( V_{2}\right) \)\(V\in L^{\infty }(\mathbb {R}^{N})\) and \( \liminf _{\left| x\right| \rightarrow +\infty }V(x)=v_{\infty }>0\);

  • \(\left( H_{1}\right) \int _{\mathbb {R}^{N}}\left[ \left| \nabla \phi \right| ^{2}+ V(x)\phi ^{2}\right] dx<0\) for some \(\phi \in C_{c}^{\infty }(\mathbb {R}^{N})\);

  • \(\left( V_{3}\right) \ \ 0\in \sigma \left( -\Delta +V\right) \);

  • \(\left( A_{0}'\right) \)\(a\in L^{\infty }(\mathbb {R}^{N}) \ \ \text{ and } \ \ \limsup _{\left| x\right| \rightarrow +\infty }a(x)=a_{\infty }<0\);

  • \((F_{1})\)\(f\in L^{2}(\mathbb {R}^{N})\);

  • \(\left( N^{\pm }\right) \) \( \int _{\mathbb {R}^{N}} a(x)\left| \varphi \right| ^{q+1}> 0 \ \ \left( \text{ or } \ \ <0 \right) \ \ \forall \varphi \in \text{ Ker }\,\left( -\Delta +V\right) , \ \ \varphi \ne 0.\)

Under these assumptions, by using bounded domain approximation techniques, Tehrani [37] proved the existence of at least one solution.

In this paper, we prove the existence of two solutions for problem (2.2), provided that both a(x) and V(x) change sign in \(\mathbb {R}^{N}\). We consider two classes of assumptions on the indefinite non-symmetric potentials a(x) and V(x).

First class: We suppose that V satisfies \(\left( H_{1}\right) \) and the following hypotheses:

  • \(\left( H_{3}\right) \ \ V\in L^{\infty }(\mathbb {R}^{N})\) and there exist \(R_{0}, \beta >0\) such that

    $$\begin{aligned} V(x)\ge \beta , \ \ \forall \ \ \left| x\right| \ge R_{0}. \end{aligned}$$

  • \(\left( H_{4}\right) \ \ \) the linear problem

    $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+V(x)u=0,\\ u\in H^{1}(\mathbb {R}^{N}) \end{array}\right. \end{aligned}$$

has not a nontrivial solution.

We suppose that a satisfies:

  • \(\left( A_{1}\right) \ \ a\in L^{\infty }(\mathbb {R}^{N})\)

and there exist \(\alpha ,R_{1}>0\) such that

$$\begin{aligned} a(x)\le -\alpha , \ \ \forall \left| x\right| \ge R_{1}. \end{aligned}$$

Second class: We assume that a and V satisfy \(\left( H_{1}\right) \), \(\left( H_{3}\right) \), \(\left( N^{\pm }\right) \) and the following conditions:

  • \(\left( \overline{A_{1}}\right) \ \ a\in L^{\frac{2}{1-q}}(\mathbb {R}^{N}), \ \ 0<q<1.\)

  • \(\left( H_{2}\right) \ \ 0\in \sigma _{d}\left( -\Delta +V\right) \),

where \(\sigma _{d}\) is the discrete spectrum of \(L=-\Delta +V(x)\).

We consider the functional \(I: Y \rightarrow \mathbb {R}\), where \(Y= H^{1}(\mathbb {R}^{N})\cap L^{q+1}\left( \mathbb {R}^{N}\right) \) or \(Y= H^{1}(\mathbb {R}^{N})\). The critical points of I are weak solutions of problem (2.2).

The main results in this paper are the following.

Theorem 2.1

Assume that hypotheses \(\left( F\right) , \left( A_{1}\right) , \left( H_{1}\right) ,\left( H_{3}\right) \) and \(\left( H_{4}\right) \) hold. Then there exists \(m_{0}>0\) such that if

$$\begin{aligned} \left\| f\right\| _{2}^{2}+\left\| a^{+}+\chi _{B\left( 0,R_{1}\right) }\right\| _{\frac{2^{*}}{2^{*}-q-1}}^{\frac{2}{1-q}}<m_{0}, \end{aligned}$$

then problem (2.2) has two solutions \(U_{1},U_{2}\in E \) with \(I(U_{1})<0\) and \(I(U_{2})>0\). One of these solutions is nonnegative.

Theorem 2.2

Assume that \(\left( F\right) \), \( \left( \overline{A_{1}}\right) \), \(\left( H_{1}\right) \), \(\left( H_{2}\right) \), \(\left( H_{3}\right) \) and \(\left( N^{\pm }\right) \) hold. Then there exists \(m_{1}>0\) such that if

$$\begin{aligned} \left\| f\right\| _{2}^{2}+\left\| a^{+}+\chi _{B\left( 0,R_{0}\right) }\right\| _{\frac{2}{1-q}}^{\frac{2}{1-q}}<m_{1}, \end{aligned}$$

then problem (2.2) admits two solutions \(V_{1},V_{2}\in H^{1}(\mathbb {R}^{N})\) with \(I(V_{1})<0\) and \(I(V_{2})>0\). One of these solutions is nonnegative.

Remark 2.3

In fact, the assumptions

$$\begin{aligned} \left\| f\right\| _{2}^{2}+\left\| a^{+}+\chi _{B\left( 0,R_{1}\right) }\right\| _{\frac{2^{*}}{2^{*}-q-1}}^{\frac{2}{1-q}}<m_{0} \end{aligned}$$

and

$$\begin{aligned} \left\| f\right\| _{2}^{2}+\left\| a^{+}+\chi _{B\left( 0,R_{0}\right) }\right\| _{\frac{2}{1-q}}^{\frac{2}{1-q}}<m_{1} \end{aligned}$$

are necessary only to guarantee that \(I(u)\ge c>0\) on the sphere in Y (see Lemmas 3.3 and 3.4). Thus these hypotheses can be removed if there are other ways to get \(I(u)\ge c>0\) on the sphere in Y.

We divide our paper into four sections. In Sect. 2, we give some notations and preliminary results. In Sects. 3 and 4, we prove Theorems 2.1 and 2.2.

The main difficulties that arise in treating this class of nonhomogeneous Schrödinger equations (2.2) are the following: (i) the lack of compactness due to the unboundedness of the domain; (ii) the sign-changing of potentials a(x) and V(x). To avoid the first difficulty, we employ the Del Pino and Felmer method [23]. To overcome the second difficulty, we control the positive mass in relation to the negative mass of the potentials a(x) and V(x). The key tool for obtaining the multiplicity of solutions is a suitable recurrent variational method.

3 Notations and preliminaries

We will use the following notations:

\(\left\| u\right\| _{m}=\left( \int _{\mathbb {R}^{N}}\left| u\right| ^{m}dx\right) ^{\frac{1}{m}}, \ \ \forall 1\le m < \infty \);

\(B\left( 0,R\right) \) denotes the ball centered in zero of radius \(R>0\) and \(B^{c}\left( 0,R\right) =\mathbb {R}^{N}\backslash B\left( 0,R\right) \);

\(C_{S}\) is the best Sobolev constant:

$$\begin{aligned} \forall u\in H^{1}(\mathbb {R}^{N}), \ \ \left\| u\right\| _{2^{*}}\le C_{S} \left\| \nabla u\right\| _{2}; \end{aligned}$$

\(F'\): the Fréchet derivative of F at u.

Let

$$\begin{aligned} E=H^{1}(\mathbb {R}^{N})\cap L^{q+1}\left( \mathbb {R}^{N}\right) . \end{aligned}$$

If we equip E with the norm

$$\begin{aligned} \left\| u\right\| =\left\| \nabla u\right\| _{2}+\left\| u\right\| _{q+1}, \end{aligned}$$

then E becomes a reflexive Banach space. On the Sobolev space \(H^{1}(\mathbb {R}^{N})\) we consider the usual norm

$$\begin{aligned} \left\| u\right\| _{H}=\left\| u\right\| _{2}+\left\| \nabla u\right\| _{2}. \end{aligned}$$

Define the following energy functional on Y (\(Y=E\) or \(H^{1}(\mathbb {R}^{N})\)):

$$\begin{aligned} I\left( u\right)= & {} \frac{1}{2}\int _{\mathbb {R}^{N}}\left( \left| \nabla u\right| ^{2} + V(x)u^{2}\right) dx- \frac{1}{q+1}\int _{\mathbb {R}^{N}} a(x) \left| u\right| ^{q+1} dx\nonumber \\&\quad -\int _{\mathbb {R}^{N}} f(x) u dx. \end{aligned}$$
(3.1)

Under suitable assumptions on a, V and f (to be fixed later), the functional I is well defined, of class \(C^{1}\) on Y and any critical point of I is a weak solution of problem (2.2).

We recall that a Palais–Smale sequence for the functional I, for short we write (PS)-sequence, is a sequence \(\left( u_{n}\right) _{n}\in Y\) such that

$$\begin{aligned} \left( I(u_{n})\right) _{n} \ \ \text{ is } \text{ bounded } \ \ \text{ and }\ \ \left\| I'(u_{n})\right\| _{Y'}\rightarrow 0. \end{aligned}$$

The functional I is said to satisfy the Palais–Smale condition if any (PS)-sequence has a convergent subsequence in Y.

In the sequel we need the following auxiliary results.

Lemma 3.1

Let x and y be arbitrary real numbers. Then there exists \(c>0\) such that

$$\begin{aligned} \left| \left| x+y\right| ^{q+1}-\left| x\right| ^{q+1}-\left| y\right| ^{q+1}\right| \le c \left| x\right| ^{q}y, \end{aligned}$$
(3.2)

for \(0<q<1\).

Proof

If \(x=0\), the inequality (3.2) is trivial.

Suppose that \(x\ne 0\). Let \(f: \mathbb {R}\backslash \left\{ 0\right\} \mapsto \mathbb {R}\) defined by

$$\begin{aligned} f(t)=\frac{\left| 1+t\right| ^{q+1}-\left| t\right| ^{q+1}-1}{\left| t\right| }. \end{aligned}$$

Then \(\lim _{\left| t\right| \rightarrow +\infty }f(t)=0\) and \(\lim _{t\rightarrow 0^{\pm }}f(t)=\pm \left( q+1\right) \), so there exists constant \(c>0\) such that \(\left| f(t)\right| \le c, \ \ \forall t\in \mathbb {R}\backslash \left\{ 0\right\} \). In particular \(\left| f\left( \frac{y}{x}\right) \right| \le c\), so

$$\begin{aligned} \left| \left| 1+\frac{y}{x}\right| ^{q+1}-\left| \frac{y}{x}\right| ^{q+1}-1\right| \le c \left| \frac{y}{x}\right| . \end{aligned}$$

Multiplying by \(\left| x\right| ^{q+1}\), we obtain the desired result. \(\square \)

Lemma 3.2

Assume that \(\left( \overline{A_{1}}\right) \) holds. If \(u_{n}\rightharpoonup u\) weakly in \(H^{1}(\mathbb {R}^{N})\), then there exists a subsequence of \(\left( u_{n}\right) \in H^{1}(\mathbb {R}^{N})\), also denoted \(\left( u_{n}\right) \), such that

$$\begin{aligned} \lim _{n\rightarrow +\infty } \int _{\mathbb {R}^{N}}a(x)\left| u_{n}-u\right| ^{q+1}dx=0. \end{aligned}$$

Proof

Since \(a\in L^{\frac{2}{1-q}}(\mathbb {R}^{N})\), then for every \(\epsilon >0\) there exists \(R_{2}>0\) such that

$$\begin{aligned} \left( \int _{\left| x\right| >R_{2}}\left| a(x)\right| ^{\frac{2}{1-q}}dx\right) ^{\frac{1-q}{2}}<\epsilon . \end{aligned}$$

Since \(u_{n}\rightharpoonup u\) weakly in \(H^{1}(\mathbb {R}^{N})\), \(u_{n}\rightarrow u\) strongly in \(L_{loc}^{s}(\mathbb {R}^{N})\), \(2 \le s < 2^{*}\), then

$$\begin{aligned} \left( \int _{\left| x\right|<R_{2}}\left| u_{n}-u\right| ^{2}dx\right) ^{\frac{q+1}{2}}<\epsilon . \end{aligned}$$

Observe that by Hölder’s inequality we have

$$\begin{aligned} \int _{\mathbb {R}^{N}}a(x)\left| u_{n}-u\right| ^{q+1}dx= & {} \int _{\left| x\right| \le R_{2}}a(x)\left| u_{n}-u\right| ^{q+1}dx\\&\quad + \int _{\left| x\right| >R_{2}}a(x)\left| u_{n}-u\right| ^{q+1}dx<c \epsilon , \end{aligned}$$

hence \(\lim _{n\rightarrow +\infty } \int _{\mathbb {R}^{N}}a(x)\left| u_{n}-u\right| ^{q+1}dx=0.\)\(\square \)

Lemma 3.3

Assume that hypotheses \(\left( A_{1}\right) , \ \left( H_{3}\right) \) and \(\left( F\right) \) hold. Then there exist \(\rho _{0}, \, \alpha _{0}\), \( m_{0}>0\) such that \(I(u)\ge \alpha _{0}>0\) for all \(u\in E\), \(\left\| u\right\| =\rho _{0}\) and \(\left\| f\right\| _{2}^{2}+\left\| a^{+}+\chi _{B\left( 0,R_{1}\right) }\right\| _{\frac{2^{*}}{2^{*}-q-1}}^{\frac{2}{1-q}}<m_{0}.\)

Proof

Let \(u\in E\). By Hölder and Sobolev inequalities, we have

$$\begin{aligned} \begin{array}{ll} \left\| u\right\| _{2}^{2} &{}\le \left\| u\right\| _{q+1}^{2r} \left\| u\right\| _{2^{*}}^{2\left( 1-r\right) }\le C_{s}^{2\left( 1-r\right) }\left\| u\right\| _{q+1}^{2r}\left\| \nabla u\right\| _{2}^{2\left( 1-r\right) }\\ &{} \le \frac{\min (\alpha , 1)}{\left( q+1\right) \left( \left\| V\right\| _{\infty }+1\right) } \left\| u\right\| _{q+1}^{q+1}+c \left\| \nabla u\right\| _{2}^{p}, \end{array} \end{aligned}$$
(3.3)

with

$$\begin{aligned}r=\frac{\left( 2^{*}-2\right) \left( q+1\right) }{2\left( 2^{*}-q-1\right) }\in ]0,1[ \ \ \text{ and } \ \ p=\frac{2\left( 1-r\right) \left( q+1\right) }{q+1-2r}>2. \end{aligned}$$

Then from \(\left( F\right) \), \(\left( H_{3}\right) \) and \(\left( A_{1}\right) \) we infer that

$$\begin{aligned} I(u)&=\frac{1}{2}\int _{\mathbb {R}^{N}}\left( \left| \nabla u\right| ^{2}+V(x)u^{2}\right) dx-\frac{1}{q+1} \int _{\mathbb {R}^{N}}\left( a^{+}+\chi _{B\left( 0, R_{1}\right) }\right) (x) \left| u\right| ^{q+1} dx \\&\quad + \frac{1}{q+1} \int _{\mathbb {R}^{N}} \left( a^{-}+\chi _{B\left( 0, R_{1}\right) }\right) (x) \left| u\right| ^{q+1} dx- \int _{\mathbb {R}^{N}}f(x) u dx\\&\ge \frac{1}{2}\left( \left\| \nabla u\right\| _{2}^{2}-\left( \left\| V\right\| _{\infty }+1\right) \left\| u\right\| _{2}^{2}\right) +\frac{\min (\alpha , 1)}{q+1} \left\| u\right\| _{q+1}^{q+1}\\&\quad -\frac{1}{q+1}\left\| a^{+}+\chi _{B\left( 0, R_{1}\right) }\right\| _{\frac{2^{*}}{2^{*}-q-1}} \left\| u\right\| _{2^{*}}^{q+1}-\frac{1}{2}\left\| f\right\| _{2}^{2} \\&\ge \frac{1}{2}\left\| \nabla u\right\| _{2}^{2}\!+\!\frac{\min (\alpha , 1)}{2\left( q\!+\!1\right) }\left\| u\right\| _{q+1}^{q+1}\!-\!c \left\| \nabla u\right\| _{2}^{p}-\frac{C_{s}^{q+1}}{q+1} \left\| a^{+}+\chi _{B\left( 0, R_{1}\right) }\right\| _{\frac{2^{*}}{2^{*}-q-1}}\left\| \nabla u\right\| _{2}^{q+1} \\&\quad - \frac{1}{2}\left\| f\right\| _{2}^{2}\\&\ge \frac{1}{4} \left\| \nabla u\right\| _{2}^{2}+\frac{\min (\alpha , 1)}{2\left( q+1\right) }\left\| u\right\| _{q+1}^{q+1}- c \left\| \nabla u\right\| _{2}^{p}\\&\quad -\left( 2\left( q+1\right) \right) ^{\frac{q+1}{1-q}}\left( \frac{C_{s}^{q+1}}{q+1}\right) ^{\frac{2}{1-q}}\left( \frac{1-q}{2}\right) \left\| a^{+}+\chi _{B\left( 0, R_{1}\right) }\right\| _{\frac{2^{*}}{2^{*}-q-1}}^{\frac{2}{1-q}}-\frac{1}{2}\left\| f\right\| _{2}^{2}. \end{aligned}$$

Since \(p>2\), for \(\left\| \nabla u\right\| _{2} \le s \) (small enough) we have

$$\begin{aligned} \frac{1}{4}\left\| \nabla u\right\| _{2}^{2}-c \left\| \nabla u\right\| _{2}^{p}\ge \frac{1}{8} \left\| \nabla u\right\| _{2}^{2}. \end{aligned}$$

Next, using the inequality

$$\begin{aligned} \frac{1}{2} \left( x+y\right) ^{2}\le x^{2}+y^{q+1} \ \ \ \forall x\ge 0, \ \ 0\le y \le 1, \end{aligned}$$

we obtain that for \(\left\| \nabla u\right\| _{2}\le s\) and \(\left\| u\right\| _{q+1}\le 1\), we have

$$\begin{aligned} \begin{array}{ll} I(u) &{} \ge \frac{\left\| \nabla u\right\| _{2}^{2}}{8}+ \frac{\min (\alpha , 1)}{2\left( q+1\right) } \left\| u\right\| _{q+1}^{q+1}- \frac{1}{2} \left\| f\right\| _{2}^{2}\\ &{}\quad - \left( 2\left( q+1\right) \right) ^{\frac{q+1}{1-q}}\left( \frac{C_{s}^{q+1}}{q+1}\right) ^{\frac{2}{1-q}}\left( \frac{1-q}{2}\right) \left\| a^{+}+\chi _{B\left( 0, R_{1}\right) }\right\| _{\frac{2^{*}}{2^{*}-q-1}}^{\frac{2}{1-q}}\\ &{}\ge c_{0}\left\| u\right\| ^{2}-\frac{1}{2}\left\| f\right\| _{2}^{2}\\ &{} \quad -\left( 2\left( q+1\right) \right) ^{\frac{q+1}{1-q}}\left( \frac{C_{s}^{q+1}}{q+1}\right) ^{\frac{2}{1-q}}\left( \frac{1-q}{2}\right) \left\| a^{+}+\chi _{B\left( 0, R_{1}\right) }\right\| _{\frac{2^{*}}{2^{*}-q-1}}^{\frac{2}{1-q}}, \end{array} \end{aligned}$$
(3.4)

where \(c_{0}=\min \left( \frac{1}{16}, \frac{\min \left( \alpha ,1\right) }{4\left( q+1\right) }\right) \).

Set \(\rho _{0}=\min \left( s,1\right) \). Then by (3.4) and for

$$\begin{aligned} \left\| f\right\| _{2}^{2}+\left\| a^{+}+\chi _{B\left( 0,R_{1}\right) }\right\| _{\frac{2^{*}}{2^{*}-q-1}}^{\frac{2}{1-q}}< \frac{c_{0}}{\max \left( 1, \left( 1-q\right) \left( 2\left( q+1\right) \right) ^{\frac{q+1}{1-q}}\left( \frac{C_{s}^{q+1}}{q+1}\right) ^{\frac{2}{1-q}}\right) }\rho _{0}^{2}=m_{0}, \end{aligned}$$

we infer that

$$\begin{aligned} I(u)\ge c_{0}\frac{\rho _{0}^{2}}{2}=\alpha _{0}, \ \ \text{ for } \ \ \left\| u\right\| =\rho _{0}. \end{aligned}$$

The proof is completed. \(\square \)

Lemma 3.4

Suppose that hypotheses \(\left( F\right) \), \(\left( \overline{A_{1}}\right) \) and \( \left( H_{3}\right) \) are satisfied. Then there exist \(\rho _{1}, \ \ \alpha _{1}, \ \ m_{1}>0\) such that \(I(u)\ge \alpha _{1}>0\) for all \(u\in H^{1}(\mathbb {R}^{N})\), \(\left\| u\right\| _{H}=\rho _{1}\) and

$$\begin{aligned} \left\| f\right\| _{\frac{2N}{N+2}}^{2}+\left\| a^{+}+\chi _{B\left( 0,R_{0}\right) }\right\| _{\frac{2}{1-q}}^{\frac{2}{1-q}}<m_{1}. \end{aligned}$$

Proof

Combining hypothesis \(\left( H_{3}\right) \) with Hölder and Sobolev inequalities, we obtain

$$\begin{aligned} \begin{array}{ll} &{} \int _{\mathbb {R}^{N}} V^{-}(x)u^{2}dx \le \left\| V\right\| _{\infty } \int _{\mathbb {R}^{N}} \left( \left( a^{-}+\chi _{B\left( 0,R_{0}\right) }\right) (x)\right) ^{\frac{2r}{q+1}}u^{2} dx\\ &{} \quad \le \left\| V\right\| _{\infty } \left( \int _{\mathbb {R}^{N}} \left( a^{-}+\chi _{B\left( 0,R_{0}\right) }\right) (x) \left| u\right| ^{q+1}dx\right) ^{\frac{2r}{q+1}} \left\| u\right\| _{2^{*}}^{2\left( 1-r\right) }\\ &{} \quad \le \left\| V\right\| _{\infty } C_{s}^{2\left( 1-r\right) } \left( \int _{\mathbb {R}^{N}} \left( a^{-}+\chi _{B\left( 0,R_{0}\right) }\right) (x)\left| u\right| ^{q+1}dx\right) ^{\frac{2r}{q+1}} \left\| \nabla u\right\| _{2}^{2\left( 1-r\right) }\\ &{} \quad \le \frac{1}{ \left( q+1\right) } \int _{\mathbb {R}^{N}} \left( a^{-}+\chi _{B\left( 0,R_{0}\right) }\right) (x)\left| u\right| ^{q+1}dx+c\left\| \nabla u \right\| _{2}^{p} \end{array} \end{aligned}$$
(3.5)

and

$$\begin{aligned} \begin{array}{ll} &{}\int _{\mathbb {R}^{N}}\left( \left| \nabla u\right| ^{2}+u^{2}\right) dx= \int _{\mathbb {R}^{N}}\left| \nabla u\right| ^{2}+ \int _{B\left( 0,R_{0}\right) }u^{2}dx+ \int _{B^{c}\left( 0, R_{0}\right) }u^{2}dx\\ &{} \quad \le \int _{\mathbb {R}^{N}}\left| \nabla u\right| ^{2}dx+C_{s}^{2}\left( \text{ meas }\,\left( B\left( 0,R_{0}\right) \right) \right) ^{\frac{2}{N}} \int _{\mathbb {R}^{N}}\left| \nabla u\right| ^{2}dx\\ &{} \qquad +\frac{1}{\beta } \int _{B^{c}\left( 0,R_{0}\right) }V^{+}(x)u^{2}dx \\ &{} \quad \le \max \left\{ \max \left( 1,C_{s}^{2}\left( \text{ meas }\,\left( B\left( 0,R_{0}\right) \right) \right) ^{\frac{2}{N}}\right) , \frac{1}{\beta }\right\} \\ &{} \qquad \times \left( \int _{\mathbb {R}^{N}}\left| \nabla u\right| ^{2}+ \int _{B^{c}\left( 0,R_{0}\right) }V^{+}(x)u^{2}dx\right) . \end{array} \end{aligned}$$
(3.6)

Using (3.5), we have

$$\begin{aligned} I\left( u\right)&= \frac{1}{2} \int _{\mathbb {R}^{N}}\left( \left| \nabla u\right| ^{2}+V^{+}(x) u^{2}\right) dx -\frac{1}{2} \int _{\mathbb {R}^{N}}V^{-}(x)u^{2}dx\\&\quad +\frac{1}{q+1} \int _{\mathbb {R}^{N}} \left( a^{-}+\chi _{B\left( 0,R_{0}\right) }\right) (x)\left| u\right| ^{q+1}dx\\&\quad -\frac{1}{q+1} \int _{\mathbb {R}^{N}} \left( a^{+}+\chi _{B\left( 0,R_{0}\right) }\right) (x)\left| u\right| ^{q+1} dx- \int _{\mathbb {R}^{N}} f(x)u dx\\&\ge \frac{1}{4} \left\| \nabla u\right\| _{2}^{2}-c \left\| \nabla u\right\| _{2}^{p}+\frac{1}{2} \int _{B^{c}_{\left( 0,R_{0}\right) }}V^{+}(x)u^{2}dx\\&\quad +\frac{1}{2\left( q+1\right) } \int _{\mathbb {R}^{N}}\left( a^{-}+\chi _{B\left( 0,R_{0}\right) }\right) (x)\left| u\right| ^{q+1} dx\\&\quad -\left\| a^{+}+\chi _{B\left( 0,R_{0}\right) }\right\| _{\frac{2}{1-q}} \left\| u\right\| _{H}^{q+1}-C_{s}^{2}\left\| f\right\| _{\frac{2N}{N+2}}^{2}\\&\ge \frac{1}{4} \left\| \nabla u\right\| _{2}^{2}-c \left\| \nabla u\right\| _{2}^{p}+\frac{1}{2} \int _{B^{c}\left( 0,R_{0}\right) }V^{+}(x)u^{2}dx\\&\quad -\left\| a^{+}+\chi _{B\left( 0,R_{0}\right) }\right\| _{\frac{2}{1-q}} \left\| u\right\| _{H}^{q+1}-C_{s}^{2}\left\| f\right\| _{\frac{2N}{N+2}}^{2}. \end{aligned}$$

Using (3.6) and for \(\left\| \nabla u\right\| _{2}\le \rho _{1}\) (\(\rho _{1}\) small enough), we infer that

$$\begin{aligned}\begin{array}{ll} I(u) &{}\ge \frac{1}{8}\left( \int _{\mathbb {R}^{N}}\nabla u^{2}dx+ \int _{B^{c}\left( 0,R_{0}\right) }V^{+}(x)u^{2}dx\right) \\ &{} \quad -\left\| a^{+}+\chi _{B\left( 0,R_{0}\right) }\right\| _{\frac{2}{1-q}} \left\| u\right\| _{H}^{q+1}-C_{s}^{2}\left\| f\right\| _{\frac{2N}{N+2}}^{2}\\ &{}\ge c_{1}\left\| u\right\| _{H}^{2}-\frac{1-q}{2}\left( \frac{2}{c_{1}}\right) ^{\frac{q+1}{1-q}}\left( \frac{q+1}{2}\right) ^{\frac{1+q}{1-q}}\left\| a^{+}+\chi _{B\left( 0,R_{0}\right) }\right\| _{\frac{2}{1-q}}^{\frac{2}{1-q}}-C_{s}^{2}\left\| f\right\| _{\frac{2N}{N+2}}^{2}, \end{array} \end{aligned}$$

where

$$\begin{aligned} c_{1}=\frac{1}{8\max \left( \max \left( 1,C_{s}^{2}\left( \text{ meas }\,\left( B\left( 0,R_{0}\right) \right) \right) ^{\frac{2}{N}}\right) , \frac{1}{\beta }\right) }. \end{aligned}$$

Set

$$\begin{aligned} m_{1}=\frac{c_{1}}{2\max \left( 2C_{s}^{2}, \left( 1-q\right) \left( \frac{2}{c_{1}}\right) ^{\frac{1+q}{1-q}}\left( \frac{q+1}{2}\right) ^{\frac{1+q}{1-q}}\right) }. \end{aligned}$$

If \(\left\| f\right\| _{\frac{2N}{N+2}}^{2}+\left\| a^{+}+\chi _{B(0,R_{0})}\right\| _{\frac{2}{1-q}}^{\frac{2}{1-q}}<m_{1}\), we have

$$\begin{aligned} I(u)\ge \frac{c_{1}}{2}\rho _{1}^{2}=\alpha _{1}, \ \ \text{ for } \ \ \left\| u\right\| _{H}=\rho _{1}. \end{aligned}$$

This completes the proof. \(\square \)

4 Proof of Theorem 2.1

We split the proof into several steps. We first establish the existence of first solution with negative energy. Next, we show that problem (2.2) has a weak solution with positive energy.

4.1 Existence of a nonnegative solution

Consider the minimization problem

$$\begin{aligned} M_{1}= \inf _{u\in B\left( 0,\rho _{0}\right) }I(u), \end{aligned}$$
(4.1)

where \(\rho _{0}\) is defined in Lemma 3.3.

Lemma 4.1

Assume that hypotheses \(\left( A_{1}\right) , \left( H_{3}\right) \) and \(\left( F\right) \) hold. Then \(-\infty<M_{1}<0.\)

Proof

By Lemma 3.3 we have \(M_{1}>-\infty \).

Using hypothesis \(\left( F\right) \), there is a function \(\psi \in E\) such that \( \int _{\mathbb {R}^{N}}f(x)\psi dx>0\). For \(t>0\), we have

$$\begin{aligned} \begin{array}{ll} I(t\psi )&{}= \frac{t^{2}}{2} \int _{\mathbb {R}^{N}}\left( \left| \nabla \psi \right| ^{2}+V(x)\psi ^{2}\right) dx\\ &{} \quad -\frac{t^{q+1}}{q+1} \int _{\mathbb {R}^{N}} a(x)\left| \psi \right| ^{q+1}dx-t \int _{\mathbb {R}^{N}} f(x) \psi dx\\ &{} <0 \ \ \text{ for } \ \ t>0 \ \ \text{ small } \text{ enough }. \end{array} \end{aligned}$$
(4.2)

Hence \(M_{1}<0.\) Thus we conclude the proof. \(\square \)

Theorem 4.2

Assume that hypotheses \(\left( F\right) , \left( A_{1}\right) \) and \(\left( H_{3}\right) \) hold. Then there exists a weak nonnegative solution \(U_{1}\in E\) of problem (2.2), \(I(U_{1})=M_{1}\).

Proof

Let \(\left( u_{n}\right) _{n}\) be a minimizing sequence of problem (4.1). Since \(\left( u_{n}\right) \in B{\left( 0,\rho _{0}\right) }\), we can extract a subsequence, also denoted by \(\left( u_{n}\right) \), such that \(u_{n}\rightharpoonup U_{1} \ \ \text{ in } \ \ E\), \(u_{n}\rightarrow U_{1}\) in \(L^{m}_{loc}(\mathbb {R}^{N})\) for all \( 1 \le m < 2^{*}\) and \(u_{n}\rightarrow U_{1}\) a.e. in \(\mathbb {R}^{N}\).

Setting \(v_{n}=u_{n}-U_{1}\rightharpoonup 0\) in E, we have

$$\begin{aligned} \begin{array}{ll} I(u_{n})&{}= \frac{1}{2}\int _{\mathbb {R}^{N}}\left( \left| \nabla v_{n}\right| ^{2}+V(x) v_{n}^{2}\right) dx+ \frac{1}{2} \int _{\mathbb {R}^{N}}\left( \left| \nabla U_{1}\right| ^{2}+V(x) U_{1}^{2}\right) dx \\ &{}\quad +\int _{\mathbb {R}^{N}}\left( \nabla v_{n}\nabla U_{1}+ V(x) v_{n}U_{1}\right) dx\\ &{}\quad - \frac{1}{q+1}\int _{\mathbb {R}^{N}}a(x)\left( \left| u_{n}\right| ^{q+1}-\left| U_{1}\right| ^{q+1}\right) dx-\frac{1}{q+1}\int _{\mathbb {R}^{N}}a(x)\left| U_{1}\right| ^{q+1} dx\\ &{}\quad -\int _{\mathbb {R}^{N}} f(x)v_{n} dx-\int _{\mathbb {R}^{N}} f(x) U_{1} dx. \end{array} \end{aligned}$$
(4.3)

Since \(v_{n}\rightharpoonup 0\) in E, we obtain

$$\begin{aligned} \int _{\mathbb {R}^{N}}\left( \nabla v_{n}\nabla U_{1}+ V(x) v_{n} U_{1} \right) dx \rightarrow 0 \ \ \text{ and } \ \ \int _{\mathbb {R}^{N}} f(x)v_{n}dx\rightarrow 0. \end{aligned}$$
(4.4)

By Lemma 3.1 we have

$$\begin{aligned} \int _{\mathbb {R}^{N}} \left| a(x)\right| \left| \left| u_{n}\right| ^{q+1}-\left( \left| U_{1}\right| ^{q+1}+\left| v_{n}\right| ^{q+1}\right) \right| dx \le c \int _{\mathbb {R}^{N}} \left| a(x)\right| \left| U_{1}\right| ^{q} \left| v_{n}\right| \rightarrow 0. \end{aligned}$$
(4.5)

From \(\left( H_{3}\right) \) and \(\left( A_{1}\right) \), we deduce that

$$\begin{aligned} \int _{\mathbb {R}^{N}}V^{-}(x)v_{n}^{2} dx\rightarrow 0 \ \ \text{ and } \ \ \int _{\mathbb {R}^{N}}\left( a^{+}(x)+\chi _{B(0, R_{1})}\right) \left| v_{n}\right| ^{q+1} dx\rightarrow 0. \end{aligned}$$
(4.6)

Using relations (4.4), (4.5), (4.6) and passing to the limit in (4.3), we get

$$\begin{aligned} M_{1}&=I(U_{1})\!+ \!\lim _{n\rightarrow \!+\!\infty }\left( \frac{1}{2}\int _{\mathbb {R}^{N}}\left| \nabla v_{n}\right| ^{2}+V^{+}(x) v_{n}^{2}+\frac{1}{q+1}\left( a^{-}(x)+\chi _{B\left( 0, R_{1}\right) }\right) \left| v_{n}\right| ^{q+1}\right) dx\\&\ge I(U_{1})+\lim _{n\rightarrow +\infty }\left( \frac{1}{2}\left\| \nabla v_{n}\right\| _{2}^{2}+\frac{\min \left( \alpha , 1\right) }{q+1}\left\| v_{n}\right\| _{q+1}^{q+1}\right) \ge M_{1}. \end{aligned}$$

It follows that \(v_{n}\rightarrow 0\) in E, \(M_{1}=I(U_{1})<0\), \(U_{1}\in B\left( 0,\rho _{0}\right) \), \(I'(U_{1})=0\). Thus \(U_{1}\) is a weak solution of problem (2.2). Since \(M_{1} \le I(\left| u_{n}\right| )\le I(u_{n})\), then \(\left( \left| u_{n}\right| \right) \) is also a minimizing sequence of problem (4.1). We deduce that we can suppose that \(U_{1}\ge 0\) a.e. in \(\mathbb {R}^{N}\). \(\square \)

4.2 Existence of a second solution with positive energy solution

We start this subsection by showing that the functional I satisfies the Palais–Smale condition. For this purpose, we need the following auxiliary properties.

Lemma 4.3

Assume that hypotheses \(\left( A_{1}\right) \), \(\left( H_{3}\right) \), \( \left( H_{4}\right) \) and \(\left( F\right) \) hold. Then any (PS)-sequence of I is bounded in E.

Proof

Let \(\left( u_{n}\right) \in E\) be a (PS)-sequence of I. We argue by contradiction, assuming that \(\left\| u_{n}\right\| =t_{n}\rightarrow +\infty \). Re-normalizing, we set \(v_{n}=\frac{u_{n}}{t_{n}}\). Thus up to a subsequence, \(v_{n}\rightharpoonup v\) in E.

We claim that \(v=0\). For this purpose, we take \(\varphi \in C_{c}^{\infty }(\mathbb {R}^{N})\). Since \(\left( u_{n}\right) \) is a \(\left( PS\right) \)-sequence of I, we have

$$\begin{aligned} \begin{array}{ll} \langle I'(u_{n}), \varphi \rangle &{} =\int _{\mathbb {R}^{N}}\left( \nabla u_{n}\nabla \varphi + V(x)u_{n}\varphi \right) dx-\int _{\mathbb {R}^{N}} a(x)\left| u_{n}\right| ^{q-1}u_{n} \varphi dx\\ &{} \quad -\int _{\mathbb {R}^{N}} f(x)\varphi dx=o(1)\left\| \varphi \right\| .\end{array} \end{aligned}$$
(4.7)

Dividing relation (4.7) by \(t_{n}\), we obtain

$$\begin{aligned} \int _{\mathbb {R}^{N}}\left( \nabla v_{n} \nabla \varphi + V(x) v_{n}\varphi \right) dx-\int _{\mathbb {R}^{N}}a(x) \frac{\left| u_{n}\right| ^{q-1} u_{n}}{t_{n}} \varphi dx-\int _{\mathbb {R}^{N}}\frac{f(x)\varphi }{t_{n}}dx=o(1).\nonumber \\ \end{aligned}$$
(4.8)

Using \(\left( A_{1}\right) \) and Hölder’s inequality, we deduce that

$$\begin{aligned} \int _{\mathbb {R}^{N}} a(x) \frac{\left| u_{n}\right| ^{q-1}u_{n}}{t_{n}} \varphi dx \le \left\| a\right\| _{\infty } \frac{\left\| u_{n}\right\| _{2^{*}}^{q}}{t_{n}} \left\| \varphi \right\| _{\frac{2^{*}}{2^{*}-q}}\le c \left\| a\right\| _{\infty } \frac{\left\| u_{n}\right\| ^{q}}{t_{n}} \left\| \varphi \right\| _{\frac{2^{*}}{2^{*}-q}}. \end{aligned}$$

Thus since \(0<q<1\), we have

$$\begin{aligned} \int _{\mathbb {R}^{N}} a(x) \frac{\left| u_{n}\right| ^{q-1}u_{n}}{t_{n}} \varphi dx\rightarrow 0. \end{aligned}$$
(4.9)

As a consequence of hypothesis \(\left( F\right) \), we have

$$\begin{aligned} \int _{\mathbb {R}^{N}}\frac{f(x) \varphi }{t_{n}} dx \rightarrow 0 \ \ \text{ as } \ \ n \rightarrow +\infty . \end{aligned}$$
(4.10)

From (4.9), (4.10) and passing to the limit in relation (4.8), we infer that

$$\begin{aligned} \int _{\mathbb {R}^{N}}\left( \nabla v\nabla \varphi +V(x)v \varphi \right) dx=0. \end{aligned}$$

Using now \(\left( H_{4}\right) \), we conclude that \(v=0\) a.e. in \(\mathbb {R}^{N}\). This proves our claim.

Substituting \(\varphi =v_{n}\) in relation (4.8), we obtain

$$\begin{aligned}&\int _{\mathbb {R}^{N}}\left| \nabla v_{n}\right| ^{2}dx+\int _{\mathbb {R}^{N}}\left( V^{+}+\chi _{B\left( 0,R_{0}\right) }\right) (x)v_{n}^{2} dx -\int _{\mathbb {R}^{N}} \left( V^{-}(x)+\chi _{B\left( 0,R_{0}\right) }(x)\right) v_{n}^{2} dx\\&\quad -\int _{\mathbb {R}^{N}} \frac{a^{+}(x)\left| u_{n}\right| ^{q+1}}{t_{n}^{2}}dx+\int _{\mathbb {R}^{N}} \frac{a^{-}(x)\left| u_{n}\right| ^{q+1}}{t_{n}^{2}}dx -\int _{\mathbb {R}^{N}}\frac{f(x)v_{n}}{t_{n}}=o(1). \end{aligned}$$

It follows that

$$\begin{aligned} \begin{array}{ll} &{} \left\| \nabla v_{n}\right\| _{2}^{2}-\int _{\mathbb {R}^{N}} \left( V^{-}(x)+\chi _{B\left( 0,R_{0}\right) }(x)\right) v_{n}^{2} dx\\ &{}\quad -\int _{\mathbb {R}^{N}} \frac{a^{+}(x)\left| u_{n}\right| ^{q+1}}{t_{n}^{2}}dx-\int _{\mathbb {R}^{N}}\frac{f(x)v_{n}}{t_{n}}\le o(1). \end{array} \end{aligned}$$
(4.11)

Using now \(\left( A_{1}\right) \), \(\left( H_{3}\right) \) and \(\left( F\right) \), we obtain

$$\begin{aligned} \begin{array}{ll} &{}\int _{\mathbb {R}^{N}} \frac{a^{+}(x)\left| u_{n}\right| ^{q+1}}{t_{n}^{2}}dx \rightarrow 0\\ &{}\int _{\mathbb {R}^{N}}\left( V^{-}+\chi _{B\left( 0, R_{0}\right) }\right) (x) v_{n}^{2} dx \rightarrow 0 \\ &{}\int _{\mathbb {R}^{N}} \frac{f(x)v_{n}}{t_{n}}\rightarrow 0.\end{array} \end{aligned}$$
(4.12)

Combining (4.11) and (4.12), we conclude that

$$\begin{aligned} \lim _{n\rightarrow +\infty }\left\| \nabla v_{n}\right\| _{2}=0. \end{aligned}$$
(4.13)

On the other hand, since \((u_{n})\) is a (PS)-sequence of I, we have

$$\begin{aligned} \frac{ \int _{\mathbb {R}^{N}}\left( \left| \nabla u_{n}\right| ^{2}+V(x)u_{n}^{2}\right) dx}{t_{n}^{q+1}}=\frac{ \int _{\mathbb {R}^{N}}a(x)\left| u_{n}\right| ^{q+1}dx}{t_{n}^{q+1}}+\frac{ \int _{\mathbb {R}^{N}}f(x)u_{n}}{t_{n}^{q+1}}+o(1). \end{aligned}$$

Therefore

$$\begin{aligned} \begin{array}{ll} o(1)=\frac{I(u_{n})}{t_{n}^{q+1}}&{}=\left( \frac{1}{2}-\frac{1}{q+1}\right) \int _{\mathbb {R}^{N}}a(x)\left| v_{n}\right| ^{q+1}dx-\frac{1}{2t_{n}^{q}} \int _{\mathbb {R}^{N}}f(x)v_{n}\\ &{}=\left( \frac{1}{q+1}-\frac{1}{2}\right) \int _{\mathbb {R}^{N}}\left( a^{-}+\chi _{B\left( 0,R_{1}\right) }\right) (x)\left| v_{n}\right| ^{q+1}dx\\ &{}\quad -\left( \frac{1}{q+1}-\frac{1}{2}\right) \int _{\mathbb {R}^{N}}\left( a^{+}+\chi _{B\left( 0,R_{1}\right) }\right) (x)\left| v_{n}\right| ^{q+1}dx\\ &{} \quad -\frac{1}{2t_{n}^{q}} \int _{\mathbb {R}^{N}}f(x)v_{n}. \end{array} \end{aligned}$$
(4.14)

From \(\left( A_{1}\right) \), \(\left( F\right) \) and (4.14), we infer that

$$\begin{aligned} \lim _{n\rightarrow +\infty }\left\| v_{n}\right\| _{q+1}=0. \end{aligned}$$
(4.15)

Combining (4.13) and (4.15), we conclude that \(v_{n}\rightarrow 0\) in E as \(n\rightarrow +\infty \), which contradicts \(\left\| v_{n}\right\| =1\). The proof is completed. \(\square \)

Lemma 4.4

Assume that hypotheses \(\left( A_{1}\right) \), \(\left( H_{1}\right) \), \(\left( H_{3}\right) \), \(\left( H_{4}\right) \) and \(\left( F\right) \) hold. Then I satisfies the (PS) condition.

Proof

Let \(\left( u_{n}\right) \in E\) be a \(\left( PS\right) \) sequence such that

$$\begin{aligned} I(u_{n})\rightarrow M_{2} \ \ \ \text{ and } \ \ \ I'(u_{n})\rightarrow 0 \ \ \ \text{ in } \ \ E'. \end{aligned}$$

Using Lemma 4.3, \(\left( u_{n}\right) \) is bounded in E. Then up to a subsequence, \(u_{n}\rightharpoonup U_{2}\) in E, \(u_{n}\rightarrow U_{2}\) in \(L^{m}_{loc}(\mathbb {R}^{N})\) for all \(1\le m < 2^{*}\) and \(u_{n}\rightarrow U_{2}\) a.e. in \(\mathbb {R}^{N}\). According to [23], it is sufficient to prove that for any \(\epsilon >0\), there exist \(R_{2}>0\) and \(n_{0}\in \mathbb {N}^{*}\) such that

$$\begin{aligned} \int _{\left| x\right| \ge R_{2}}\left( \left| \nabla u_{n}\right| ^{2}+\left| u_{n}\right| ^{q+1}\right) dx \le \epsilon , \ \ \ \text{ for } \text{ all } \ \ R\ge R_{2} \ \ \text{ and } \ \ n\ge n_{0}. \end{aligned}$$

Let \(R\ge 1\) and \(\phi _{R}\) be a smooth cut-off function such that \(\phi _{R}=0\) on \(B\left( 0, \frac{R}{2}\right) \), \(\phi _{R}=1\) on \(B^{c}\left( 0,R\right) \), \(0\le \phi _{R} \le 1\) and there exists \(c>0\) such that

$$\begin{aligned} \left| \nabla \phi _{R}\right| \le \frac{c}{R}, \ \ \text{ for } \text{ all } \ \ x\in \mathbb {R}^{N}. \end{aligned}$$
(4.16)

We remark that for any \(u\in E\),

$$\begin{aligned} \left\| \phi _{R} u\right\| \le c \left\| u\right\| . \end{aligned}$$
(4.17)

Since \(I'(u_{n})\rightarrow 0\) in \(E'\) then for any \(\epsilon >0\) there exists \(n_{0}>0\) such that

$$\begin{aligned} \left| \left\langle I'(u_{n}), \phi _{R}u_{n}\right\rangle \right| \le c \left\| I'(u_{n})\right\| _{E'} \left\| u_{n}\right\| \le \frac{\epsilon }{3}, \ \ \forall n\ge n_{0}. \end{aligned}$$

Thus for \(n\ge n_{0}\),

$$\begin{aligned} \begin{array}{ll} &{}\int _{\mathbb {R}^{N}}\left( \left| \nabla u_{n}\right| ^{2}+\left( a^{-}+\chi _{B\left( 0, R_{1}\right) }\right) (x)\left| u_{n}\right| ^{q+1}\right) \phi _{R}dx \\ &{} \quad \le \int _{\mathbb {R}^{N}} f(x)u_{n}\phi _{R}dx+\int _{\mathbb {R}^{N}} V^{-}(x)u_{n}^{2}\phi _{R}dx \\ &{} \qquad -\int _{\mathbb {R}^{N}}u_{n} \nabla u_{n} \nabla \phi _{R}dx \\ &{} \qquad + \int _{\mathbb {R}^{N}}\left( a^{+}(x)+\chi _{B\left( 0,R_{1}\right) }\right) \left| u_{n}\right| ^{q+1}\phi _{R} dx+\frac{\epsilon }{3}. \end{array} \end{aligned}$$
(4.18)

By the Hölder inequality and (4.16), there exists \(R_{3}>0\) such that

$$\begin{aligned} \int _{\mathbb {R}^{N}} u_{n}\nabla u_{n}\nabla \phi _{R}dx \le \frac{c}{R}< \frac{\epsilon }{3}, \ \ \text{ for } \text{ all } \ \ R\ge R_{3}. \end{aligned}$$
(4.19)

From \(\left( F\right) \), \(\left( H_{3}\right) \) and \(\left( A_{1}\right) \), there exists \(R_{4}>0\) such that

$$\begin{aligned} \begin{array}{ll} &{}\int _{\mathbb {R}^{N}} f(x)u_{n}\phi _{R} dx +\int _{\mathbb {R}^{N}} V^{-}(x)u_{n}^{2} \phi _{R} dx +\int _{\mathbb {R}^{N}}\left( a^{+}+\chi _{B\left( 0,R_{1}\right) }\right) (x)\left| u_{n}\right| ^{q+1} dx\\ &{}\quad \le c\left( \left\| f \phi _{R}\right\| _{2}+\left\| V^{-} \phi _{R}\right\| _{\frac{N}{2}}+\left\| \left( a^{+} +\chi _{B\left( 0, R\right) }\right) \phi _{R}\right\| _{\frac{2^{*}}{2^{*}-q-1}}\right) \\ &{}\quad \le \frac{\epsilon }{3} \ \ \text{ for } \ \ R\ge R_{4}. \end{array} \end{aligned}$$
(4.20)

Set \(R_{2}=\max \left( R_{3},R_{4}\right) \). Then by (4.18), (4.19) and (4.20) we deduce that

$$\begin{aligned} \min \left( 1,\min \left( \alpha ,1\right) \right) \int _{\mathbb {R}^{N}}\left( \left| \nabla u_{n}\right| ^{2}+\left| u_{n}\right| ^{q+1}\right) \phi _{R} dx\le \epsilon , \ \ \forall n\ge n_{0} \ \ \text{ and } \ \ \forall R\ge R_{2}. \end{aligned}$$

This ends the proof. \(\square \)

By (4.2) and \(\left( H_{1}\right) \), there exists \(\varphi _{0}=t\phi \in B^{c}\left( 0,\rho _{0}\right) \), with t large enough, such that \(I(\varphi _{0})<0\) (\(\rho _{0}\) is fixed by Lemma 3.3).

Consider the problem

$$\begin{aligned} M_{0}= \inf _{\gamma \in \Gamma } \max _{t\in \left[ 0,1\right] }I(\gamma (t)), \end{aligned}$$

where

$$\begin{aligned} \Gamma =\left\{ \gamma \in C\left( \left[ 0,1\right] ,E\right) , \ \gamma (0)=0 \ \ \text{ and } \ \ \gamma (1)=\varphi _{0}\right\} . \end{aligned}$$

We observe that \(M_{0}>0\). Indeed for \(\gamma \in \Gamma \), we have \(\gamma (0)=0\) and \(\gamma (1)\notin B\left( 0,\rho _{0}\right) \). Then there exists \(r_{0}\in \left[ 0,1\right] \) such that \(\left\| \gamma (r_{0})\right\| =\rho _{0}\). By Lemma 3.3 we have \(I(\gamma (r_{0}))>\alpha _{0}\) and so \(M_{0}>0\). Then, by Mountain-Pass theorem [5] and Lemma 4.4, we prove the existence of \(U_{0}\in E\) such that \(I(U_{0})=M_{0}>0\). \(U_{0}\) is a weak solution of problem (2.2).

5 Proof of Theorem 2.2

We first observe that \(\left( H_{2}\right) \) and \(\left( H_{3}\right) \) imply that the Schrödinger operator \(L=-\Delta +V(x)\) is defined on \(H^{2}(\mathbb {R}^{N})\) and 0 is an isolated eigenvalue of finite multiplicity.

Let \(H^{-}, \ H^{0}\) and \(H^{+}\) denote the negative, null and positive spaces of the quadratic form associated to the operator L. More precisely,

$$\begin{aligned} H^{-}= \oplus _{\lambda _{i}<0}\text{ Ker }\left( L-\lambda _{i}I\right) , \ \ H^{0}=\text{ Ker }\left( L\right) , \ \ H^{+}= \oplus _{\lambda _{i}>0}\text{ Ker }\left( L-\lambda _{i}I\right) . \end{aligned}$$

We have the orthogonal decomposition \(H^{1}(\mathbb {R}^{N})=H^{-}\oplus H^{0}\oplus H^{+}.\)

For \(u\in E\), we denote by \(u^{\mp }, \ u^{0}\) the orthogonal projections of u on \(H^{\mp }, \ H^{0}\) respectively. Then \(u=u^{-}+u^{0}+u^{+}\). Moreover, there is an equivalent norm \(\left\| \, \cdot \, \right\| _{X}\) on \(H^{1}(\mathbb {R}^{N})\) such that

$$\begin{aligned} I(u)=\frac{1}{2}\left( \left\| u^{+}\right\| _{X}^{2}-\left\| u^{-}\right\| _{X}^{2}\right) -\frac{1}{q+1} \int _{\mathbb {R}^{N}}a(x)\left| u \right| ^{q+1} dx- \int _{\mathbb {R}^{N}}f(x) u dx. \end{aligned}$$

For more details, we refer the reader to Costa and Tehrani [22].

5.1 Existence of a nonnegative solution of problem (2.2)

Consider the problem

$$\begin{aligned} M'_{0}=\inf _{u\in B\left( 0,\rho _{1}\right) }I\left( u\right) , \end{aligned}$$
(5.1)

where \(\rho _{1}\) is defined in Lemma 3.4.

Remark 5.1

By the same arguments used in the proof of Lemma 4.1, we conclude that \(-\infty<M'_{0}<0.\)

Theorem 5.2

Assume that hypotheses \(\left( \overline{A_{1}}\right) \), \(\left( H_{1}\right) \), \(\left( H_{3}\right) \) and \(\left( F\right) \) hold. Then there exists \(V_{0}\in H^{1}(\mathbb {R}^{N})\) such that \(I\left( V_{0}\right) =M'_{0}\). \(V_{0}\) is a weak solution of problem 2.2.

Proof

It is sufficient to replace Lemma 3.1 by Lemma 3.2 in the proof of Theorem 4.2. \(\square \)

5.2 Existence of a second solution of problem (2.2)

Lemma 5.3

Suppose that hypotheses \(\left( \overline{A_{1}}\right) \), \(\left( H_{2}\right) \), \(\left( H_{3}\right) \), \(\left( F\right) \) and \(\left( N^{\mp }\right) \) hold. Then any (P-S)-sequence of I is bounded in \(H^{1}(\mathbb {R}^{N})\).

Proof

Let \(\left( u_{n}\right) \) be a \(\left( \text{ PS }\right) \)-sequence of I.

Case 1. We suppose that a satisfies \(\left( N^{+}\right) \). Arguing by contradiction, suppose that \(\left\| u_{n}\right\| _{X}\rightarrow +\infty \). Then there exist \(c>0\) and \(d>0\) such that for n large enough we have

$$\begin{aligned} \left\| u_{n}^{\mp }\right\| _{X}\le c \left\| u_{n}\right\| _{X}^{q}+d. \end{aligned}$$
(5.2)

Indeed, from \(\left( \overline{A_{1}}\right) \) and for n large enough we obtain

$$\begin{aligned} \left\| u_{n}^{+}\right\| _{X}\ge \left\langle I'\left( u_{n}\right) , u_{n}^{+}\right\rangle \ge \left\| u_{n}^{+}\right\| _{X}^{2}-c\left\| a\right\| _{\frac{2}{1-q}}\left\| u_{n}\right\| _{X}^{q} \left\| u_{n}^{+}\right\| _{X}-c\left\| f\right\| _{2}\left\| u_{n}^{+}\right\| _{X}, \end{aligned}$$

hence

$$\begin{aligned} \left\| u_{n}^{+}\right\| _{X}\le c\left\| a\right\| _{\frac{2}{1-q}} \left\| u_{n}\right\| _{X}^{q}+c\left\| f\right\| _{2}. \end{aligned}$$

Similarly, from \(\left( \overline{A_{1}}\right) \), we have

$$\begin{aligned} -\left\| u_{n}^{-}\right\| _{X}\le \left\langle I'\left( u_{n}\right) , u_{n}^{-}\right\rangle \le -c\left\| u_{n}^{-}\right\| _{X}^{2}+c\left\| a\right\| _{\frac{2}{1-q}}\left\| u_{n}\right\| _{X}^{q} \left\| u_{n}^{-}\right\| _{X}+c\left\| f\right\| _{2} \left\| u_{n}^{-}\right\| _{X}, \end{aligned}$$

for n sufficiently large. Therefore

$$\begin{aligned} \left\| u_{n}^{-}\right\| _{X}\le c \left\| u_{n}\right\| _{X}^{q}+d. \end{aligned}$$

Using (5.2) we deduce

$$\begin{aligned} \left\| u_{n}-u_{n}^{0}\right\| _{X}=\left\| u_{n}^{+}+u_{n}^{-}\right\| _{X}\le c \left\| u_{n}\right\| _{X}^{q}+d \end{aligned}$$
(5.3)

and

$$\begin{aligned} \lim _{n \rightarrow +\infty } \frac{\left\| u_{n}^{0}\right\| _{X}}{\left\| u_{n}\right\| _{X}}=1. \end{aligned}$$
(5.4)

On the other hand, combining relations (5.2), (5.3), (5.4), \(\overline{\left( A_{1}\right) }\) with the mean value theorem and the Hölder inequality, we obtain

$$\begin{aligned} \begin{array}{ll} I\left( u_{n}\right) &{} = \frac{1}{2}\left\| u_{n}^{+}\right\| _{X}^{2}-\frac{1}{2}\left\| u_{n}^{-}\right\| _{X}^{2}-\frac{1}{q+1} \int _{\mathbb {R}^{N}} a(x)\left[ \left| u_{n}\right| ^{q+1}-\left| u_{n}^{0}\right| ^{q+1}\right] dx \\ {} &{}\quad -\frac{1}{q+1} \int _{\mathbb {R}^{N}} a(x)\left| u_{n}^{0}\right| ^{q+1}dx- \int _{\mathbb {R}^{N}} f(x) u_{n} dx\\ &{}\le c \left\| u_{n}\right\| _{X}^{2q}+d-\frac{1}{q+1} \int _{\mathbb {R}^{N}} a(x)\left[ \left| u_{n}\right| ^{q+1}-\left| u_{n}^{0}\right| ^{q+1}\right] dx \\ &{} \quad -\frac{1}{q+1} \int _{\mathbb {R}^{N}} a(x)\left| u_{n}^{0}\right| ^{q+1}dx +c\left\| f\right\| _{2}\left\| u_{n}\right\| _{X} \end{array} \end{aligned}$$
(5.5)

and

$$\begin{aligned} \begin{array}{ll} &{}\left| \int _{\mathbb {R}^{N}} a(x)\left[ \left| u_{n}\right| ^{q+1}-\left| u_{n}^{0}\right| ^{q+1}\right] dx\right| \\ &{} \quad \le c \left\| a\right\| _{\frac{2}{1-q}} \left( \left\| u_{n}\right\| _{X}+\left\| u_{n}^{0}\right\| _{X}\right) ^{q} \left\| u_{n}-u_{n}^{0}\right\| _{X}\\ &{} \quad \le c \left( \left\| u_{n}\right\| _{X}+\left\| u_{n}^{0}\right\| _{X}\right) ^{q} \left( \left\| u_{n}\right\| _{X}^{q}+1\right) \\ &{} \quad \le c \left( \left( \left\| u_{n}\right\| _{X}+\left\| u_{n}^{0}\right\| _{X}\right) ^{q}+\left\| u_{n}\right\| _{X}^{2q}+ \left( \left\| u_{n}\right\| _{X}+\left\| u_{n}^{0}\right\| _{X}\right) ^{2q}\right) \\ &{} \quad \le c \left( \left\| u_{n}\right\| _{X}^{2q}+\left\| u_{n}\right\| _{X}^{q}\right) +c \left\| u_{n}^{0}\right\| _{X}^{2q}, \ \ \text{ for } \ \ n \ \ \text{ sufficiently } \text{ large }. \end{array} \end{aligned}$$
(5.6)

Using (5.5) and (5.6) we deduce that

$$\begin{aligned} I\left( u_{n}\right)\le & {} c \left( \left\| u_{n}\right\| _{X}^{2q}+\left\| u_{n}^{0}\right\| _{X}^{2q}+\left\| u_{n}\right\| _{X}^{q}\right) \nonumber \\&\quad -\frac{1}{q+1} \int _{\mathbb {R}^{N}} a(x) \left| u_{n}^{0}\right| ^{q+1}dx+c\left\| f\right\| _{2} \left\| u_{n}\right\| _{X}+d. \end{aligned}$$
(5.7)

As a consequence of \(\left( N^{+}\right) \) and Lemma 3.2, we obtain for all \(0<d<q+1\)

$$\begin{aligned} \frac{1}{\left\| u_{n}^{0}\right\| _{X}^{d}} \int _{\mathbb {R}^{N}} a(x) \left| u_{n}^{0}\right| ^{q+1}dx=\frac{\left\| u_{n}^{0}\right\| _{X}^{q+1}}{\left\| u_{n}^{0}\right\| _{X}^{d}} \int _{\mathbb {R}^{N}} a(x) \left( \frac{\left| u_{n}^{0}\right| }{\left\| u_{n}^{0}\right\| _{X}}\right) ^{q+1}dx\rightarrow +\infty . \end{aligned}$$
(5.8)

Let us first assume that \(\frac{1}{2}<q<1\). Relations (5.4) and (5.8) yield

$$\begin{aligned} I\left( u_{n}\right)\le & {} \left\| u_{n}^{0}\right\| _{X}^{2q}\left[ c \left( \frac{\left\| u_{n}\right\| _{X}^{2q}}{\left\| u_{n}^{0}\right\| _{X}^{2q}}+\frac{\left\| u_{n}\right\| _{X}^{q}}{\left\| u_{n}^{0}\right\| _{X}^{2q}}+1\right) \right. \\&\quad \left. -\frac{1}{\left\| u_{n}^{0}\right\| _{X}^{2q}} \int _{\mathbb {R}^{N}} a(x) \left| u_{n}^{0}\right| ^{q+1}dx+c\left\| f\right\| _{2} \frac{\left\| u_{n}\right\| _{X}}{\left\| u_{n}^{0}\right\| _{X}^{2q}}\right] \rightarrow -\infty . \end{aligned}$$

Therefore we reach contradiction and we conclude that the sequence \(\left( u_{n}\right) \) is bounded in \(H^{1}(\mathbb {R}^{N})\).

Next, we assume that \(0<q<\frac{1}{2}\). For \(1<d<q+1\) and using (5.4) and (5.8), we obtain

$$\begin{aligned}\begin{array}{ll} I\left( u_{n}\right) &{} \le c\,\left\| u_{n}^{0}\right\| _{X}^{d} \left( \frac{\left\| u_{n}\right\| _{X}^{2q}}{\left\| u_{n}^{0}\right\| _{X}^{d}}+\frac{\left\| u_{n}^{0}\right\| _{X}^{2q}}{\left\| u_{n}^{0}\right\| _{X}^{d}} +\frac{\left\| u_{n}\right\| _{X}^{q}}{\left\| u_{n}^{0}\right\| _{X}^{d}}\right) \\ &{} \quad - \int _{\mathbb {R}^{N}} a(x) \left| u_{n}^{0}\right| ^{q+1}dx+\left\| f\right\| _{2} \,\left\| u_{n}^{0}\right\| _{X}^{d}\, \frac{\left\| u_{n}\right\| _{X}}{\left\| u_{n}^{0}\right\| _{X}^{d}}\rightarrow -\infty .\end{array} \end{aligned}$$

This yields to a contradiction and we conclude that the sequence \(\left( u_{n}\right) \) is bounded in \(H^{1}(\mathbb {R}^{N})\).

Case 2. We suppose that a satisfies \(\left( N^{-}\right) \). We have

$$\begin{aligned} I\left( u_{n}\right)&\ge -\left\| u_{n}^{-}\right\| _{X}^{2}-\frac{1}{q+1} \int _{\mathbb {R}^{N}} a(x)\left[ \left| u_{n}\right| ^{q+1}-\left| u_{n}^{0}\right| ^{q+1}\right] dx \\&\quad -\frac{1}{q+1} \int _{\mathbb {R}^{N}} a(x)\left| u_{n}^{0}\right| ^{q+1}dx- \int _{\mathbb {R}^{N}} f(x) u_{n} dx\\&\ge -c \left\| u_{n}\right\| _{X}^{2q}-c -c\left\| a\right\| _{\frac{2}{1-q}}\left( \left\| u_{n}\right\| _{X}+\left\| u_{n}^{0}\right\| _{X}\right) ^{q} \left\| u_{n}-u_{n}^{0}\right\| _{X}\\&\quad -\frac{1}{q+1} \int _{\mathbb {R}^{N}} a(x)\left| u_{n}^{0}\right| ^{q+1}dx-c\left\| f\right\| _{2} \left\| u_{n}\right\| _{X} \end{aligned}$$

and

$$\begin{aligned} \lim _{n \rightarrow +\infty }\frac{1}{\left\| u_{n}^{0}\right\| _{X}^{d}} \int _{\mathbb {R}^{N}} a(x) \left| u_{n}^{0}\right| ^{q+1}dx=-\infty , \ \ \forall \ \ 0<d<q+1. \end{aligned}$$
(5.9)

Using the above inequalities and similar to Case 1, we conclude that \(\left( u_{n}\right) \) is bounded in \(H^{1}(\mathbb {R}^{N})\). \(\square \)

Theorem 5.4

We suppose that hypotheses \((\overline{A_1})\), \( (H_{2})\), \((H_{3})\) and (F) hold. Then I satisfies the (PS) condition.

Proof

Let \(\left( u_{n}\right) \subset H^{1}\left( \mathbb {R}^{N}\right) \) be a (PS)-sequence such that

$$\begin{aligned} I\left( u_{n}\right) \rightarrow M'_{1} \ \ \text{ and } \ \ I'\left( u_{n}\right) \rightarrow 0 \ \ \text{ in } \ \ \left( H^{1}\left( \mathbb {R}^{N}\right) \right) '. \end{aligned}$$

By Lemma 5.3, \(\left( u_{n}\right) \) is bounded in \(H^{1}\left( \mathbb {R}^{N}\right) \). Thus up to a subsequence, \(u_{n}\rightharpoonup V_{1}\) in \(H^{1}\left( \mathbb {R}^{N}\right) \), \(u_{n}\rightarrow V_{1}\) in \(L^{m}_{\text{ loc }}\left( \mathbb {R}^{N}\right) \) for all \( 1\le m< 2^{*}\) and \(u_{n}\rightarrow V_{1}\) a.e. in \(\mathbb {R}^{N}\).

By Del Pino’s argument [23], it is sufficient to prove that for any \(\epsilon >0\), there exist \(R_{2}>0\) and \(n_{0}>0\) such that

$$\begin{aligned} \int _{\left| x\right| \ge R}\left( \left| \nabla u_{n}\right| ^{2}+\left| u_{n}\right| ^{2}\right) dx<\epsilon , \ \ \text{ for } \text{ all } \ \ R\ge R_{2} \ \ \text{ and } \ \ n\ge n_{0}. \end{aligned}$$

Let \(\phi _{R}\) be the cut-off function defined in (4.16). Since \(I'\left( u_{n}\right) \rightarrow 0\) as \(n\rightarrow +\infty \) in \(H^{1}\left( \mathbb {R}^{N}\right) '\), for \(\epsilon >0\), there exists \(n_{0}>0\) such that

$$\begin{aligned} \left| \left\langle I'(u_{n}), \phi _{R}u_{n}\right\rangle \right| \le c \left\| I'(u_{n})\right\| \left\| u_{n}\right\| _{H}\le \frac{\epsilon }{3}, \ \ \forall n\ge n_{0}. \end{aligned}$$

It follows that

$$\begin{aligned} \begin{array}{ll} &{} \int _{\mathbb {R}^{N}}\left( \left| \nabla u_{n}\right| ^{2}+\left( V^{+}+\chi _{B\left( 0,R_{0}\right) }\right) (x) u_{n}^{2}\right) \phi _{R}dx \\ &{} \quad \le \int _{\mathbb {R}^{N}} \left( V^{-}+\chi _{B\left( 0,R_{0}\right) }\right) (x)\phi _{R}u_{n}^{2}dx+ \int _{\mathbb {R}^{N}}a(x)\phi _{R}\left| u_{n}\right| ^{q+1} dx \\ &{} \qquad + \int _{\mathbb {R}^{N}} f(x)\phi _{R}(x)u_{n} dx- \int _{\mathbb {R}^{N}} u_{n}\nabla u_{n}\nabla \phi _{R}dx+\frac{\epsilon }{3}. \end{array} \end{aligned}$$
(5.10)

Therefore

$$\begin{aligned} \begin{array}{ll} &{}\min \left( \beta ,1\right) \int _{\mathbb {R}^{N}}\left( \left| \nabla u_{n}\right| ^{2}+ u_{n}^{2}\right) \phi _{R}dx \\ &{} \quad \le \int _{\mathbb {R}^{N}} \left( V^{-}+\chi _{B\left( 0,R_{0}\right) }\right) (x)\phi _{R}u_{n}^{2}dx+ \int _{\mathbb {R}^{N}}a(x)\phi _{R}\left| u_{n}\right| ^{q+1} dx \\ &{} \qquad + \int _{\mathbb {R}^{N}} f(x)\phi _{R}(x)u_{n} dx- \int _{\mathbb {R}^{N}} u_{n}\nabla u_{n}\nabla \phi _{R}dx+\frac{\epsilon }{3}, \end{array} \end{aligned}$$
(5.11)

From \(\left( F\right) \), \(\left( H_{3}\right) \), \(\overline{\left( A_{1}\right) }\) and (4.16), there exists \(R_{3}>0\) such that

$$\begin{aligned} c(\left\| f \phi _{R}\right\| _{2}+\left\| \left( V^{-}+\chi _{B\left( 0,R_{0}\right) }\right) \phi _{R}\right\| _{\frac{N}{2}}+\left\| a\phi _{R}\right\| _{\frac{2}{1-q}}+\frac{1}{R})\le \frac{\epsilon }{3} \ \ \text{ for } \ \ \left| x\right| \ge R_{4}. \end{aligned}$$

The rest of the proof is similar to the second step of the proof of Lemma 4.4. \(\square \)

Using (4.2) in combination with hypothesis \(\left( H_{1}\right) \), we construct \(\varphi _{2}=t\phi \in B^{c}\left( 0,\rho _{1}\right) \) for t large enough, such that \(I(\varphi _{2})<0\) (\(\rho _{1}\) is fixed by Lemma 3.4). Consider the mini-max problem

$$\begin{aligned} M^{''}_{0}= \inf _{\gamma \in \Gamma ^{''}_{0}}\max _{t\in \left[ 0,1\right] }I(\gamma (t)), \end{aligned}$$

where

$$\begin{aligned} \Gamma ^{''}_{0}= \{\gamma \in C ([0,1], H^{1}(\mathbb {R}^{N})), \ \gamma (0)=0 \ \text{ and } \ \gamma (1)=\varphi _{2}\}. \end{aligned}$$

Combining the mountain pass theorem [5] and Lemma 5.4, we deduce the existence of \(W_{0}\in H^{1}(\mathbb {R}^{N})\) such that \(M_{0}^{''}=I\left( W_{0}\right) \), \(I(W_{0})>0\) and \(W_{0}\) is a weak solution of problem (2.2).

5.3 Concluding remarks

The content of this paper is in relationship with the recent contributions of Bégout and Díaz [9,10,11,12,13] for the understanding of the nonlinear stationary Schrödinger equation. We expect that the methods developed by Díaz [24, 25] in the case of an unbounded potential V(x) of Hardy type can be applied to show that the nonnegative solutions of problem (2.2) established in Theorems 2.1 and 2.2 have a compact support (if f(x) is also with compact support). This case corresponds to the abstract setting of infinite well type potentials.