Abstract
We consider the following nonlinear Schrödinger equation
where V is a non-symmetric bounded potential, a is an indefinite weight, \(0<q<1\) and \(f\ne 0\) is a nonnegative perturbation such that \(f\in L^{2}(\mathbb {R}^{N})\cap L^{\frac{2N}{N+2}}(\mathbb {R}^{N})\). Using variational methods, we prove the existence of two solutions with negative and positive energies, one of these solutions being nonnegative.
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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
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
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
Using the Ansatz (1.2), we reduce the nonlinear Schrödinger equation (1.1) to the semilinear elliptic equation
The change of variable \(y=\hbar ^{-1}x\) (and replacing y by x) yields
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
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
It seems that the first reference dealing with the limit case, the so-called infinite well potential,
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
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
has a bounded solution. Their analysis showed that such a solution exists for potentials like
while no solution exists if
Consider the following class of sublinear Schrödinger equations
where
![figure a](http://media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs13398-018-0541-9/MediaObjects/13398_2018_541_Figa_HTML.gif)
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
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
where \(\lambda _{k}\) denotes the kth eigenvalue of the Laplace operator. They proved that a sufficient condition for the existence of a solution is
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
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
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
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
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
and
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:
\(F'\): the Fréchet derivative of F at u.
Let
If we equip E with the norm
then E becomes a reflexive Banach space. On the Sobolev space \(H^{1}(\mathbb {R}^{N})\) we consider the usual norm
Define the following energy functional on Y (\(Y=E\) or \(H^{1}(\mathbb {R}^{N})\)):
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
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
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
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
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
Proof
Since \(a\in L^{\frac{2}{1-q}}(\mathbb {R}^{N})\), then for every \(\epsilon >0\) there exists \(R_{2}>0\) such that
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
Observe that by Hölder’s inequality we have
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
with
Then from \(\left( F\right) \), \(\left( H_{3}\right) \) and \(\left( A_{1}\right) \) we infer that
Since \(p>2\), for \(\left\| \nabla u\right\| _{2} \le s \) (small enough) we have
Next, using the inequality
we obtain that for \(\left\| \nabla u\right\| _{2}\le s\) and \(\left\| u\right\| _{q+1}\le 1\), we have
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
we infer that
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
Proof
Combining hypothesis \(\left( H_{3}\right) \) with Hölder and Sobolev inequalities, we obtain
and
Using (3.5), we have
Using (3.6) and for \(\left\| \nabla u\right\| _{2}\le \rho _{1}\) (\(\rho _{1}\) small enough), we infer that
where
Set
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
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
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
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
Since \(v_{n}\rightharpoonup 0\) in E, we obtain
By Lemma 3.1 we have
From \(\left( H_{3}\right) \) and \(\left( A_{1}\right) \), we deduce that
Using relations (4.4), (4.5), (4.6) and passing to the limit in (4.3), we get
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
Dividing relation (4.7) by \(t_{n}\), we obtain
Using \(\left( A_{1}\right) \) and Hölder’s inequality, we deduce that
Thus since \(0<q<1\), we have
As a consequence of hypothesis \(\left( F\right) \), we have
From (4.9), (4.10) and passing to the limit in relation (4.8), we infer that
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
It follows that
Using now \(\left( A_{1}\right) \), \(\left( H_{3}\right) \) and \(\left( F\right) \), we obtain
Combining (4.11) and (4.12), we conclude that
On the other hand, since \((u_{n})\) is a (PS)-sequence of I, we have
Therefore
From \(\left( A_{1}\right) \), \(\left( F\right) \) and (4.14), we infer that
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
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
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
We remark that for any \(u\in E\),
Since \(I'(u_{n})\rightarrow 0\) in \(E'\) then for any \(\epsilon >0\) there exists \(n_{0}>0\) such that
Thus for \(n\ge n_{0}\),
By the Hölder inequality and (4.16), there exists \(R_{3}>0\) such that
From \(\left( F\right) \), \(\left( H_{3}\right) \) and \(\left( A_{1}\right) \), there exists \(R_{4}>0\) such that
Set \(R_{2}=\max \left( R_{3},R_{4}\right) \). Then by (4.18), (4.19) and (4.20) we deduce that
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
where
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,
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
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
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
Indeed, from \(\left( \overline{A_{1}}\right) \) and for n large enough we obtain
hence
Similarly, from \(\left( \overline{A_{1}}\right) \), we have
for n sufficiently large. Therefore
Using (5.2) we deduce
and
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
and
Using (5.5) and (5.6) we deduce that
As a consequence of \(\left( N^{+}\right) \) and Lemma 3.2, we obtain for all \(0<d<q+1\)
Let us first assume that \(\frac{1}{2}<q<1\). Relations (5.4) and (5.8) yield
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
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
and
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
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
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
It follows that
Therefore
From \(\left( F\right) \), \(\left( H_{3}\right) \), \(\overline{\left( A_{1}\right) }\) and (4.16), there exists \(R_{3}>0\) such that
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
where
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.
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Acknowledgements
V. D. Rădulescu acknowledges the support through a grant of the Romanian Ministry of Research and Innovation, CNCS–UEFISCDI, project number PN-III-P4-ID-PCE-2016-0130, within PNCDI III. He was also supported by the Slovenian Research Agency Grants P1-0292, J1-8131, J1-7025, N1-0064, and N1-0083.
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Bahrouni, A., Ounaies, H. & Rădulescu, V.D. Bound state solutions of sublinear Schrödinger equations with lack of compactness. RACSAM 113, 1191–1210 (2019). https://doi.org/10.1007/s13398-018-0541-9
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DOI: https://doi.org/10.1007/s13398-018-0541-9