 Research
 Open Access
Existence of solutions for perturbed fractional equations with two competing weighted nonlinear terms
 Yujuan Tian^{1}Email author and
 Shasha Zhao^{1}
 Received: 16 July 2018
 Accepted: 24 September 2018
 Published: 1 October 2018
Abstract
This paper deals with a perturbed nonlocal equation of fractional Laplacian type involving two competing nonlinear terms with weights f and h. Under two kinds of integrability conditions on the ratio of f to h, we show some different existence results in this setting by using variational methods. Our results are extension of some problems studied by Carboni and Mugnai (J. Differ. Equ. 262(3):2393–2413, 2017) and Xiang et al. (J. Differ. Equ. 260(2):1392–1413, 2016).
Keywords
 Fractional Laplacian
 Weighted nonlinearities
 Variational methods
 Existence
MSC
 35A01
 35J20
 35J60
 35R11
1 Introduction and main results
 (H_{1}):

g is a Carathéodory function, and there exist two positive constants \(b_{1}\) and \(b_{2}\) such thatwhere \(1< q< r<2_{s}^{*} \), and \(2^{*}_{s}= \frac{2N}{N2s}\) is the Sobolev fractional critical exponent;$$ b_{1}t^{r}\leq g(x,t)t\leq b_{2}t^{r}\quad \mbox{for a.e. } x\in\Omega \mbox{ and all } t\in\mathbb{R}, $$(1.2)
 (H_{2}):

\(f,h:\Omega\rightarrow\mathbb{R}_{+}\) are positive weights satisfying \(f\in L^{\frac{2_{s}^{*}}{2_{s}^{*}q}}(\Omega)\) and \(h\in L^{\frac{2_{s}^{*}}{2_{s}^{*}r}}(\Omega)\). Moreover, \(f(x)\) and \(h(x)\) are related by the condition$$ \biggl(\frac{f^{r}}{h^{q}} \biggr)^{\frac {1}{rq}}\in L^{1}(\Omega); $$(1.3)
 (H_{3}):

\(v:\Omega\rightarrow\mathbb{R}\) is a perturbed weight such that \(v(x)\geq0\) for a.e. \(x\in\Omega\).
We now state our main results. Our first result covers both convex–concave and concave–concave cases.
Theorem 1.1
 (i)
only the trivial solution if \(\lambda<\lambda_{0}\);
 (ii)
at least one nontrivial solution of definite sign if \(\lambda>\lambda_{*}\);
 (iii)
at least two nontrivial solutions if \(\lambda>\lambda^{*}\); one them is positive, and one is negative.
Theorem 1.2
 (i)
the results of Theorem 1.1 still hold, and the number \(\lambda_{0}>0\);
 (ii)
if \(\lambda=\lambda_{0}\), then problem (\({P}_{\lambda }\)) has at least one nontrivial solution; if \(\lambda=\lambda_{*}\), then problem (\({P}_{\lambda }\)) has at least one nontrivial solution of definite sign; if \(\lambda=\lambda^{*}\), then problem (\({P}_{\lambda }\)) has at least two nontrivial solutions; one of them is positive, and one is negative;
 (iii)
there exist a number \(\tilde{\lambda }\geq\lambda^{*}\) such that problem (\({P}_{\lambda }\)) has at least four nontrivial solutions for every \(\lambda\geq\tilde{\lambda }\); two of them are positive, and two are negative.
Our work are an extension of problems studied in [14], where \(s=\frac{1}{2}\), \(f(x)\equiv1\), and \(v(x)\equiv0\). Even in this case, our assumptions on \(h(x)\) are weaker than those in [14] since \(h\in L^{\frac{2_{s}^{*}}{2_{s}^{*}r}}(\Omega)\), which is allowed to satisfy condition (1.4) here. On the other hand, the problem studied in this paper covers both convex–concave and concave–concave cases, and owing to the different approach used here, the assumptions on perturbed weight can be relaxed to be nonnegative. From this point, our results extend some results of [1]. In addition, using variational methods, we present more complete results in the convex–concave case by describing the problem at all threshold values \(\lambda_{0}\), \(\lambda _{*}\), \(\lambda^{*}\), and λ̃.
Many other types of fractional and nonlocal problems are also actively studied in recent years. For example, we refer to [15–21] for fractional Kirchhoftype problems. Studies of fractional Schrödinger problems can be found in [22–24]. Fractional problems with special nonlinearities have been considered in [25–27]. Results on the Brézis–Nirenberg problem for a fractional operator were presented in [28, 29].
The remaining of this paper is organized as follows. In Sect. 2, we recall some preliminary results and main embedding results for fractional Sobolev spaces. In Sect. 3 and 4, we prove Theorems 1.1 and 1.2, respectively.
2 Preliminaries
In this section, we recall some necessary definitions and properties of the fractional Laplacian operator. For further details, we refer to [2, 30, 31] and references therein.
Lemma 2.1
Proof
Lemma 2.2
The lemma can be proved similarly as Theorem 2.1 in [1], which we will not repeat here.
Now, in view of the previous considerations, we have the definition of weak solution to problem (\({P}_{\lambda }\)).
Definition 2.1
Finally, let us recall an inequality that we will use afterward.
Lemma 2.3
(see [1])
3 Proof of Theorem 1.1
3.1 Nonexistence of solutions
3.2 Existence of solutions
Step 1. We prove that there exist a nonnegative critical point \(u_{1}\) for the functional \(\Psi_{+}\) and a nonpositive critical point \(u_{2}\) for the functional \(\Psi_{}\).
Let us first focus on the functional \(\Psi_{+}\). We have the following lemma.
Lemma 3.1
The functional \(\Psi_{+}\) is coercive and weakly lower semicontinuous in E.
Proof
We may apply similar arguments to prove that the functional \(\Psi_{}\) has a minimizer in E, that is, that there exists \(u_{2}\in E\) such that \(\Psi_{}(u_{2})=\inf_{u\in E}\Psi_{}(u)\). Moreover, we can proceed with \(\varphi=u_{2}^{+}\) similarly as in (3.4). Noting that \((ab)(a^{+}b^{+})\geq(a^{+}b^{+})^{2}\), we get that \(u_{2}\leq 0\) a.e. in Ω and that \(u_{2}\) is also a weak solution of (\({P}_{\lambda }\)).
Step 2. We show that if λ is large enough, then \(u_{1}\) and \(u_{2}\) are nontrivial with \(\Psi(u_{1})<0\) and \(\Psi(u_{2})<0\).
Lemma 3.2
For any \(\lambda>\lambda_{+}^{*}\), (\({P}_{\lambda }\)) admits at least one nontrivial positive solution.
Proof
Fix \(\lambda>\lambda_{+}^{*}\). By the definition of \(\lambda_{+}^{*}\), there exists \(\mu\in(\lambda_{+}^{*},\lambda)\) such that (\(P_{\mu}\)) has a nontrivial solution \(u_{\mu}\in E\) with \(u_{\mu}>0\) in Ω. Obviously, \(u_{\mu}\) is a subsolution of (\({P}_{\lambda }\)).
Thus by the maximum principle (see [33], Lemma 5.2) we get the existence of a nontrivial positive solution u of (\({P}_{\lambda }\)) with \(u_{\mu}\leq u\leq\tilde{u}\). □
4 Proof of Theorem 1.2
4.1 The proof of statement (i)
So far, all the necessary conditions are satisfied, and we can follow the same lines as in Sect. 3 to get the results of Theorem 1.1.
4.2 The cases \(\lambda=\lambda_{0},\lambda_{*},\lambda^{*}\)
For \(\lambda=\lambda_{*}\), without loss of generality, we can assume that \(\lambda_{*}=\lambda_{+}^{*}<\lambda_{}^{*}\). Then there also exists a decreasing sequence \(\{\lambda_{n}\}\) converging to \(\lambda _{*}\). Let \(u_{n}\in E\) be a nontrivial positive solution of problem (\(P_{\lambda_{n}}\)). It suffices to repeat the previous arguments to conclude that (\(P_{\lambda_{*}}\)) has at least one nontrivial positive solution.
For \(\lambda=\lambda^{*}\), assuming that \(\lambda^{*}=\lambda _{+}^{*}>\lambda_{}^{*}\), \(\{P_{\lambda_{*}}\}\) admits at least one nontrivial positive solution. Furthermore, since \(\lambda ^{*}>\lambda_{}^{*}\), we deduce that \(\{P_{\lambda_{*}}\}\) also admits a nontrivial negative solution. So, the second statement of Theorem 1.2 follows.
4.3 The case \(\lambda\geq\tilde{\lambda}\)
Lemma 4.1
There exist \(\rho\in(0,\u_{1}\_{E})\) and \(\alpha>0\) such that \(\Psi _{+}(u)\geq\alpha\) for all \(u\in E\) with \(\u\_{E}=\rho\).
Proof
Next, we prove that the Palais–Smale condition holds for every sequence \(\{u_{n} \}\subset E\).
Lemma 4.2
Assume that a sequence \(\{u_{n} \}\subset E\) such that \(\{\Psi _{+}(u_{n})\}\) is bounded and \(\Psi'_{+}(u_{n})\rightarrow0\) in \(E^{*}\) as \(n\rightarrow\infty\). Then \(\{u_{n}\}\) has a convergent subsequence in E.
Proof
As a consequence, the mountain pass theorem guarantees the existence of a critical point \(u_{3}\) for \(\Psi_{+}\) with \(\Psi_{+}(u_{3})\geq\alpha >0\), which implies that \(u_{3}\) is nontrivial and different from \(u_{1}\) and \(u_{2}\). Moreover, working as we did for \(u_{1}\), we can show that \(u_{3}\geq0\). So, \(u_{3}\) is a critical point of Ψ. The third nontrivial solution of problem (\({P}_{\lambda }\)) is obtained.
Similarly, we apply the mountain pass theorem to \(\Psi_{}\) to prove the existence of the fourth nontrivial weak solution of problem (\({P}_{\lambda }\)) having negative sign and different from the previous ones. Thus, when \(\lambda>\tilde{\lambda}\), four nontrivial solutions have been obtained.
If \(\lambda=\tilde{\lambda}\), then we can assume that \(\tilde {\lambda}=\lambda_{+}>\lambda_{}\). Let \(\lambda_{n}=\tilde{\lambda }+\frac{1}{n}\). Then \(\lambda_{n}\rightarrow\tilde{\lambda}\) as \(n\rightarrow\infty\) and \(\lambda_{n}>\tilde{\lambda}\) for \(n\in\mathbb{N}_{+}\). So, for every \(n\in\mathbb{N}_{+}\), there exist two nontrivial positive solutions \(u_{n_{1}}\) and \(u_{n_{2}}\) of problem (\(P_{\lambda_{n}}\)) with \(\Psi(u_{n_{1}})<0\) and \(\Psi(u_{n_{2}})\geq\alpha>0\). Proceeding as in Sect. 4.2, we get that there exist \(u_{\tilde{\lambda}_{1}}\geq0\) and \(u_{\tilde {\lambda}_{2}}\geq0\) such that \(u_{n_{1}}\rightharpoonup u_{\tilde {\lambda}_{1}}\) in E and \(u_{n_{2}}\rightharpoonup u_{\tilde {\lambda}_{2}}\) in E. Furthermore, we can verify that \(u_{\tilde {\lambda}_{1}}\) and \(u_{\tilde{\lambda}_{2}}\) are two weak solutions of problem (\(P_{\tilde{\lambda}}\)) with \(\Psi ({u_{\tilde{\lambda}_{1}}})<0\) and \(\Psi({u_{\tilde{\lambda }_{2}}})\geq\alpha>0\), which implies that \(u_{\tilde{\lambda }_{1}}\neq u_{\tilde{\lambda}_{2}}\). On the other hand, since \(\tilde{\lambda}>\lambda_{}\), (\(P_{\tilde{\lambda}}\)) admits two different nontrivial negative solutions. Thus, for \(\lambda=\tilde{\lambda}\), (\(P_{\tilde{\lambda}}\)) has at least four nontrivial solutions.
Declarations
Acknowledgements
The authors would like to thank the referees for their useful suggestions.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Funding
This work is supported by National Natural Science Foundation of China (Grant Nos. 11501333, 11571208).
Authors’ contributions
Both authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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