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Existence of positive solutions for a fractional elliptic problems with the Hardy-Sobolev-Maz’ya potential and critical nonlinearities
Boundary Value Problems volume 2017, Article number: 181 (2017)
Abstract
In this paper, we consider the study of a fractional elliptic problem with the Hardy-Sobolev-Maz’ya potential and critical nonlinearities. By means of variational methods and suitable technique, a positive solution to this problem is obtained.
1 Introduction and main result
In this paper, we consider the existence of the solutions for the following problem:
where Ω is a smooth bounded domain in \(\mathbb{R}^{N}= \mathbb{R}^{k}\times \mathbb{R}^{N-k}\) with \(N\geq 3\) and \(2\leq k< N\), \(0\leq \mu < a_{k,s}:=2^{2s}\Gamma^{2}(\frac{k+2s}{4})/\Gamma^{2}( \frac{k-2s}{4})\), \(s\in (0,1)\), \(\Gamma (t)=\int_{0}^{+\infty }\tau ^{t-1}e^{-\tau }\,d\tau \). A point \(x\in \mathbb{R}^{N}\) is denoted as \(x=(z,w)\in \mathbb{R}^{N}=\mathbb{R}^{k}\times \mathbb{R}^{N-k}\). \(\lambda >0\) is a real parameter, the number \(2_{s}^{*}(\alpha)=2(N-\alpha)/(N-2s)\) is a critical Hardy-Sobolev exponent with \(s\in (0,1)\) and \(\alpha \in [0,2s)\). The nonlinearity term f is continuous function and satisfies suitable hypotheses. Here, \((-\Delta)^{s}\) is the fractional Laplace operator (see [1, 2]) defined, up to a normalization factor, by
In recent years, much attention has been focused on the study of the problems involving fractional operators. The fractional operators appear in several applications to some models related to probability, mathematical, finances or fluid mechanics, soft thin films, stratified materials, multiple scattering and minimal surfaces (see [3–6]). When \(\mu =0\) and \(\alpha =0\), problem (1) reduces to critical fractional equation. Abundant results have been accumulated (see [7–12]).
For a class of fractional elliptic problems with the Hardy potential
Abdellaoui and Medina et al. in [13] gave the solvability of the problem (2) for the linear case \(g(x,t)=g(x)\) and the nonlinear case \(g(x,t)=\frac{h(x)}{t^{\sigma }}\), respectively. For critical case, a positive solution was obtained in [14] with by the Lagrange multipliers technique. Moreover, the authors in [15] have studied the solvability of problem (2) for the case \(g(x,t)\) involving concave-convex nonlinearities.
Recently, Jiang and Tang in [16] had considered the problem (1) for the case \(s=1\), they supposed the nonlinearity term \(f\in C(\overline{\Omega }\times \mathbb{R}^{+},\mathbb{R}^{+})\) satisfies the following conditions:
- \((f_{1})\) :
-
\(f(x,t)=0\) for \(t\leq 0\) uniformly for \(x\in \overline{ \Omega }\). There exists a nonempty open subset \(\Omega_{0}\subset \Omega \) with \((0,w^{0})\in \mathbb{R}^{k}\times \mathbb{R}^{N-k} \in \Omega_{0}\), such that \(f(x,t)\geq 0\) for almost everywhere \(x\in \Omega \) and all \(t>0\); \(f(x,t)>0\) for almost \(x\in \Omega_{0}\) and all \(t>0\).
- \((f_{2})\) :
-
\(\lim_{t\rightarrow 0^{+}}\frac{f(x,t)}{t}=0\) and \(\lim_{t\rightarrow +\infty }\frac{f(x,t)}{t^{2_{s}^{*}(\alpha)-1}}=0\) uniformly for \(x\in \overline{\Omega }\).
For \(\lambda >0\) large enough, they obtained the existence of positive solutions of problem (1) for \(s=1\) by using variational methods. For the case \(s=1\) and \(\lambda =1\), Ding and Tang in [17] obtained the existence of positive solutions for problem (1) by the variational methods and some analysis techniques with f satisfying the (AR) condition. For related papers on the semilinear elliptic equations with Hardy-Sobolev critical exponents of (1) for \(s=1\), we just mention [18, 19] and the references therein.
To the best of our knowledge, there is no result in the literature on the fractional elliptic problem with Hardy-Sobolev-Maz’ya potential and critical nonlinearities. Motivated by the above papers, our aim is to study the existence of positive solutions for problem (1) and our main result of this paper is as follows.
Theorem 1
Assume that conditions \((f_{1})\) and \((f_{2})\) hold. Then there exists \(\lambda^{*}>0\) such that \(\lambda \geq \lambda^{*}\), problem (1) admits a positive solution.
2 Functional setting and useful tools
We will denote by \(H^{s}(\mathbb{R}^{N})\) the usual fractional Sobolev space endowed with the natural norm
We consider the function space
with the norm
which is equivalent to its general norm due to the Hardy inequality
where \(Q=\mathbb{R}^{2N}\setminus (C\Omega \times C\Omega)\) with \(C\Omega =\mathbb{R}^{N}\setminus \Omega \). We can introduce the best fractional critical Hardy-Sobolev constant \(S_{\mu,\alpha }\), given by
From [20], we know that \(S_{\mu,\alpha }\) is attained by functions
Let \(u^{+}=\max \{u,0\}\), the energy functional \(J_{\lambda }:X_{0} ^{s}\rightarrow \mathbb{R}\) associated to the problem (1) is defined as
for all \(u\in X_{0}^{s}\), where \(F(x,t)\) is a primitive function of \(f(x,t)\) defined by \(F(x,t)=\int_{0}^{t}f(x,\tau)\,d\tau \). Obviously, \(J_{\lambda }\) is a \(C^{1}(X_{0}^{s})\) functional, and it is well known that the solutions of problem (1) are the critical points of the energy functional \(J_{\lambda }\). In fact, if u is a weak solution of problem (1), we have
for all \(\varphi \in X_{0}^{s}\). Now, we will give some essential lemmas as follows.
Lemma 1
Let \(\lambda >0\) and f satisfies assumptions \((f_{1})\) and \((f_{2})\). We can deduce that:
-
(i)
there exist \(\varsigma,\rho >0\) such that \(J_{\lambda }(u)\geq \varsigma >0\) for any \(u\in X_{0}^{s}\), with \(\Vert u\Vert _{X_{0} ^{s}}=\rho \);
-
(ii)
there exists \(e\in X_{0}^{s}\), with \(e\geq 0\) in \(\mathbb{R}^{N}\) such that \(J_{\lambda }(e)<0\) and \(\Vert e\Vert _{X_{0}^{s}} > \rho \).
Proof of Lemma 1
(i) Fixing \(\lambda >0\), from \((f_{2})\), for \(\varepsilon >0\), there exists \(C_{1}>0\), one has
It is evident that \(X_{0}^{s}\hookrightarrow L^{q}(\Omega)\) for \(1\leq q\leq 2_{s}^{*}(\alpha)\), then there exists \(C_{2}>0\) such that
Take \(u\in X_{0}^{s}\). Combining (4), (5), (7) and (8), we have
where \(C_{i}\), \(i=3,4,5\), are positive constants. For \(\varepsilon >0\) small and according to the fact \(2< 2^{*}(s)\), then there exists \(\rho >0\) small enough such that \(J_{\lambda }(u)\geq \varsigma >0\), for any \(\Vert u\Vert _{X_{0}^{s}}=\rho \).
(ii) Given \(\lambda >0\). Take \(v \in X_{0}^{s}\), with \(v\geq 0\) in \(\mathbb{R}^{N}\) and \(\Vert v\Vert _{X_{0}^{s}}=1\). From \((f_{1})\), we get
then \(J_{\lambda }(tv)\rightarrow -\infty \) as \(t\rightarrow +\infty \). Choosing \(e=t_{*}v\) with \(t_{*}>0\) large enough, we get \(\Vert e\Vert _{X _{0}^{s}}>\rho \) and \(J_{\lambda }(e)<0\). This completes the proof of the Lemma 1. □
We recall that a sequence \(\{u_{j}\}_{j\in \mathbb{N}} \subset X_{0} ^{s}\) is a Palais-Smale sequence for the functional \(J_{\lambda }\) at level \(c_{\lambda }\) if
as \(j\rightarrow \infty \). We say that \(J_{\lambda }\) satisfies the Palais-Smale condition if every Palais-Smale sequence of \(J_{\lambda }\) has a convergent subsequence in \(X_{0}^{s}\). Now put
where
Obviously, \(c_{\lambda }>0\) from Lemma 1. Next, we introduce an asymptotic condition for the level \(c_{\lambda }\).
Lemma 2
Under the conditions of Lemma 1, \(\lim_{\lambda \rightarrow \infty } c_{\lambda }=0\).
Proof of Lemma 2
Fix \(\lambda >0\). Since the functional \(J_{\lambda }\) satisfies the Mountain pass geometry, there exists \(t_{\lambda }>0\) verifying \(J_{\lambda }(t_{\lambda }e)=\max_{t\geq 0}J _{\lambda }(te)\), where \(e \in X_{0}^{s}\) is the function given in Lemma 1. Hence, by (6), we have \(\langle J'_{\lambda }(t_{\lambda }e),t _{\lambda }e\rangle =0\), that is,
From \((f_{1})\), we have
which implies that \(\{t_{\lambda }\}\) is bounded. Hence, there exist a number \(t_{0}\geq 0\) and a subsequence of \(\{\lambda_{j}\}_{j\in \mathbb{N}}\), which we still denote by \(\{\lambda_{j}\}_{j\in \mathbb{N}}\), such that \(\lambda_{j} \rightarrow +\infty \) and \(t_{\lambda_{j}}\rightarrow t_{0}\) as \(j\rightarrow \infty \). So by (9) there exists \(D>0\) such that \(t_{\lambda }^{2}\Vert e\Vert _{X_{0} ^{s}}^{2}\leq D \) for any \(j\in \mathbb{N}\), then
If \(t_{0}>0\), by \((f_{1})\) and the Lebesgue dominated convergence theorem, we obtain
Recalling that \(\lambda_{j}\rightarrow \infty \), we have
which contradicts (10). Thus \(t_{0}=0\) for \(\lambda_{j}\rightarrow \infty \). Now, let us consider the path \(g(t)=te\), for \(t\in [0,1]\), which belongs to Γ. By Lemma 1 and \((f_{1})\), we get
Notice that \(t_{\lambda_{j}}\rightarrow t_{0}=0\) as \(j\rightarrow \infty \), one has
which leads to \(\lim_{\lambda \rightarrow \infty } c_{\lambda }=0\). This completes the proof of the Lemma 2. □
Lemma 3
Assume that conditions \((f_{1})\) and \((f_{2})\) hold. If \(\{u_{j}\}\subset X_{0}^{s}\) is a \((PS)_{c_{\lambda }}\) condition of \(J_{\lambda }\), then \(\{u_{j}\}\) is bounded in \(X_{0}^{s}\).
Proof of Lemma 3
By \((f_{2})\) and the boundedness of Ω, for any \(\varepsilon >0\), there exists \(T>0\), such that
for \(C_{i}(\varepsilon)>0\), \(i=6,7\). Furthermore, for any \((x,t)\in \Omega \times \mathbb{R}^{+}\), we have
Then, for \(\xi \in (2,2_{s}^{*}(\alpha))\), one has
for \(C_{8}(\varepsilon)>0\) and any \((x,t)\in \overline{\Omega } \times \mathbb{R}^{+}\). Set \(l(x,t):=\frac{\vert t\vert ^{2_{s}^{*}(\alpha)-1}}{\vert z\vert ^{ \alpha }}+\lambda f(x,t)\), we claim that \(l(x,t)\) satisfies the \(\mathrm{(AR)}\) condition. By (11), one easily gets
where \(L(x,t)=\int_{0}^{t}l(x,\tau)\,d\tau \). Thus, for a fixed \(\lambda >0\) and \(\varepsilon >0\) sufficiently small, there exists \(T'_{\lambda }>0\), such that
Moreover, by \((f_{2})\), we obtain
for any \(0\leq t\leq T'_{\lambda }\). Notice that \(\xi <2_{s}^{*}( \alpha)\), we obtain \(T_{\lambda }>0\). It follows from the above inequalities that
Combining \((f_{2})\), (6) and (12), it follows that
Hence, we obtain \(\{u_{j}\}_{j\in \mathbb{N}}\) is bounded in \(X_{0}^{s}\). This completes the proof of Lemma 3. □
Lemma 4
Assume that conditions \((f_{1})\) and \((f_{2})\) hold. Then \(J_{\lambda }\) satisfies the \((PS)_{c_{\lambda }}\) condition with \(c_{\lambda }<\frac{2s-\alpha }{2(N-\alpha)} S_{ \mu,\alpha }^{\frac{N-\alpha }{2s-\alpha }} -T_{\lambda }\vert \Omega \vert \), where \(T_{\lambda }\) is a bounded constant given in Lemma 3.
Proof of Lemma 4
Let \(\{u_{j}\}_{j\in \mathbb{N}}\subset X_{0}^{s}\) be a \((PS)_{c_{\lambda }}\) sequence of \(J_{\lambda }\). From Lemma 3, we know that \(\{u_{j}\}_{j\in \mathbb{N}}\) is bounded. Thus there exist a subsequence (still denoted by \(\{u_{j}\}_{j\in \mathbb{N}}\)) and \(u_{\lambda }\in X_{0}^{s}\) such that
Due to the continuity of embedding \(X_{0}^{s}\hookrightarrow H^{s}( \Omega)\hookrightarrow L^{\nu }(\Omega)\) and the boundedness of \(\{u_{j}\}_{j\in \mathbb{N}}\), there exists a constant \(C_{\nu }\) such that \(\Vert u\Vert _{\nu }\leq \Vert u\Vert _{X_{0}^{s}}\leq C_{\nu }\) for all \(u\in X_{0}^{s}\) and \(\nu \in [2,2_{s}^{*}(\alpha)]\). Now, according to \((f_{2})\), for any \(\varepsilon >0\), there exists a \(a(\varepsilon)>0\) such that
Set \(\delta =\frac{\varepsilon }{2a(\varepsilon)}>0\). When \(E\subset \Omega \), meas \(E<\delta \), one gets
Obviously, \(\{\int_{\Omega }F(x,u_{j}^{+})\,dx,j\in N\}\) is equi-absolutely continuous. It follows from Vitali’s convergence theorem that
Similarly, we get
for any \(v \in X_{0}^{s}\). That is, \(\langle J'_{\lambda }(u_{\lambda }),v\rangle =0\) for any \(v \in X_{0}^{s}\). Then \(u_{\lambda }\) is a critical point of \(J_{\lambda }\), thus \(u_{\lambda }\) is a solution of problem (1). It follows from (6) and (12) that
for \(\xi \in (2,2_{s}^{*}(\alpha))\). Now, let \(w_{j}=u_{j}-u_{\lambda }\), by the Brezis-Lieb lemma [21], we have
Since \(J_{\lambda }(u_{j})=c_{\lambda }+o(1)\), by (14) and (16)-(18), we obtain
According to \(\langle J'_{\lambda }(u_{j}),u_{j}\rangle =o(1)\), (15) and (16)-(18), we get
Assume that \(\Vert w_{j}\Vert _{X_{0}^{s}}\rightarrow l\), it follows from (20) that
as \(j\rightarrow \infty \). From (4), one has
We get \(l\geq S_{\mu,\alpha }^{\frac{N-\alpha }{2s-\alpha }}\). It follows from (19) and (20) that
which contradicts \(c_{\lambda }<\frac{2s-\alpha }{2(N-\alpha)} S_{ \mu,\alpha }^{\frac{N-\alpha }{2s-\alpha }}-T_{\lambda }\vert \Omega \vert \). Therefore, we have \(l=0\), which implies that \(u_{j}\rightarrow u_{ \lambda }\) in \(X_{0}^{s}\). This completes the proof of the Lemma 4. □
3 Proof of Theorem 1
Thanks to Lemmas 1, 2, 3 and Lemma 4, the functional \(J_{\lambda }\) satisfies all the assumptions of the mountain pass theorem for any \(\lambda \geq \lambda^{*}\), with \(\lambda^{*}>0\). This guarantees the existence of a critical point \(u_{\lambda }\in X_{0}^{s}\) for \(J_{\lambda }\) at level \(c_{\lambda }\). Since \(J_{\lambda }(u_{\lambda })=c_{\lambda }>0=J_{\lambda }(0)\) we have \(u_{\lambda }\not \equiv 0\). By [22, Lemma 8], we have \(u_{\lambda }^{-}\in X_{0}^{s}\) and we let \(\varphi =u_{\lambda }^{-}\) in (6), we get
Moreover, for a.e. \(x,y\in \mathbb{R}^{N}\), one has
by the fact that \(\mu < a_{k,s}\). Hence, according to \((f_{1})\), \(f(x,u_{\lambda }(x))u_{\lambda }^{-}(x)=0\) for \(x\in \mathbb{R}^{N}\), we obtain
which implies that \(u_{\lambda }^{-}\equiv 0\). Hence, \(u_{\lambda } \geq 0\). It implies \((-\Delta)^{s} u_{\lambda }\geq 0\). Then, by the strong maximum principle, we obtain \(u_{\lambda }\) is a positive solution of problem (1). This completes the proof of Theorem 1.
4 Conclusion
In this paper, we devoted our study to the existence of solutions for a fractional elliptic problems with the Hardy-Sobolev-Maz’ya potential and critical nonlinearities. The approach of this paper is by the well-known mountain pass theorem. The nonlinear term f satisfies assumptions \((f_{1})\), \((f_{2})\) without the (AR) conditions. We established a new term \(l(x,t)\), which satisfies the (AR) conditions combined with the critical term \(\frac{\vert t\vert ^{2_{s}^{*}(\alpha)-1}}{\vert z\vert ^{ \alpha }}\) by using some analysis techniques. Then we overcame the compactness and obtained a positive solution of problem (1). Our results are new and the work established in this paper is of quite a general nature.
References
Nezza, ED, Palatucci, G, Valdinoci, E: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521-573 (2012)
Silvestre, L: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60(1), 67-112 (2007)
Servadei, R, Valdinoci, E: Mountain Pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389(2), 887-898 (2012)
Guliyev, VS, Omarova, MN, Ragusa, MA, Scapellato, A: Commutators and generalized local Morrey spaces. J. Math. Anal. Appl. 457(2), 1388-1402 (2018)
Scapellato, A: On some qualitative results for the solution to a Dirichlet problem in Local Generalized Morrey Spaces. 1798, UNSP 020138 (2017) doi:10.1063/1.4972730
Scapellato, A: Some properties of integral operators on generalized Morrey spaces. AIP Conf. Proc. 1863(21), 510004 (2017). doi:10.1063/1.4992662
Luo, H, Tang, X, Li, S: Mountain pass and linking type sign-changing solutions for nonlinear problems involving the fractional Laplacian. Bound. Value Probl. 2017, 108 (2017).
Shang, X, Zhang, J, Yang, Y: Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent. Commun. Pure Appl. Anal. 13(2), 567-584 (2014)
Servadei, R, Valdinoci, E: The Brezis-Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. 367(1), 67-102 (2015)
Fiscella, A, Bisci, GM, Servadei, R: Bifurcation and multiplicity results for critical nonlocal fractional Laplacian problems. Bull. Sci. Math. 140(1), 14-35 (2016)
Servadei, R, Valdinoci, E: Fractional Laplacian equations with critical Sobolev exponent. Rev. Mat. Complut. 28(3), 1-22 (2015)
Fiscella, A, Bisci, GM, Servadei, R: Multiplicity results for fractional Laplace problems with critical growth. Manuscr. Math. 1-20 (2016)
Abdellaoui, B, Medina, M, Peral, I, Primo, A: The effect of the Hardy potential in some Calderón-Zygmund properties for the fractional Laplacian. J. Differ. Equ. 260(11), 8160-8206 (2016)
Dipierro, S, Montoro, L, Peral, I, Sciunzi, B: Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential. Calc. Var. Partial Differ. Equ. 55(4), 1-29 (2016)
Barrios, B, Medina, M, Peral, I: Some remarks on the solvability of non-local elliptic problems with the Hardy potential. Commun. Contemp. Math. 16(4), 1350046 (2014)
Jiang, RT, Tang, CL: Semilinear elliptic problems involving Hardy-Sobolev-Maz’ya potential and Hardy-Sobolev critical exponents. Electron. J. Differ. Equ. 2016, 12 (2016)
Ding, L, Tang, CL: Existence and multiplicity of solutions for semilinear elliptic equations with Hardy terms and Hardy-Sobolev critical exponents. Appl. Math. Lett. 20(12), 1175-1183 (2007)
Bouchekif, M, Matallah, A: Multiple positive solutions for elliptic equations involving a concave term and critical Sobolev-Hardy exponent. Appl. Math. Lett. 22(2), 268-275 (2009)
Shang, YY: Existence and multiplicity of positive solutions for some Hardy-Sobolev critical elliptic equation with boundary singularities. Nonlinear Anal. 75(5), 2724-2734 (2012)
Cotsiolis, A, Tavoularis, N: Best constants for Sobolev inequalities for higher order fractional derivatives. J. Math. Anal. Appl. 295, 225-236 (2004)
Brézis, H, Lieb, E: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88(3), 486-490 (1983)
Fiscella, A, Valdinoci, E: A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 94, 156-170 (2014)
Acknowledgements
The authors thank the anonymous referee for the careful reading and some helpful comments.
Funding
This paper is supported by National Natural Science Foundation of China (No. 11661021), Innovation Group Major Program of Guizhou Province (No. KY[2016]029), Science and Technology Foundation of Guizhou Province (No. J[2014]2088).
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Cai, Z., Chu, C. & Lei, C. Existence of positive solutions for a fractional elliptic problems with the Hardy-Sobolev-Maz’ya potential and critical nonlinearities. Bound Value Probl 2017, 181 (2017). https://doi.org/10.1186/s13661-017-0912-8
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DOI: https://doi.org/10.1186/s13661-017-0912-8
MSC
- 35A15
- 35B09
- 35B33
- 35R11
Keywords
- fractional operator
- Hardy-Sobolev-Maz’ya potential
- critical nonlinearities
- positive solution