Abstract
Using Brouwer’s fixed point theorem, we prove the existence of solutions for some nonlinear problem with subcritical Sobolev exponent in \(S_{+}^{4}\).
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1 Introduction and the main result
The exponent Lebesgue space \(L^{p}(\Omega)\) is defined by
This space is endowed with the norm
The Sobolev space \(W^{1,p}(\Omega)\) is defined by
The corresponding norm for this space is
Define \(W_{0}^{1}(\Omega)=H_{0}^{1}(\Omega)\) as the closure of \(C_{c}^{\infty}(\Omega)\) with respect to the \(W^{1,p}(\Omega)\) norm which is a Hilbert space [1].
We consider the problem of the scalar curvature on the standard four dimensional half sphere under minimal boundary conditions \((S)\):
where \(S^{4}_{+}= \{ x\in \mathbb{R}^{5} / \vert x\vert = 1, x_{5} >0 \}\), g is the standard metric, and K is a \(C^{3}\) positive Morse function on \(\overline{S_{+}^{4}}\).
The scalar curvature problem on \(S^{n}\) and also on \(S^{n}_{+}\) was the subject of several works in recent years, we can cite for example [2–12].
Recall that the embedding of \(H^{1}(S^{4}_{+})\) into \(L^{4}(S^{4}_{+})\) is noncompact. For this reason, we have focused our study on the family of subcritical problems \((S_{\varepsilon})\)
where ε is a small positive parameter.
Note that the solutions of problem \((S)\) can be the limit as \(\varepsilon\to0\) of some solutions \((u_{\varepsilon})\) for \((S_{\varepsilon})\).
Djadli et al. [13] studied this problem in the case of the three dimensional half sphere. Assuming that the critical points of \(K_{1}\) verify \((\partial K/\partial\nu)(a_{i})>0\) they demonstrated that there exist solutions \((u_{\varepsilon})\) concentrated at the points \((a_{1}, \ldots, a_{p})\). Moreover, in [14], we established the existence of another type of solutions \((u_{\varepsilon})\) of \((S_{\varepsilon})\) such that is concentrated at two points \(a_{1}\in \partial S^{4}_{+}\) and \(a_{2}\in S^{4}_{+}\).
In this work, we aim to construct some positive solutions of \((S_{\varepsilon})\) which are concentrated at two different points of the boundary. To state our result, we will give the following notations. For \(a\in \overline{S_{+}^{4}}\) and \(\lambda>0\), let
where d is the geodesic distance on \((\overline{S_{+}^{4}},g)\) and \(c_{0}\) is chosen so that \(\delta_{(a,\lambda) }\) is the family of solutions of the following problem:
The space \(H^{1}(S_{+}^{4})\) is equipped with the norm \(\Vert \cdot \Vert \) and its corresponding inner product \(\langle\cdot ,\cdot \rangle\):
Theorem 1
Let \(z_{1}\) and \(z_{2}\) be a nondegenerate critical points of \(K_{1}=K_{\mid\partial S^{4}_{+}}\) with \((\partial K/\partial\nu)(z_{i}) >0\), \(i=1,2\). Then there exists \(\varepsilon_{0}>0\) such that, for each \(\varepsilon\in(0,\varepsilon_{0})\), problem \((S_{\varepsilon})\) has a solution \((u_{\varepsilon})\) of the form
where, as \(\varepsilon\rightarrow0\), \(\alpha_{i}\rightarrow K(z_{i})^{-1/2}\); \(\Vert v\Vert \rightarrow0\); \(x_{i}\rightarrow z_{i}\); \(x_{i}\in\partial S^{4}_{+}\); \(\lambda_{i}\rightarrow+\infty\); \(\lambda_{1}=c\lambda_{2}(1+o(1))\).
The rest of this work is summarized as follows. In Section 2, we present a classical preliminaries and we perform a useful estimations of functional \((I_{\varepsilon})\) associated to the problem \((S_{\varepsilon})\) for \((\varepsilon> 0)\) and its gradient. Section 3 is devoted to the construction of solutions and the proof of our result.
2 Useful estimations
We introduce the structure variational associated to the problem \((S_{\varepsilon})\) for \(\varepsilon> 0\)
It is well known that there is an equivalence between the existence of solutions for \((S_{\varepsilon})\) and the positive critical point of \(I_{\varepsilon}\). Moreover, in order to reduce our problem to \(\mathbb{R}^{4}_{+}\) we will perform some stereographic projection. We denote \(D^{1,2}(\mathbb{R}^{4}_{+})\) for the completion of \(C^{\infty}_{c} (\overline{\mathbb{R}^{4}_{+}})\) with respect to the Dirichlet norm. Recall that an isometry \(\i: H^{1}(S^{4}_{+}) \to D^{1,2}(\mathbb{R}^{4}_{+})\) is induced by the stereographic projection \(\pi_{a}\) about a point \(a \in\partial S^{4}_{+}\) following the formula
For every \(\phi\in H^{1}(S^{4}_{+})\), one can check that the following holds true:
Furthermore, using (3) with \(\pi_{-a}\), it is easy to see that \(\i\delta_{(a,\lambda)}\) is given by
\(\delta_{(a,\lambda)}\) will be written instead of \(\i\delta_{(a,\lambda)}\) in the sequel.
Let
where \(\nu_{0}\) is a small positive constant, σ, \(c_{0}\), \(d_{0}\) are some suitable positive constants, and
Here, \(x^{j}_{i}\) denotes the jth component of \(x_{i}\). Also
In the sequel, we will write \(\delta_{i}\) instead of \(\delta_{(x_{i},\lambda_{i})}\) and \(u=\alpha_{1}\delta_{1}+\alpha_{2}\delta _{2}+v\) for the sake of simplicity.
In the remainder of this section, we will give expansions of the gradient of the functional \(I_{\varepsilon}\) associated to \((S_{\varepsilon})\) for \(\varepsilon>0\). Thus estimations are needed in Section 3. We need to recall that [15] proved the following remark when \(n= 3\), but the same argument is available for the dimension 4.
Remark 2
For \(\varepsilon>0\) and \(\delta _{(a,\lambda)}\) defined in (1), we have
Now, explicit computations, using Remark 2, yield the following propositions.
Proposition 3
Let \((\alpha,\lambda,x,v) \in M_{\varepsilon}\). Then, for \(u=\alpha_{1}\delta_{(x_{1},\lambda _{1})}+\alpha_{2}\delta_{(x_{2},\lambda_{2})}+v\), we have the following expansion:
where
Proof
We have
A computation similar to the one performed in [16] shows that, for \(i=1,2\),
and
For the other integral, we write
We also write
Expansions of K around \(x_{i}\) and \(x_{j}\) give
Combining (5)-(12), we derive our proposition. □
Proposition 4
Let \((\alpha,\lambda,x,v) \in M_{\varepsilon}\). Then, for \(u=\alpha_{1}\delta_{(x_{1},\lambda _{1})}+\alpha_{2}\delta_{(x_{2},\lambda_{2})}+v\), we have
where
Proof
Observe that (see [16])
For the other part, we have the expansions of K around \(x_{i}\) and using Remark 2,
Combining (5), (13), (14), (15), (16), and (17), the proof follows. □
Proposition 5
Let \((\alpha,\lambda,x,v) \in M_{\varepsilon}\). Then, for \(u=\alpha_{1}\delta_{(x_{1},\lambda _{1})}+\alpha_{2}\delta_{(x_{2},\lambda_{2})}+v\), we have the following expansion:
where
Proof
We have
For the other part
3 Construction of the solution
The method of this type of theorem was followed first by Bahri, Li and Rey [17] when they studied an approximation problem of the Yamabe-type problem on domains. Many authors used this idea to construct some solutions to other problems. The method becomes standard. Here we will follow the idea of [17] and take account of the new estimates since we have an equation different from the one studied in [17]. From the idea of [17], using the coefficients of Euler-Lagrange, we obtain
Proposition 6
Let A point \(m=(\alpha,\lambda,x,v)\in M_{\varepsilon}\) is a critical point of the function \(\Psi_{\varepsilon}\) if and only if \(u=\alpha_{1}\delta_{1}+\alpha_{2}\delta_{2}+v\) is a critical point of functional \(I_{\varepsilon}\), which means the existence of some \((A,B,C )\in \mathbb{R}^{2}\times\mathbb{R}^{2}\times (\mathbb{R}^{4} )^{2}\) with the following:
Now, by a careful study of equation \((E_{v})\), we get the following.
Proposition 7
[12] For any \((\varepsilon,\alpha,\lambda,x)\) with \((\alpha,\lambda,x,0)\in M_{\varepsilon}\), there exists a smooth map which associates \(\overline{v} \in E_{(x,\lambda)}\) with \(\Vert \overline{v}\Vert <\nu_{0}\) and equation (26) in the previous proposition is verified for some \((A,B,C )\in\mathbb{R}^{2}\times\mathbb{R}^{2}\times (\mathbb{R}^{4} )^{2}\). Such a v̅ is unique, minimizes \(\Psi_{\varepsilon}(\alpha ,\lambda,x,v)\) with respect to v in \(\{v\in E_{(x,\lambda)}/\Vert v\Vert <\nu_{0}\}\), and
Proof of Theorem 1
Once v̅ is defined by Proposition 7, we estimate the corresponding numbers A, B, C by taking the scalar product in \(H^{1}(S^{4}_{+})\) of \((E_{v})\) with \(\delta_{i}\), \({\partial\delta _{i}}/{\partial \lambda_{i}}\), \({\partial\delta_{i}}/{\partial x_{i}}\) for \(i=1,2\), respectively. So we get the following coefficients of a quasi-diagonal system:
with \(\vert x_{1}-x_{2}\vert \geq c >0\) and \(\Gamma_{1}\), \(\Gamma_{2}\) are positive constants.
We have also
Using Propositions 3, some computations yield
with \(\beta_{i}=\alpha_{i}-1/K(z_{i})^{\frac{1}{2}}\) and
In the same way, using Propositions 4, we get
where \(c_{2}\) and \(c_{3}\) are defined in Proposition 4 and
Lastly, using Propositions 5, we have
where
From these estimates, we deduce
By solving the system in A, B, and C, we find
Now, we can evaluate the right hand side in \((E_{\lambda_{i}})\) and \((E_{x_{i}})\),
where
Now, we consider a point \((z_{1},z_{2})\in \partial S^{4}_{+}\times\partial S^{4}_{+}\) such that \(z_{1}\) and \(z_{2}\) are nondegenerate critical points of \(K_{1}\). We set
where \(\zeta_{i} \in\mathbb{R}\) and \((\xi_{1},\xi_{2})\in\mathbb {R}^{3}\times\mathbb{R}^{3}\) are assumed to be small.
Using (28) and these changes of variables, \((E_{\alpha_{i}})\) becomes
Also, we use (30), we have
Combining this with (31), then \((E_{\lambda_{i}})\) becomes
Using (32), (33), and (36), \((E_{x_{i}})\) is equivalent to
Observe that the functions \(V_{\alpha_{i}}\), \(V_{\lambda_{i}}\), and \(V_{x_{i}}\) are smooth.
We can also write the system as
where L is a fixed linear operator on \(\mathbb{R}^{8}\) defined by
and V, W are smooth functions satisfying
Now, by an easy computation, we see that the determinant of the linear operator L is not 0. Hence L is invertible, and according to Brouwer’s fixed point theorem, there exists a solution \((\beta^{\varepsilon},\zeta^{\varepsilon},\xi^{\varepsilon})\) of (40) for ε small enough, such that
Hence, we have constructed \(m^{\varepsilon}= (\alpha_{1}^{\varepsilon},\alpha_{2}^{\varepsilon},\lambda_{1}^{\varepsilon}, \lambda_{2}^{\varepsilon},x_{1}^{\varepsilon},x_{2}^{\varepsilon})\) such that \(u_{\varepsilon}:= \sum\alpha_{i}^{\varepsilon}\delta_{(x_{i}^{\varepsilon},\lambda _{i}^{\varepsilon})}+\overline{v_{\varepsilon}}\), verifies (23)-(27). From Proposition 6, \(u_{\varepsilon}\) is a critical point of \(I_{\varepsilon}\), which implies that \(u_{\varepsilon}\) verify
We multiply equation (41) by \(u_{\varepsilon}^{-}=\max(0,-u_{\varepsilon})\) and we integrate on \(S^{4}_{+}\), we get
We know also from the Sobolev embedding theorem that
Equations (42) and (43) imply that either \(u_{\varepsilon}^{-}\equiv 0\), or \(\vert u_{\varepsilon}^{-}\vert _{4-\varepsilon}\) is far away from zero. Since \(m^{\varepsilon}\in M^{\varepsilon}\), we have \(\Vert \overline{v_{\varepsilon}} \Vert <\nu_{0}\), where \(\nu_{0}\) is a small positive constant (see the definition of \(M_{\varepsilon}\)). This implies that \(\vert u_{\varepsilon}^{-}\vert _{4-\varepsilon}\) is very small. Thus, \(u_{\varepsilon}^{-}\equiv0\) for ε small enough. Then \(u_{\varepsilon}\) is a non-negative function which satisfies (41). Finally, the maximum principle completes the proof of our theorem. □
4 Conclusion
Thus it has been concluded that under some assumptions on the function K, there exist solutions of the nonlinear problem \((S_{\varepsilon})\) which are concentrated at two different points of the boundary.
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Acknowledgements
I would like to thank Deanship of Scientific Research at Taibah University for the financial support of this research project.
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Jebril, I.H. Nonlinear problem with subcritical exponent in Sobolev space. J Inequal Appl 2016, 305 (2016). https://doi.org/10.1186/s13660-016-1245-3
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DOI: https://doi.org/10.1186/s13660-016-1245-3