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On conformal invariants for elliptic systems with multiple critical exponents. (English) Zbl 1150.35053

The article addresses the following question: {it Prescribed Curvatures Problem (PCP)} Let \((M,g)\) be a compact Riemannian manifold of dimension \(n>2\) with boundary \(\partial M\). Let the corresponding scalar curvature of \(M\) be \(4\frac{n-1}{n-2}r\) and the mean curvature of boundary \(\partial M\) be \(\frac{2}{n-2}h\). Suppose that a function \(R\) is defined on \(M\) and a function \(H\) is defined on \(\partial M\). Can one find a related conformal metric \(g'=\rho g\) with \(\rho >0\) on \(M\) such that in this metric the scalar curvature of \(M\) is equal to \(4\frac{n-1}{n-2}R\) and the mean curvature of \(\partial M\) is equal to \(\frac{2}{n-2}H\)?
For compact manifolds without boundary the PCP corresponds to the Yamabe Problem. In [J. Differ. Geom. 35, No. 1, 21–84 (1992; Zbl 0771.53017); Ann. Math. (2) 136, No. 1, 1–50 (1992; Zbl 0766.53033); Calc. Var. Partial Differ. Equ. 4, No. 6, 559–592 (1996; Zbl 0867.53034)], J. F. Escobar studied the PCP problem for the following three cases: (i) \(R=\text{const}\), \(H=0\); (ii) \(H=\)const, \(R=0\); (iii) \(H=0\), \(R=f(x)\) and the energy \(E(u)\) defined below is positive.
The energy functional is \[ E(u)=\int_{M}\left(| \nabla u|^{2} - r(x)| u|^{2}\right) \,d\mu_{g}- \int_{\partial M}h(x)| u|^{2}\,d\nu_{g}, \] where \(d\mu_{g}\) and \(d\nu_{g}\) are the Riemannian measures induced by \(g\) on \(M\) and \(\partial M\), respectively. There is always a function \(u\in C^{\infty}(M)\) such that \(E(u)>0\). We suppose that:
Condition A. The sign of \(E\) is indefinite, i.e., there exists \(u\in C^{\infty}(M)\) such that \(E(u)<0\).
The crucial element of Escobar’s articles is the Yamabe Invariants that are defined when \(R\neq 0\), \(H=0\) as (they are Sobolev quotients for the energy) \[ \begin{alignedat}{2} Q(M,g) &= \inf_{u\in C^{\infty}(M)}\frac{E(u)}{\left(\int_{M}R(x)| u|^{n^{*}}\,d\mu_{g} \right)^{\frac{n-2}{n}}}, \quad&n^{*} &=\frac{2n}{n-2},\tag{1}\\ Q(\partial M,g) &= \inf_{u\in C^{\infty}(M)}\frac{E(u)}{\left(\int_{\partial M}H(x)| u|^{n^{**}}\,d\nu_{g} \right)^{\frac{n-2}{n-1}}}, \quad&n^{**} &=\frac{2(n-1)}{n-2}.\tag{2} \end{alignedat} \]
They both are conformal invariants. Note that \(n^{*}\) is the critical Sobolev exponent for the embedding \(W^{1,2}(M)\subset L^{p}(M)\) and \(n^{**}\) is one for the trace-embedding \(W^{1,2}(M)\subset L^{p}(\partial M)\). The authors also address the question: Are there other conformal invariants corresponding to the energy functional?, mainly for the case when \(R\neq 0\), \(H\neq 0\) and when the energy can change sign.
In the cases considered by Escobar in [loc. cit.], the Yamabe invariants \(Q(M,g)\) and \(Q(\partial M,g)\) are bounded below by a positive constant. The case when \(R=0\) and \(E(u)>0\) has been studied in [loc. cit.]. In the latter case the Yamabe invariant \(Q(\partial M,g)\) can vanish.
The sign of the first-eigenvalue \(\lambda_{g}\) of the problem \[ \begin{alignedat}{2} -\Delta_{g}u+r(x)u &= 0 \quad&&\text{in }M, \\ \frac{\partial u}{\partial N} +h(x)u &= \lambda_{g}u,\quad&&\text{in }\partial M, \end{alignedat} \] is a conformal invariant [loc. cit.]. Moreover, under the Condition A, we have \(\lambda_{g}<0\).
The authors extended the boundary value problem in a more general setting, when \(R\) and \(H\) are arbitrary continuous functions and the energy can change sign. Namely, the function \(u\) defining the conformal map is a positive solution to the elliptic problem \[ \begin{alignedat}{2} -\Delta_{g}u+r(x)u &= R(x)u^{n^{*}-1} \quad&&\text{in }M,\\ \frac{\partial u}{\partial n} = h(x)u + H(x)u^{n^{**}-1} &= \lambda_{g}u \quad&&\text{in }\partial M, \end{alignedat}\tag{3} \] where \(\Delta_{g}\) is the Laplace-Beltrami operator, \(\nabla_{g}\) is the gradient in the metric \(g\), \(\frac{\partial }{\partial n}\) is the normal derivative with respect to the outward normal \(n\) on \(\partial M\) and the metric \(g\), moreover, \(R,r\in C^{\infty}(M)\) and \(h,H\in C^{\infty}(\partial M)\).
The elliptic boundary problem (3) can be stated in the variational form since the solution of the problem corresponds to a critical point of the Euler functional \[ I(u)=\frac{1}{2}E(u) - \frac{1}{n^{*}}B(u)-\frac{1}{n^{*}}F(u),\tag{4} \] where \[ \begin{aligned} F(u) &= \int_{M}R(x)| u|^{n^{*}}\,d\mu_{g}, \\ B(u) &= \int_{\partial M} H(x)| u|^{n^{**}}\,d\nu_{g}. \end{aligned} \]
Thus the functional \(I\) does not satisfy the Palais-Smale condition on all levels. In the cases studied by Escobar, the limiting Palais-Smale levels \(C_{P-S}\) are expressed in terms of the Sobolev quotients \(Q(S^{n})\) and \(Q(\partial S^{n,+})\) for thew sphere \(S^{n}\) and the semi-sphere \(S^{n,+}\), respectively. The authors consider the problem of finding the limiting Palais-Smale levels \(C_{P-S}\) in the general case, which in turn addresses the question “What is the Sobolev quotient in the cases when both critical nonlinear terms present on the manifold and on the boundary?”
It is also discussed the problem of the existence of multiple positive solutions to the geometrical problem on prescribed curvatures, which yields the new conformal invariants, important for the classification of manifolds.
In order to enunciate the main results some technical conditions must be described:
Condition B. \(R(x)\equiv 0\).
Condition C. \(H(x)=\mu H^{+}(x)-H^{-}(x)\), where \(H^{+}(x)>0\) as \(x\in \partial M\), \(\mu\geq 0\), and the set \(\{x\in M\mid H^{+}(x)>0\}\) is non-empty.
Condition D. \(n>5\), \(H(x)\) achieves a global maximum at a nonumbilic point of the boundary \(\partial M\ni 0\), where \(\nabla H(0)=0\), the second derivatives \(\frac{\partial^{2}H} {\partial x_{i}\partial x_{j}}(0)\) are defined and \[ | \Delta H(0)| \leq c(n)\|\pi(0)-h(0)g(0)\|, \] where \(c(n)\) is a suitable constant (recall that \(p\in\partial M\) is umbilic if the tensor \((\pi-hg)(p)=0\); \(\pi\) is the second fundamental form of \(\partial M\)).
Let \(B^{\pm}(u)=\int_{\partial M}H^{\pm}(x)| u|^{n^{**}}d\nu_{g}\). Consider \[ \mu_{1}=\inf_{u\in C^{\infty}(M)}\left(\frac{B^{-}(u)}{B^{+}(u)}:E(u)\leq 0\right). \] It follows that \(0\leq \mu_{1}\leq \infty\), and \(\mu_{1}=\infty\) if, and only if, \(\partial M^{+}=\{x\in\partial M\mid H(x)>0\}=\emptyset\).
Assume \(\partial M^{+}\neq \emptyset\). Let \(\lambda_{g}(\partial M^{+})\) be the first eigenvalue of the problem \[ \begin{alignedat}{2} -\Delta_{g}\psi(\partial M^{+}) + r(x)\psi(\partial M^{+}) &=0 \quad&&\text{in }M,\\ \frac{\partial \psi(\partial M^{+})}{\partial n} + h(x)\psi(\partial M^{+}) &= \lambda_{g}(\partial M^{+})\psi(\partial M^{+}) \quad&&\text{on }\partial M^{+},\\ \psi(\partial M^{+}) &=0 \quad&&\text{on }\partial M^{-}. \end{alignedat}\tag{5} \] The sign of \(\lambda_{g}(\partial M^{+})\) is a conformal invariant, as shown in [loc. cit.].
The first and the second main results are
Lemma. Suppose that \(n\geq 3\). Assume that conditions B–C hold. Then
(i)
the characteristic value \(\mu_{1}\) is a conformal invariant;
(ii)
\(\mu_{1} >0\) if, and only if, \(\lambda_{g}(\partial M^{+})>0\);
(iii)
if \(\mu_{1}=0\), then the problem (1) has no positive solution for any \(\mu>\mu^{*}\).
Theorem. Suppose that \(n\geq 3\). Assume A–C hold and \(\lambda_{g}(\partial M^{+})>0\). Then,
(i)
for every \(\mu\in(0,\mu_{1}]\) the problem (1) has a positive solution \(u^{1}_{\mu}\) such that \(I_{\mu}(u^{1}_{\mu})<0\);
(ii)
if condition D holds too, then there exists \(\varepsilon>0\) such that for every \(\mu\in(\mu_{1}-\varepsilon,\mu_{1}]\) the problem (1) has another positive solution \(u^{2}_{\mu}\) such that \(I_{\mu}(u^{2}_{\mu})>0\) if \(\mu<\mu_{1}\) and \(I_{\mu_{1}}(u^{2}_{\mu_{1}}=0\).
Condition E. \(R(x)\geq 0\) on \(M\), \(\int_{M}R(x)\,dx>0\).
The authors consider the problem with parameters \(\lambda\geq 0\) at \(R(x)\) and \(\mu>\geq 0\) at \(H^{+}(x)\), i.e , they consider the critical point of \[ I(u)=\frac{1}{2}E(u) - \frac{1}{n^{*}}B_{\mu}(u)-\frac{1}{n^{*}}\lambda F(u). \] Letting \(\mu>0\), they consider \[ \Lambda^{1}_{\mu}= \inf_{u\in C^{\infty}(M)}\left(\frac{-B^{2}_{\mu}(u)}{4E(u)F(u)}\;\bigg|\;E(u)<0\right) \] in order to prove the third main result.
Theorem. Suppose that \(n\geq 3\) and A–C, E hold. Assume that \(\mu_{1}>0\). Then for every \(\mu\in (0,\mu_{1}]\)
(i)
the characteristic value \(\Lambda^{1}_{\mu}\) is a conformal invariant;
(ii)
for every \(\lambda\in[0,\Lambda^{1}_{\mu}]\) the problem (1) has a psotive solution \(u^{1}_{\lambda,\mu}\) such that \(I_{\lambda,\mu}(u^{1}(\lambda,\mu)<0\) and this value is a conformal invariant;
(iii)
if condition D holds too, then there exists \(\varepsilon>0\) such that for every \(\mu\in(\mu_{1}-\varepsilon,\mu_{1}]\) there exists \(\lambda_{\mu}>0\), such that for every \(\lambda\in (0,\lambda_{\mu})\) problem (1) has a second positive solution \(u^{2}_{\lambda,\mu}\) for which \(I_{\lambda,\mu}(u^{2}(\lambda,\mu)>0\) and this value is a conformal invariant.
Along the prove the two new conformal invariants \(Q^{1}_{\lambda,\mu}(M,\partial M,g)\) and \(Q^{2}_{\lambda,\mu}(M,\partial M,g)\) are introduced. In the cases studied by Escobar, and for certain values of \(\mu\) and \(\lambda\), the invariant \(Q^{2}_{\lambda,\mu}(M,\partial M,g)\) coincide with the Yamabe invariant \(Q(M,g)\) or \(Q(\partial M,g)\).

MSC:

35J70 Degenerate elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
58J32 Boundary value problems on manifolds
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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