##
**Perturbation methods and semilinear elliptic problems on \(\mathbb R^n\).**
*(English)*
Zbl 1115.35004

Progress in Mathematics 240. Basel: Birkhäuser (ISBN 3-7643-7321-0/hbk). ix, 183 p. (2006).

The introductory chapter 1 is devoted to the statements of the main nonlinear variational problems which are studied in more detail in the monograph. These are the following.

1\(^\circ\). The existence of solutions problem of elliptic equation on \(\mathbb R^n\), \(-\Delta u+u=b(x)u^p\), \(0<u\in W^{1,2}(\mathbb R^n)\), \(n\geq 3\), \(1<p<\frac{n+2}{n-2}\) subcritical case and \(p=\frac{n+2}{n-2}\) – critical case, where the main difficulties is the lack of the Palais-Smale (PS) compactness condition;

2\(^\circ\). Bifurcation from the essential spectrum for the equation \(F(\lambda,x)=0\), \(F:\mathbb R^n\times H\to H\), \(F(\lambda,0)\equiv 0\), \(\forall\lambda\in\mathbb R\), in a Hilbert space \(H\), when unlike to the compact case, in the presence of the essential spectrum of \(F'(\lambda,0)\) one tires to show that the infimum of such a spectrum is still a bifurcation point.

Typical example of such a situation is the problem \(\psi''+\lambda\psi+h(x)|\psi|^{p-1}\psi=0\), \(\lim_{|x|\to 0}\psi(x)=0\), \(p>1\), where from the bottom \(\lambda=0\) of the essential spectrum of \(\psi''+\lambda\psi\), \(\psi\in W^{1,2}(\mathbb R)\), bifurcates a family of solutions \((\lambda,\psi_\lambda)\), \(\lambda<0\) of the nonlinear equation with \((\lambda,\psi_\lambda)\to (0,0)\) as \(\lambda\downarrow 0\).

3\(^\circ\). Semiclassical standing waves of nonlinear Schrödinger equation (NLS) \(\frac{\partial\psi}{\partial t}=-\hbar^2\Delta\psi+(a_0+Q(x))\psi+a_1|\psi|^{p-1}\psi\), \(a_1<0\), arising in plasma physics, in nonlinear optics at the presence of self-focusing material and in the Ginzburg-Landau theory with the nonlinearity of the form \(|\psi|^2\psi-|\psi|^4\psi\) etc. Here various perturbation problems with solutions concentrating at higher-dimensional manifolds arise.

4\(^\circ\). Other problems whose main feature is that they possess solutions concentrating at points or at manifolds, such as Neumann singularly perturbed problems and radial problems with concentration on spheres.

5\(^\circ\). The abstract setting of the above mentioned variational problems.

In Ch. 2 some abstract results on the existence of critical points of perturbed functionals \(I_\varepsilon\) on a Hilbert space \(H^1\) are proved. An review on critical point theory is given containing various variants of the Mountain-Pass lemma (M-PTh) and introduced by P.L. Lions’ Concentration-Compactness principle (CCP).

Sec. 2.2 deals with critical points of the perturbed functionals \(I_\varepsilon u=I_0(u)+\varepsilon G(u)\), where \(I_0\in C^2(H,\mathbb R)\) and \(G\in C^2(H,\mathbb R)\) is the perturbation. It is supposed the existence of smooth \(d\)-dimensional, \(0<d<\infty\), critical manifold \(Z\) of \(I_0\) not satisfying in general PS condition, but for all \(z\in Z\) \(I_0''(z)\) is an index \(0\) Fredholm map. It is suggested a finite-dimensional reduction by the Lyapunov-Schmidt method revisited. On the base of the reduced functional \(\Phi_\varepsilon=I_{\varepsilon}(z+w_\varepsilon (z)):Z\to\mathbb R\) introduction the series of critical points existence theorems is proved.

In Sec. 2.3 critical points of the more general class of perturbed functionals \(I_\varepsilon(u)=I_0(u)+G(\varepsilon,u)\) are studied with relevant existence theorems.

In Ch. 3 the perturbation techniques in particular those discussed in Sec. 2.3 are applied to study the problems concerning the bifurcation from the infimum of the essential spectrum with applications to nonlinear optics.

Ch. 4 is devoted to the elliptic problem on \(\mathbb R^n\) \(-\Delta u+u=(1+\varepsilon h(x))u^p\), \(0<u\in W^{1,2}(\mathbb R^n)\), \(n\geq 3\), in the case of subcritical growth \(1<p<\frac{n+2}{n-2}\), where several existence results are proved.

Ch. 5 studies elliptic problems \(-\Delta u=u^{(n+2)/(n-2)}+\varepsilon k(x)u^q\), \(1\leq q\leq (n+2)/(n-2)\), first the unperturbed problem and further the critical case \(q=(n+2)/(n-2)\). The existence results for the critical case are applied in Chapters 5, 6 to the problems arising in differential geometry connected with Yamabe-Like equation.

Ch. 6 is completely devoted to the Yamabe problem of finding on a Riemannian manifold \((M,g)\), \(n\geq 3\) a conformal metric with constant scalar curvature \(R_0\in\mathbb R\), that is equivalent to the finding solutions to the equation \(-2c_n\Delta_gu+R_gu=R_0u^{\frac{n+2}{n-2}}\), \(u>0\) on \(M^l\) (*), where \(R_g=R_{kj}g^{kj}\), \(g^{ij}=(g)^{-1}_{ij}\), is the scalar curvature, \(c_n=2(n-1)/(n-2)\). After description of various predecessors results the authors starting from the sphere \(S^n\) in high dimensions and then by improving the technique, to obtain noncompactness of solutions in the case of some metrics of class \(C^k\) on \(S^n\) prove the following two theorems.

Theorem 6.2: Let \(n\geq 6\) and \(l\geq 2\). Then there exists a family of smooth metrics \(\bar{g}_\varepsilon\) on \(S^n\), converging ( in \(C^\infty(S^n)\)) to \(\bar{g}_0\) as \(\varepsilon\to 0\) such that, for every small enough \(\varepsilon\), the problem (*) on \((S^n,\bar{g}_\varepsilon)\) possesses at least \(l\) solutions.

Theorem 6.3: Let \(k\geq 2\) and \(n\geq 4k+3\). Then there exists a family of \(C^k\) metrics \(\bar{g}_\varepsilon\) on \(S^n\), with \(\|\bar{g}_\varepsilon-\bar{g}_0\|_{C^k(S^n)}\to 0\) as \(\varepsilon\to 0\) with the following property: for every small enough \(\varepsilon>0\) the problem (*) on \((S^n,\bar{g}_\varepsilon)\) possesses a sequence of solutions \(v^i_\varepsilon\) with \(\|v^i_\varepsilon\|\to 0\) as \(i\to\infty\).

Ch. 7 surveys some other problems arising in conformal geometry, first paying attention on the scalar curvature problem for the standard sphere, next to some problem on manifold with boundary.

In Ch. 8 standing waves of the NLS equation are studied, namely solutions to the following problem \(-\varepsilon^2\Delta u+V(x)u=u^p\) in \(\mathbb{R}^n\), \(0<u\in W^{1,2}(\mathbb R^n)\), where \(p>1\) is subcritical and \(V\) is a smooth bounded potential, interesting in the behaviour of the solutions as \(\varepsilon\to 0^+\) (semiclassical limit). It is shown that there exist spikes, i.e. solutions concentrating at single points of \(\mathbb R^n\).

Ch. 9 is devoted to singularly perturbed Neumann Problems on a bounded domain \(\Omega\subset\mathbb R^n\): \(-\varepsilon^2\Delta u+u=u^p\) in \(\Omega\) (**), \(\frac{\partial u}{\partial V}\) on \(\partial\Omega\), \(u>0\) in \(\Omega\), where \(p>1\) is subcritical. The following result is obtained: suppose \(\Omega\subseteq\mathbb R^n\), \(n\geq 2\) is a smooth bounded domain, \(1<p<\frac{n+2}{n-2}\) (\(1<p<\infty\) if \(n=2\)) and \(X_0\in\partial\Omega\) is a local strict maximum or minimum, or a non-degenerate critical point of the mean curvature \(H\) of \(\partial\Omega\); then for sufficiently small \(\varepsilon>0\) the problem (**) admits a solution concentrating at \(X_0\).

In the final Ch. 10 the results on the existence of solutions of NLS and singularly perturbed Neumann problems concentrating at spheres, in the radial case, are presented.

1\(^\circ\). The existence of solutions problem of elliptic equation on \(\mathbb R^n\), \(-\Delta u+u=b(x)u^p\), \(0<u\in W^{1,2}(\mathbb R^n)\), \(n\geq 3\), \(1<p<\frac{n+2}{n-2}\) subcritical case and \(p=\frac{n+2}{n-2}\) – critical case, where the main difficulties is the lack of the Palais-Smale (PS) compactness condition;

2\(^\circ\). Bifurcation from the essential spectrum for the equation \(F(\lambda,x)=0\), \(F:\mathbb R^n\times H\to H\), \(F(\lambda,0)\equiv 0\), \(\forall\lambda\in\mathbb R\), in a Hilbert space \(H\), when unlike to the compact case, in the presence of the essential spectrum of \(F'(\lambda,0)\) one tires to show that the infimum of such a spectrum is still a bifurcation point.

Typical example of such a situation is the problem \(\psi''+\lambda\psi+h(x)|\psi|^{p-1}\psi=0\), \(\lim_{|x|\to 0}\psi(x)=0\), \(p>1\), where from the bottom \(\lambda=0\) of the essential spectrum of \(\psi''+\lambda\psi\), \(\psi\in W^{1,2}(\mathbb R)\), bifurcates a family of solutions \((\lambda,\psi_\lambda)\), \(\lambda<0\) of the nonlinear equation with \((\lambda,\psi_\lambda)\to (0,0)\) as \(\lambda\downarrow 0\).

3\(^\circ\). Semiclassical standing waves of nonlinear Schrödinger equation (NLS) \(\frac{\partial\psi}{\partial t}=-\hbar^2\Delta\psi+(a_0+Q(x))\psi+a_1|\psi|^{p-1}\psi\), \(a_1<0\), arising in plasma physics, in nonlinear optics at the presence of self-focusing material and in the Ginzburg-Landau theory with the nonlinearity of the form \(|\psi|^2\psi-|\psi|^4\psi\) etc. Here various perturbation problems with solutions concentrating at higher-dimensional manifolds arise.

4\(^\circ\). Other problems whose main feature is that they possess solutions concentrating at points or at manifolds, such as Neumann singularly perturbed problems and radial problems with concentration on spheres.

5\(^\circ\). The abstract setting of the above mentioned variational problems.

In Ch. 2 some abstract results on the existence of critical points of perturbed functionals \(I_\varepsilon\) on a Hilbert space \(H^1\) are proved. An review on critical point theory is given containing various variants of the Mountain-Pass lemma (M-PTh) and introduced by P.L. Lions’ Concentration-Compactness principle (CCP).

Sec. 2.2 deals with critical points of the perturbed functionals \(I_\varepsilon u=I_0(u)+\varepsilon G(u)\), where \(I_0\in C^2(H,\mathbb R)\) and \(G\in C^2(H,\mathbb R)\) is the perturbation. It is supposed the existence of smooth \(d\)-dimensional, \(0<d<\infty\), critical manifold \(Z\) of \(I_0\) not satisfying in general PS condition, but for all \(z\in Z\) \(I_0''(z)\) is an index \(0\) Fredholm map. It is suggested a finite-dimensional reduction by the Lyapunov-Schmidt method revisited. On the base of the reduced functional \(\Phi_\varepsilon=I_{\varepsilon}(z+w_\varepsilon (z)):Z\to\mathbb R\) introduction the series of critical points existence theorems is proved.

In Sec. 2.3 critical points of the more general class of perturbed functionals \(I_\varepsilon(u)=I_0(u)+G(\varepsilon,u)\) are studied with relevant existence theorems.

In Ch. 3 the perturbation techniques in particular those discussed in Sec. 2.3 are applied to study the problems concerning the bifurcation from the infimum of the essential spectrum with applications to nonlinear optics.

Ch. 4 is devoted to the elliptic problem on \(\mathbb R^n\) \(-\Delta u+u=(1+\varepsilon h(x))u^p\), \(0<u\in W^{1,2}(\mathbb R^n)\), \(n\geq 3\), in the case of subcritical growth \(1<p<\frac{n+2}{n-2}\), where several existence results are proved.

Ch. 5 studies elliptic problems \(-\Delta u=u^{(n+2)/(n-2)}+\varepsilon k(x)u^q\), \(1\leq q\leq (n+2)/(n-2)\), first the unperturbed problem and further the critical case \(q=(n+2)/(n-2)\). The existence results for the critical case are applied in Chapters 5, 6 to the problems arising in differential geometry connected with Yamabe-Like equation.

Ch. 6 is completely devoted to the Yamabe problem of finding on a Riemannian manifold \((M,g)\), \(n\geq 3\) a conformal metric with constant scalar curvature \(R_0\in\mathbb R\), that is equivalent to the finding solutions to the equation \(-2c_n\Delta_gu+R_gu=R_0u^{\frac{n+2}{n-2}}\), \(u>0\) on \(M^l\) (*), where \(R_g=R_{kj}g^{kj}\), \(g^{ij}=(g)^{-1}_{ij}\), is the scalar curvature, \(c_n=2(n-1)/(n-2)\). After description of various predecessors results the authors starting from the sphere \(S^n\) in high dimensions and then by improving the technique, to obtain noncompactness of solutions in the case of some metrics of class \(C^k\) on \(S^n\) prove the following two theorems.

Theorem 6.2: Let \(n\geq 6\) and \(l\geq 2\). Then there exists a family of smooth metrics \(\bar{g}_\varepsilon\) on \(S^n\), converging ( in \(C^\infty(S^n)\)) to \(\bar{g}_0\) as \(\varepsilon\to 0\) such that, for every small enough \(\varepsilon\), the problem (*) on \((S^n,\bar{g}_\varepsilon)\) possesses at least \(l\) solutions.

Theorem 6.3: Let \(k\geq 2\) and \(n\geq 4k+3\). Then there exists a family of \(C^k\) metrics \(\bar{g}_\varepsilon\) on \(S^n\), with \(\|\bar{g}_\varepsilon-\bar{g}_0\|_{C^k(S^n)}\to 0\) as \(\varepsilon\to 0\) with the following property: for every small enough \(\varepsilon>0\) the problem (*) on \((S^n,\bar{g}_\varepsilon)\) possesses a sequence of solutions \(v^i_\varepsilon\) with \(\|v^i_\varepsilon\|\to 0\) as \(i\to\infty\).

Ch. 7 surveys some other problems arising in conformal geometry, first paying attention on the scalar curvature problem for the standard sphere, next to some problem on manifold with boundary.

In Ch. 8 standing waves of the NLS equation are studied, namely solutions to the following problem \(-\varepsilon^2\Delta u+V(x)u=u^p\) in \(\mathbb{R}^n\), \(0<u\in W^{1,2}(\mathbb R^n)\), where \(p>1\) is subcritical and \(V\) is a smooth bounded potential, interesting in the behaviour of the solutions as \(\varepsilon\to 0^+\) (semiclassical limit). It is shown that there exist spikes, i.e. solutions concentrating at single points of \(\mathbb R^n\).

Ch. 9 is devoted to singularly perturbed Neumann Problems on a bounded domain \(\Omega\subset\mathbb R^n\): \(-\varepsilon^2\Delta u+u=u^p\) in \(\Omega\) (**), \(\frac{\partial u}{\partial V}\) on \(\partial\Omega\), \(u>0\) in \(\Omega\), where \(p>1\) is subcritical. The following result is obtained: suppose \(\Omega\subseteq\mathbb R^n\), \(n\geq 2\) is a smooth bounded domain, \(1<p<\frac{n+2}{n-2}\) (\(1<p<\infty\) if \(n=2\)) and \(X_0\in\partial\Omega\) is a local strict maximum or minimum, or a non-degenerate critical point of the mean curvature \(H\) of \(\partial\Omega\); then for sufficiently small \(\varepsilon>0\) the problem (**) admits a solution concentrating at \(X_0\).

In the final Ch. 10 the results on the existence of solutions of NLS and singularly perturbed Neumann problems concentrating at spheres, in the radial case, are presented.

Reviewer: Boris V. Loginov (Ul’yanovsk)

### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35B20 | Perturbations in context of PDEs |

35J60 | Nonlinear elliptic equations |

35B25 | Singular perturbations in context of PDEs |

53C43 | Differential geometric aspects of harmonic maps |

58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |