##
**Existence of a ground state in nonlinear equations of the Klein-Gordon type.**
*(English)*
Zbl 0707.35143

Variational inequalities and complementarity problems, theory and applications, Proc. int. School Math., Erice/Sicily 1978, 35-51 (1980).

This paper is concerned with the proof of the existence of nontrivial (i.e., nonconstant) solutions \(u\in H^ 1({\mathbb R}^ N)\), to equations of the form \(-\Delta u=g(u)\) in \({\mathbb R}^ N\) where \(g\) is a function such that \(g(0)=0\). Such equations arise, for example, when one looks for solitary waves to the nonlinear Klein-Gordon or Schrödinger equations. Another motivation for such a study comes from the recent analysis of vacuum instability [see S. Coleman, Phys. Rev. D 15, 2929–2936 (1977)]. The authors restrict themselves to the case where the function \(g\) is of the form \(g(u)=\lambda u^ p-\mu u^ q-mu\). The main result of the paper is the following theorem:

Suppose \(N\geq 3\). Let \(g(t)=\lambda t^ p-\mu t^ q-mt\) where \(\lambda,m>0\), \(\mu\geq 0\) and \(1<p,q\). Suppose that (1) \(p<\max [q,(N+2)/(N-2)]\), (2) if \(p\geq (N+2)/(N-2)\) then \(\mu >0\), (3) there exists a \(\zeta >0\) such that \(G(\zeta)>0\) where \(G(s)=\int^{s}_{0}g(t)\,dt\). Then \(-\Delta u=g(u)\) has a solution \(u\in C^{\infty}({\mathbb R}^ N)\) which is strictly positive, spherically symmetric, decreases with \(r=| x|\) and is such that, for all \(| \alpha | \in {\mathbb N}\), \(| D^{\alpha}u(x)| \leq C_{\alpha}e^{-\delta_{\alpha}r}\) where \(C_{\alpha},\delta_{\alpha}>0.\)

This theorem generalizes the results of several authors; see in particular the article by S. Coleman, V. Glaser and A. Martin [Commun. Math. Phys. 58, 211–221 (1978)]. Note that the results apply in spaces of dimension not less than 3. The proof of the theorem uses the method introduced by Coleman, Galser and Martin [op. cit.]. Instead of looking directly for nontrivial critical points of \(S(u)=T(u)-V(u)\), where \(T(u)=\int_{{\mathbb R}^ N}| \nabla u(x)|^ 2\, dx\) and \(V(u)=\int_{{\mathbb R}^ N}G(u(x))\,dx\), one looks for a function \(w\in H^ 1({\mathbb R}^ N)\cap L^{p+1}({\mathbb R}^ N)\cap L^{q+1}({\mathbb R}^ N)\) which minimizes \(T(w)\) for some fixed value of \(V(w)\), say, \(V(w)=1\). As shown by the authors, if a solution to that last problem exists, then for an appropriately chosen g, it can always be transformed into a solution of \(-\Delta u=g(u)\) by a change of scale. The proof of the theorem is then reduced to that of showing the existence of a positive, spherically symmetric solution \(u\in H^ 1({\mathbb R}^ N)\) which decreases with \(r=| x|\) to the constrained minimization problem. Moreover, it is possible to show that the solution constructed in that way has S less than or equal to that of any other nontrivial solutions.

We would like to remark here that it might not be appropriate to call those solutions “ground states” since in field theory it is rather the trivial constant solution(s) that would bear that name. The omission of two dimensions in the statement of the theorem is due to the fact that the reduced problem has no solution when \(N=1\) or \(N=2\). The case \(N=1\) is treated by the authors in an appendix where they prove a very general existence result by the methods of ordinary differential equations. For \(N=2\), although the existence results have not yet attained the same generality as for the other dimensions, some progress has been made; we are referred to a forthcoming paper of the authors for details. A discussion of the “almost necessary” character of the hypotheses (1)–(3) in the theorem concludes the work.

[For the entire collection see Zbl 0476.00017.]

Suppose \(N\geq 3\). Let \(g(t)=\lambda t^ p-\mu t^ q-mt\) where \(\lambda,m>0\), \(\mu\geq 0\) and \(1<p,q\). Suppose that (1) \(p<\max [q,(N+2)/(N-2)]\), (2) if \(p\geq (N+2)/(N-2)\) then \(\mu >0\), (3) there exists a \(\zeta >0\) such that \(G(\zeta)>0\) where \(G(s)=\int^{s}_{0}g(t)\,dt\). Then \(-\Delta u=g(u)\) has a solution \(u\in C^{\infty}({\mathbb R}^ N)\) which is strictly positive, spherically symmetric, decreases with \(r=| x|\) and is such that, for all \(| \alpha | \in {\mathbb N}\), \(| D^{\alpha}u(x)| \leq C_{\alpha}e^{-\delta_{\alpha}r}\) where \(C_{\alpha},\delta_{\alpha}>0.\)

This theorem generalizes the results of several authors; see in particular the article by S. Coleman, V. Glaser and A. Martin [Commun. Math. Phys. 58, 211–221 (1978)]. Note that the results apply in spaces of dimension not less than 3. The proof of the theorem uses the method introduced by Coleman, Galser and Martin [op. cit.]. Instead of looking directly for nontrivial critical points of \(S(u)=T(u)-V(u)\), where \(T(u)=\int_{{\mathbb R}^ N}| \nabla u(x)|^ 2\, dx\) and \(V(u)=\int_{{\mathbb R}^ N}G(u(x))\,dx\), one looks for a function \(w\in H^ 1({\mathbb R}^ N)\cap L^{p+1}({\mathbb R}^ N)\cap L^{q+1}({\mathbb R}^ N)\) which minimizes \(T(w)\) for some fixed value of \(V(w)\), say, \(V(w)=1\). As shown by the authors, if a solution to that last problem exists, then for an appropriately chosen g, it can always be transformed into a solution of \(-\Delta u=g(u)\) by a change of scale. The proof of the theorem is then reduced to that of showing the existence of a positive, spherically symmetric solution \(u\in H^ 1({\mathbb R}^ N)\) which decreases with \(r=| x|\) to the constrained minimization problem. Moreover, it is possible to show that the solution constructed in that way has S less than or equal to that of any other nontrivial solutions.

We would like to remark here that it might not be appropriate to call those solutions “ground states” since in field theory it is rather the trivial constant solution(s) that would bear that name. The omission of two dimensions in the statement of the theorem is due to the fact that the reduced problem has no solution when \(N=1\) or \(N=2\). The case \(N=1\) is treated by the authors in an appendix where they prove a very general existence result by the methods of ordinary differential equations. For \(N=2\), although the existence results have not yet attained the same generality as for the other dimensions, some progress has been made; we are referred to a forthcoming paper of the authors for details. A discussion of the “almost necessary” character of the hypotheses (1)–(3) in the theorem concludes the work.

[For the entire collection see Zbl 0476.00017.]

Reviewer: Luc Vinet (MR 81i:35137)

### MSC:

35Q55 | NLS equations (nonlinear Schrödinger equations) |

35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |

35J85 | Unilateral problems; variational inequalities (elliptic type) (MSC2000) |