Homoclinic type solutions for a semilinear elliptic PDE on \(\mathbb{R}^ n\). (English) Zbl 0785.35029

Coti Zelati, Vittorio; Rabinowitz, Paul H.

Commun. Pure Appl. Math. 45, No. 10, 1217-1269 (1992).

The authors obtain existence results for a family of semilinear elliptic equations. Consider the equation \(-\Delta u+u=f(x,u)\), \(x\in\mathbb{R}^ n\), where \(f\) satisfies:
\((\text{f}_ 1)\) \(f\in C^ 2(\mathbb{R}^ n\times\mathbb{R},\mathbb{R})\); and \(f(x,z)\) is \(T_ i\)-periodic in \(x_ i\),
\((\text{f}_ 2)\) \(f(x,0)=f_ u(x,0)=0\) for all \(x \in \mathbb{R}^ n\),
\((\text{f}_ 3)\) there exist \(a_ 1,a_ 2>0\) and \(s>1\) \((s<(n+2)/(n- 2)\) if \(n>2)\) such that \(| f_ n(x,u)|\leq a_ 1+a_ 2 | u |^{s-1}\) for all \(x \in \mathbb{R}^ n\) and \(n \in \mathbb{R}\),
\((\text{f}_ 4)\) there exists a \(\mu>2\) such that \(0<\mu F(x,u):=\mu\int^ u_ 0f(x,t)dt\leq f(x,u)u\) for all \(x \in \mathbb{R}^ n\) and \(u \neq 0\).
The functional corresponding to the semilinear elliptic equation is \(I(u)={1 \over 2} \| u \|^ 2-\int_{\mathbb{R}^ n}F(x,u)dx\), and one seeks solutions of this equation in the space \(E=W^{1,2}(\mathbb{R}^ n,\mathbb{R})\).
Denote \(I^ a=\{u \in E;I(u) \leq a\}\), \(I_ a=\{u \in E;I(u) \geq a\}\), \(I^ b_ a=I_ a\cap I^ b\), where \(a,b \in \mathbb{R}\); \({\mathcal K}\) denotes the set of critical points \(u\) of \(I\) in \(E\), \({\mathcal K}^ b=I^ b \cap{\mathcal K}\), \({\mathcal K}^ b_ a =I^ b_ a \cap {\mathcal K}\), \({\mathcal K}(a)={\mathcal K}^ a_ a\).
Let \(\Gamma=\{g \in C([0,1],E);\;g(0)=0,\;g(1) \in I^ 0\backslash \{0\}\}\) and \(c=\inf_{g \in \Gamma} \max_{t \in[0,1]}I(g(t))\); \(c\) is a positive critical value of \(I\). Suppose that the condition \[ \text{there exists } \alpha>0 \text{ such that } {\mathcal K}^{c+\alpha}/\mathbb{Z}^ n\text{ is a finite set}, \tag{*} \] holds. For \(A\subset E\) denote \(N_ r(A)=\{x \in E;\;\| x-A \| \leq r\}\).
Proposition 2.6.5. asserts the existence of a finite subset \(A\) of \({\mathcal K}(c)\) having some properties.
Let \({\mathcal M} (j_ 1,\dots, j_ k,A) = \{\sum^ k_{i=1}\tau_{ji}v_ i;\;v_ i \in A\}\), \({\mathcal M}^*(j_ 1,\dots,j_ k,A)= \bigcup_{\ell \in \mathbb{N}} {\mathcal M}(\ell_{j_ 1},\dots,\ell_{j_ k},A)\). Then there is an \(r_ k=r_ k(\alpha)\) and \(n=n_ 0(A,\alpha)\) such that if \(r \leq r_ k\), \(w \in \overline N_ r({\mathcal M}^*(j_ 1, \dots,j_ k, A))\cap{\mathcal K}\) then \(w \in {\mathcal K}_{kc-\alpha}^{kc+\alpha}\).
The main result of this paper is that under \((\text{f}_ 1)-(\text{f}_ 4)\), \((*)\), and for any choice \(j_ 1,\dots,j_ k\) in \(\mathbb{Z}^ k\), with corresponding \(n_ 0(A,\alpha)\) as before, there is an \(r_ 0>0\) such that if \(r\in(0,r_ 0)\) we have \[ {\mathcal K}^{kc+\alpha}_{kc- \alpha}/\mathbb{Z}^ n\cap N_ r({\mathcal M} (\ell_{j_ 1}, \dots,\ell_{j_ k},A)) \neq \emptyset \] for all but finitely many \(\ell\in\mathbb{N}\).

Reviewer: D.M.Bors (Iaşi)

MSC:

35J60	Nonlinear elliptic equations
34C37	Homoclinic and heteroclinic solutions to ordinary differential equations

Keywords:

homoclinic type solutions; semilinear elliptic equation; existence

References:

[1]	Sobolev Spaces, Academic Press, New York, 1975. · Zbl 0314.46030
[2]	Alema, J. Diff. Eq. 96 pp 89– (1992)
[3]	The shadowing lemma for elliptic PDE, pp. 7–22 in: Dynamics of Infinite Dimensional Systems, and , eds., Springer-Verlag, New York, 1987.
[4]	Bahri, Rev. Mat. Iberoamericana 6 pp 1– (1990) · Zbl 0731.35036
[5]	Berestycki, Arch. Rat. Mech. Anal. 82 pp 313– (1983)
[6]	Berger, J. Funct. Anal. 9 pp 249– (1972)
[7]	, and , Partial Differential Equations, Interscience Publishers, New York, 1964.
[8]	Coffman, Arch. Rat. Mech. Anal. 46 pp 81– (1972)
[9]	Coti Zelati, Math. Ann. 288 pp 133– (1990)
[10]	Coti Zelati, J. Amer. Math. Soc. 4 pp 693– (1992)
[11]	Ding, Arch. Rat. Mech. Anal. 91 pp 283– (1986)
[12]	Partial Differential Equations, Holt, Rinehart, Winston, 1969.
[13]	and , Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, 1983. · Zbl 0361.35003
[14]	and , An Introduction to Variational Inequalities and Their Applications, Academic Press, 1980.
[15]	Li, J. Diff. Eq. 74 pp 34– (1988)
[16]	Lions, Analyse Nonlin. 1 pp 109– (1984)
[17]	Lions, Rev. Mat. Iberoamericana 1 pp 145– (1985) · Zbl 0704.49005
[18]	Stable and Random Motions in Dynamical Systems, Princeton University Press, 1973.
[19]	Nehari, Proc. Roy. Irish Acad. 62 pp 117– (1963)
[20]	Some aspects of semilinear elliptic equations, pp. 171–206 in: Nonlinear Diffusion Equations and Their Equilibrium States, , and , eds., Springer-Verlag, 1988.
[21]	Minimax Methods in Critical Point Theory with Applications to Differential Equations, C.B.M.S. Reg. Conf. Series in Math. No. 65. Amer. Math. Soc., Providence, RI, 1986.
[22]	A note on a semilinear elliptic equation on Rn, pp. 307–318 in: Nonlinear Analysis, a tribute in honour of Giovanni Prodi, and , eds., Quaderni Scuola Normale Superiore, Pisa, 1991.
[23]	Séré, Math. Z. 209 pp 27– (1992)
[24]	Diffeomorphisms with many periodic points, pp. 63–80 in: Differential and Combinatorial Topology, ed., Princeton University Press, 1965.
[25]	Strauss, Comm. Math. Phys. 55 pp 149– (1979)
[26]	Synge, Proc. Roy. Irish Acad. 62 pp 17– (1961)
[27]	Positive Decaying Solutions of Nonlinear Elliptic Problems, Doctoral Thesis, University of Alberta, 1990.