zbMATH — the first resource for mathematics

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

35J60 Nonlinear elliptic equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
Full Text: DOI
[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. · doi:10.1007/978-3-642-86458-2_2
[4] Bahri, Rev. Mat. Iberoamericana 6 pp 1– (1990) · Zbl 0731.35036 · doi:10.4171/RMI/92
[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 · doi:10.1007/978-3-642-61798-0
[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 · doi:10.4171/RMI/6
[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. · doi:10.1007/978-1-4613-9608-6_10
[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.