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
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