## 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)

### MSC:

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