Homoclinic-type solutions for an almost periodic semilinear elliptic equation on \(\mathbb{R}^ n\)
In this clearly written article, the existence of at least one homoclinic solution to the following semilinear elliptic equation is studied: \[ -\Delta u=-\alpha(x)g(u),\quad x\in\mathbb{R}^N,\qquad u(x)\to 0\quad\text{as }|x|\to\infty \] where \(N\geq 2\) and \(\alpha\) and \(g\) satisfy the assumptions (a1) \(\alpha \in C^{0,\gamma}(\mathbb{R}^N,\mathbb{R})\) for some \(\gamma\in (0,1)\) with \(\inf_{\alpha(x),\mathbb{R}^N}:=a>0\), (a2) \(\alpha(x)=\alpha(x_1,\dots,x_2)\) is almost periodic in \(x_1\) uniformly with respect to the other variables and \(T_i\)-periodic in \(x_i\) for \(i=1,\dots,n\), (g1) \(g\in C^1(\mathbb{R},\mathbb{R})\), (g2) there exists \(\theta>2\) such that \(0<\theta G(s):=\theta\int_0^s g(t)dt\leq g(s)s\) for all \(s\neq 0\), (g3) there exist \(a_1,a_2>0\) such that \(|g'(s)|\leq a_1+a_2|s|^{m-1}\) for all \(s\in\mathbb{R}\), where \(m\in (1,(N+ 2)/(N- 2))\) if \(N>2\). This article not only extends the article of \textit{E. Serra, M. Tarallo} and \textit{S. Terracini} [Ann. Inst. H. Poincaré, Anal. Nonlinéaire 13, No. 6, 783-812 (1996; Zbl 0873.58032)] but also can be used to slightly simplify some of its proofs.
scientific article; zbMATH DE number 1089773
scientific article