Four positive solutions for the semilinear elliptic equation: \(-\Delta u+u=a(x) u^p +f(x)\) in \(\mathbb{R}^N\)
The authors consider the existence of four positive solutions of the semilinear elliptic problem \[ \begin{cases} -\Delta u+u=a(x)u^p+f(x),\quad u>0\qquad\text{in }\mathbb R^n,\quad n\geq 1,\\ u\in H^1(\mathbb R^n),\end{cases}\tag{1} \] where \(1<p<(n+2)/(n-2)\) if \(n\geq 3\), \(1<p<\infty\) if \(n=1,2\), and under the following structural assumptions: (a) \(a\in C(\mathbb R^n)\), \(0<a(x)\leq 1\) for all \(x\in\mathbb R^n\), \(a(x)\to 1\) as \(|x|\to\infty\), \(a\not\equiv 1\) in \(\mathbb R^n\), and there are \(\delta>0\) and \(C>0\) such that \[ a(x)-1\geq-Ce^{-(2+\delta)|x|}\qquad\text{for all }x\in\mathbb R^n; \] (b) \(f\in H^{-1}(\mathbb R^n)\), \(f\geq 0\) and \(f\not\equiv 0\) in \(\mathbb R^n\). The main result of the paper is: Theorem. There exists a number \(\varepsilon_0>0\) such that if \(0<\|f\|_{H^{-1}(\mathbb R^n)}\leq\varepsilon_0\), then problem \((1)\) admits at least four positive solutions. Let \((f_j)_j\) be a sequence of functions satisfying property (b), with \(\|f_j\|_{H^{-1}(\mathbb R^n)}\to 0\) as \(j\to\infty\). Then there is a subsequence of \((f_j)_j\), still denoted by \((f_j)_j\), and four sequences \((u^{(k)}_j)_j\), \(k=1,2,3,4\), of positive solutions of problem \((1)\), with \(f\) replaced by \(f_j\), such that (i) \(\|u_j^{(1)}\|_{H^{1}(\mathbb R^n)}\to 0\) as \(j\to\infty\); (ii) there are sequences \((y^{(k)}_j)_j\) in \(\mathbb R^n\), \(k=2,3\), such that \[ |y^{(k)}_j|\to\infty\quad\text{and}\quad \|u^{(k)}_j-\omega(\cdot-y^{(k)}_j)\|_{H^{1}(\mathbb R^n)}\to 0 \quad\text{as \(j\to\infty\), for }k=2,3, \] where \(\omega\) is the unique positive radial solution of the problem \[ \begin{cases} -\Delta \omega+\omega=\omega^p,\quad \omega>0\qquad\text{in }\mathbb R^n,\\ \omega\in H^1(\mathbb R^n);\end{cases}\tag{2} \] (iii) there exists a positive solution \(v_0\) of problem \((1)\), with \(f\) replaced by \(f\equiv 0\), such that \[ \|u^{(4)}_j-v_0)\|_{H^{1}(\mathbb R^n)}\to 0 \qquad\text{as }j\to\infty. \] Moreover, if \(a(x)=a(|x|)\), \(f(x)=f(|x|)\), and (a), (b) continue to hold, then there is \(\varepsilon_1>0\) such that if \(0<\|f\|_{H^{-1}(\mathbb R^n)}\leq\varepsilon_1\), problem \((1)\) admits at least one positive solution, which is not radially symmetric. Finally, if \(a(x)=a(|x|)\), \(f(x)=f(|x|)\), \(r=|x|\), and now \[ a\geq 1,\quad a_r\leq 0,\quad f\geq 0,\quad f_r\leq 0\quad\text{in }\mathbb R^+, \] then all positive solutions of problem \((1)\) are radially symmetric. The proof technique is very delicate since problem (1) is a nontrivial perturbation of the well-known and widely studied problem (2), and relies on an interesting minimax argument introduced by Bahri and Li for the existence of at least one positive solution of problem (1) when \(f\equiv 0\). In this paper there are several difficulties to overcome and naturally the previous analysis, given for different variants of \((1)\), cannot be applied directly. Lemma 1.12 is of indepent interest.
scientific article; zbMATH DE number 1525160