Positive solutions of a semilinear elliptic equation on \(\mathbb{R}^ N\) with indefinite nonlinearity
The authors consider the existence of positive solutions of the problem \[ -\Delta u(x)=\lambda h(x)u(x)+g(x)|u(x)|^{p-2} u(x)\qquad\text{in }\mathbb{R}^n,\tag{1} \] where \(\lambda\) is a real parameter and \(n>2\), \(2<p<2^*= 2n/(n-2)\). Let \(h\) be a continuous function \(h^+\neq 0\), \(h\in L^\infty(\mathbb{R}^n)\cap L^{n/2}(\mathbb{R}^n)\), and \(g\) be a continuous function \(g\in L^\infty(\mathbb{R}^n)\cap L^q(\mathbb{R}^n)\), \(q=2n/(2n- pn+2p)\). The main result of the paper is the following Theorem. 1) For every \(\lambda\in(-\lambda_1(-h), \lambda_1(h))\), there exists at least one weak positive solution. 2) If \(\int g\varphi_h^p<0\), then for \(\lambda=\lambda_1(h)\), there exists at least one weak positive solution. Moreover, in this case for \(\lambda\) in a right neighborhood of \(\lambda_1(h)\) there exist at least two distinct weak positive solutions. 3) If \(\int g\varphi^p_{-h}<\infty\), then for \(\lambda=-\lambda_1(-h)\) there exists at least one weak positive solution. Moreover, in this case, for \(\lambda\) in a left neighborhood of \(-\lambda_1(-h)\), there exist at least two distinct weak positive solutions. Here, \(\lambda_1(h)= [\sup_v(\int hv^2/\int|\nabla v|^2)]^{-1}\), and \(\varphi\) is a solution of \(-\Delta\varphi= \lambda h\varphi\).
scientific article; zbMATH DE number 896499