Positive solutions of superlinear elliptic equations
Let \(\Omega\subset \mathbb{R}^N\) be a bounded convex domain with smooth boundary \(\Omega\) and \(f: \mathbb{R}^+\to \mathbb{R}\) be a locally Lipschitz continuous function with \(f(0)\geq 0\). The elliptic problems \[ -\Delta u= f(u),\quad u>0,\quad x\in\Omega,\quad u= 0,\quad x\in\partial\Omega;\tag{1} \] and \[ -\Delta u=\lambda f(u),\quad u>0,\quad x\in\Omega,\quad u= 0,\quad x\in\partial\Omega\tag{2} \] are considered where \(\lambda\in \mathbb{R}^+\). It is assumed that \(f\) satisfies \[ \liminf_{t\to+\infty} f(t) t^{-1}> \lambda_1\tag{H1} \] and \[ \lim_{t\to+\infty} f(t) t^{-\ell}= 0\quad\text{for }\ell= (N+ 2)/(N- 2)\quad\text{if }N\geq 3,\;\ell<\infty\quad\text{if }N= 1,2.\tag{H2} \] \(\lambda_1\) is the first eigenvalue of \(-\Delta\) with zero boundary condition. If in addition \(f\) satisfies that for some \(0\leq\Theta< 2N/(N- 2)\) \[ \limsup_{t\to\infty} {tf(t)- \Theta F(t)\over t^2f(t)^{2/N}}= 0\tag{H3} \] \((F(t)= \int^t_0 f(s) ds)\) then in the literature existence theorems for problem (1) and (2) are known. In this paper, the author proves existence results without assumption (H3) a conjecture made by \textit{P. L. Lions} [SIAM Review 24, 441-467 (1982; Zbl 0511.35033)].
scientific article; zbMATH DE number 1367851
scientific article