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Positive solutions of superlinear elliptic equations. (English) Zbl 0951.35051

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 P. L. Lions [SIAM Review 24, 441-467 (1982; Zbl 0511.35033)].

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Citations:

Zbl 0511.35033
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References:

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