Solutions of elliptic problems with nonlinearities of linear growth

Zhaoli Liu; Jiabao Su; Zhi-Qiang Wang
August 2009
Calculus of Variations & Partial Differential Equations;Aug2009, Vol. 35 Issue 4, p463
Academic Journal
In this paper, we study existence of nontrivial solutions to the elliptic equation and to the elliptic system where O is a bounded domain in $${\mathbb R^{N}}$$ with smooth boundary ?O, $${f\in C^{1}(\bar\Omega\times\mathbb{R}, \mathbb{R})}$$, f ( x, 0) = 0, $${V\in C^{2}(\bar\Omega\times\mathbb{R}^{m}, \mathbb{R})}$$ with m = 2 and $${\nabla_{u} V(x,0)=0}$$. Nontrivial solutions are obtained in the case in which the nonlinearities have linear growth. That is, for some c > 0, $${|f(x,u)|\leqslant c|u|}$$ for $${x\in\Omega}$$ and $${u\in\mathbb R}$$, and $${-cI_{m}\,\leqslant\,\nabla_{u}^{2}V(x,u)\,\leqslant\,cI_{m}}$$ for $${x\in\Omega}$$ and $${u\in\mathbb R^{m}}$$, where I m is the m � m identity matrix. In sharp contrast to the existing results in the literature, we do not make any assumptions at infinity on the asymptotic behaviors of the nonlinearity f and $${\nabla_{u} V}$$.


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