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Existence results for semilinear elliptic equations with some lack of coercivity

Published online by Cambridge University Press:  14 July 2008

A. Mercaldo
Dipartimento di Matematica e Applicazioni ‘R. Caccioppoli', Università degli Studi di Napoli ‘Federico II', Complesso Monte S. Angelo, via Cintia, 80126 Napoli, Italy (
I. Peral
Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain (


We consider the following problem:

\begin{alignat*}{2} -\text{div}(A(x,u)\nabla u)&=u^s+f(x) & \quad &\text{in }\varOmega, \\ u(x)&\ge0 & & \text{in }\varOmega, \\ u(x)&=0 & & \text{on }\p\varOmega, \end{alignat*}

where $\varOmega$ is an open bounded subset of $\mathbb{R}^N$, $N\ge3$, and

$$ A:\varOmega\times\mathbb{R}\rightarrow M_{N\times N} $$

is an elliptic matrix such that when $u\to\infty$ is non-coercive.

Research Article
2008 Royal Society of Edinburgh

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