Let \(\Omega \) be a bounded smooth domain in \(\mathbb R^n\) and \(\delta(x)\) be the distance from \(x\in\Omega\) to the boundary \(\partial\Omega\). The paper deals with the problem \[ \Delta u+\frac{\mu}{\delta^2(x)}u=f(u)\text{ in }\Omega,\quad u(x)\to\infty \text{ as } x\to\partial\Omega,\tag{1} \] where \(\frac{\mu}{\delta^2(x)}u\) is called the Hardy potential. Here, \(f:\mathbb R\to\mathbb R^+\) is a continuous and increasing function. Associated to problem (1) is the problem (2) without hardy potential: \[ \Delta U_P=f(U_P)\text{ in }\Omega,\quad U_P(x)\to\infty \text{ as } x\to\partial\Omega,\tag{2} \] to which larges solutions exist under the Keller-0sserman condition: \[ \int^{\infty}\frac{1}{\sqrt{F(s)}}ds<\infty, \text{ where } F^{'}=f. \] Existence of solutions \(u\) to (1) “comparable” to \(U_p\) is studied, separately, when \(\mu>0\), then, when \(\mu<0\). It is proved that, in each case, under suitable assumptions, problem (1) has a solution \(u(x)\) comparable to \(U_p\), which satisfies \[ U_P\leq u\leq b(U_P+C) \] for \(b\) sufficiently large, if \(\mu>0\) (Theorem 3.1). If \(\mu<0\) and \(f(t)\leq t^p,p>1,\) existence of a solution which satisfies \[b\delta^{-\frac{2}{p-1}}(x)\leq u(x) \leq U_p(x) ,\] near the boundary, is proved. Moreover, in this case \(\mu<0,\) non-existence results are presented (Theorem 4.3). Asymptotic expansions near the boundary and uniqueness of the solution are obtained on some examples.