Existence and multiplicity of solutions for a locally coercive elliptic equation

For a bounded domain \(\Omega \) $\mathrm{\Omega }$ , we establish existence and multiplicity of nontrivial solutions for the semilinear elliptic problem \(\begin{aligned} \left\{ \begin{array}{rcll} -\Delta u &{} = &{} {g(u)} - h(x) f(u), &{} \text{ in } \Omega \\ u &{} = &{} 0, &{} \text{ on } \partial \Omega ,\\ \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cccc}\hfill -\mathrm{\Delta }u& =& g(u)-h(x)f(u),\hfill & \mathrm{in}\mathrm{\Omega }\hfill \\ \hfill u& =& 0,\hfill & \mathrm{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(h\in L^\infty (\Omega )\) $h\in L^{\infty }(\mathrm{\Omega })$ is nonnegative and nontrivial, g is asymptotically linear, f is superlinear and \({g(0)}=f(0)=0\) $g(0)=f(0)=0$ . We also study the existence of solutions for the problem \(\begin{aligned} \left\{ \begin{array}{rcll} -\Delta u &{} = &{} {g(u)} - h(x)f(u)+k(x), &{} \text{ in } \Omega \\ u &{} = &{} 0, &{} \text{ on } \partial \Omega ,\\ \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cccc}\hfill -\mathrm{\Delta }u& =& g(u)-h(x)f(u)+k(x),\hfill & \mathrm{in}\mathrm{\Omega }\hfill \\ \hfill u& =& 0,\hfill & \mathrm{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ when \(k\in L^2(\Omega )\) $k\in L^2(\mathrm{\Omega })$ .