Existence and multiplicity of solutions for a locally coercive elliptic equation

Abstract

For a bounded domain $\Omega$, we establish existence and multiplicity of nontrivial solutions for the semilinear elliptic problem

$$\begin{aligned} \left\{ \begin{array}{rcll} -\Delta u &{} = &{} {g(u)} - h(x) f(u), &{} \text{ in } \Omega \\ u &{} = &{} 0, &{} \text{ on } \partial \Omega ,\\ \end{array} \right. \end{aligned}$$

where $h\in L^\infty (\Omega )$ is nonnegative and nontrivial, g is asymptotically linear, f is superlinear and ${g(0)}=f(0)=0$. We also study the existence of solutions for the problem

$$\begin{aligned} \left\{ \begin{array}{rcll} -\Delta u &{} = &{} {g(u)} - h(x)f(u)+k(x), &{} \text{ in } \Omega \\ u &{} = &{} 0, &{} \text{ on } \partial \Omega ,\\ \end{array} \right. \end{aligned}$$

when $k\in L^2(\Omega )$.