Multiplicity results for an inhomogeneous nonlinear elliptic problem

Abstract

We are concerned with the multiplicity of positive and nodal solutions of $$ \begin{align} &-\Delta u +\mu u =Q(x) |u|^{p-2} u+h(x)\quad\text{in}\,\, \Omega\\ &\qquad\qquad u \in H_0^1(\Omega), \end{align} $$ where $N\geq 3$, $2<p<\frac{2N}{N-2}$ $\mu >0$, $Q\in C(\overline{\Omega})$, and $0\not\equiv h\in L^2(\Omega)$. We show that if the maximum of $Q$ is achieved at exactly $k$ different points of $\Omega$, then for large enough $\mu$ the above problem has at least $k+1$ positive solutions and $k$ no