Multiplicity results for sublinear elliptic equations with sign-changing potential and general nonlinearity

Abstract

Abstract In this paper, we study the following elliptic boundary value problem: { − Δ u + V ( x ) u = f ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω , $$ \textstyle\begin{cases} -\Delta u+V(x)u=f(x, u),\quad x\in \Omega , \\ u=0, \quad x \in \partial \Omega , \end{cases} $$ where Ω ⊂ R N $\Omega \subset {\mathbb {R}}^{N}$ is a bounded domain with smooth boundary ∂Ω, and f is allowed to be sign-changing and is of sublinear growth near infinity in u. For both cases that V ∈ L N / 2 ( Ω ) $V\in L^{N/2}(\Omega )$ with N ≥ 3 $N\geq 3$ and that V ∈ C ( Ω , R ) $V\in C(\Omega , \mathbb {R})$ with inf Ω V ( x ) > − ∞ $\inf_{\Omega }V(x)>-\infty $ , we establish a sequence of nontrivial solutions converging to zero for above equation via a new critical point theorem.

Keywords

• Semilinear elliptic equations
• Boundary value problems
• Sublinear
• Indefinite sign
• Genus