Multi-bump type nodal solutions for a logarithmic Schrödinger equation with deepening potential well

Abstract

In this paper, we are concerned with the existence and multiplicity of multi-bump type nodal solutions for the following logarithmic Schrodinger equation $$ \left\{ \begin{array}{ll} -\Delta u+ \lambda V(x)u=u \log u^2, &{}\quad \text{ in } \quad {\mathbb {R}}^{N}, \\ u \in H^1({\mathbb {R}}^{N}), \\ \end{array} \right. $$ where $$N \ge 1$$ , $$\lambda >0$$ is a real parameter and the nonnegative continuous function $$V: {\mathbb {R}}^{N}\rightarrow {\mathbb {R}}$$ has a potential well $$\Omega : =\text {int}\, V^{-1}(0)$$ which possesses k disjoint bounded components $$\Omega =\bigcup _{j=1}^{k}\Omega _{j}$$ . Using the variational methods, we prove that if the parameter $$\lambda >0$$ is large enough, then the equation has at least $$2^{k}-1$$ multi-bump type nodal solutions.