Multiple solutions with compact support for a quasilinear Schrödinger equation with sign-changing potentials

ABSTRACT

In this paper, we study the existence of compactly supported solutions for the Schrödinger equations with indefinite potentials $-{\mathrm{\Delta }}_{p}u+V\left(x\right)u=a\left(x\right)|u{|}^{q-1}u,x\in {\mathbb{R}}^N,$ where $N\ge 2$, 1<p<2,  0<q<p−1, and $a\left(x\right),V\left(x\right)$ change sign in ${\mathbb{R}}^N$. Firstly, we establish the existence of infinitely many weak solution. Next, we study the compactness of support of classical solutions for the above equation. This paper is a continuation of the recent work established by [Bedoui N, Ounaies H. Qualitative properties and support compactness of solutions for quasilinear Schrödinger equation with sign changing potentials. Nonlinear Anal. 2020;198:111843.].

Keywords:

• Schrödinger equation
• indefinite potentials
• compactly supported solution
• variational methods

AMS Subject Classifications:

• 35J60
• 35J91
• 58E30

Disclosure statement

No potential conflict of interest was reported by the author(s).