Electron. J. Differential Equations, Vol. 2025 (2025), No. 47, pp. 1-13.

Existence of three positive solutions for a p-sublinear problem involving a Schrodinger p-Laplacian type operator

Sigifredo Herron, Emer Lopera, Diana Sanchez

Abstract:
We prove the existence of three positive solutions for the problem $$\displaylines{ -\Delta_p u + V (x)\varphi_p(u)=\lambda f(u),\quad x\in \Omega, \cr u(x)=0, \quad x\in \partial\Omega, } $$ where \(\lambda >0\), \(\Delta_p\) is the \(p\)-Laplacian operator, \(N>p>1\), \(\varphi_p (s):=|s|^{p-2}s\), \(s\in \mathbb{R}\), \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with connected and smooth boundary. In our study, \( V \in L^\infty (\Omega)\) and \(f:[0,\infty)\to \mathbb{R}\) is a \(C^1\) function. The reaction term, \(f\), is increasing and \(p\)-sublinear at infinity. Our method relies on sub-super solution techniques and the use of a theorem on the existence of multiple fixed points. We extend some results known in the literature.

Submitted March 2, 2025. Published May 8, 2025.
Math Subject Classifications: 35B09, 35B50, 35B51, 35D30, 35G30, 35J10, 35J92, 47H10.
Key Words: Subsolution; supersolution; multiple solutions; p-Laplacian; Schrodinger type operator
DOI: 10.58997/ejde.2025.47

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Sigifredo Herrón
Universidad Nacional de Colombia
Medellín, Colombia
email: sherron@unal.edu.co
Emer Lopera
Universidad Nacional de Colombia
Manizales, Colombia
email: edloperar@unal.edu.co
Diana Sánchez
Universidad Nacional de Colombia
Manizales, Colombia
email: dmsanchezm@unal.edu.co

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