Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 72, pp. 1-24.

Title: Multiplicity results for Schrodinger type fractional p-Laplacian boundary value problems

Authors: Emer Lopera (Univ. Nacional de Colombia, Manizales, Colombia)
         Leandro Recova (California State Polytechnic Univ, Pomona, CA, USA)
         Adolfo Rumbos (Pomona College,  Claremont, CA, USA)

Abstract: 
In this work, we study the existence and multiplicity of solutions 
to the  problem
$$\displaylines{
 -(\Delta)_p^s u  + V(x)|u|^{p-2}u = \lambda f(u),\quad x\in\Omega;\cr
 u=0,\quad x\in R^N\backslash\Omega,
}$$
where $\Omega\subset R^N$ is an open bounded set with Lipschitz 
boundary $\partial\Omega$, $N\geq 2$, $V\in L^{\infty}(R^N)$, 
and $(-\Delta)_p^s$ denotes the fractional p-Laplacian with 
$s\in(0,1)$, $1<p$, $sp<N$, $\lambda>0$, and $f:R\to R$ 
is a continuous function. We extend the results of Lopera et al. [22]
by  proving the existence of a second weak solution to this problem. 
We apply a variant of the mountain-pass theorem due to Hofer [15]
and infinite-dimensional Morse theory  to obtain the existence of at 
least two solutions.  


Submitted July 15, 2024. Published November 11, 2024.
Math Subject Classifications: 35J20.
Key Words:  Mountain pass theorem; Morse theory; critical groups; comparison principle.