## Differential and Integral Equations

### Multiplicity results and sign changing solutions of non-local equations with concave-convex nonlinearities

#### Abstract

In this paper, we prove the existence of infinitely many nontrivial solutions of the following equations driven by a nonlocal integro-differential operator $\mathcal{L}_K$ with concave-convex nonlinearities and homogeneous Dirichlet boundary conditions \begin{align*} \mathcal{L}_{K} u + \mu |u|^{q-1}u + \lambda |u|^{p-1}u &= 0 \quad\mbox{in}\quad \Omega, \\ u&=0 \quad\mbox{in}\quad\mathbb{R}^N\setminus\Omega, \end{align*} where $\Omega$ is a smooth bounded domain in $\mathbb R^N$, $N > 2s$, $s\in(0, 1)$, $0 < q < 1 < p\leq \frac{N+2s}{N-2s}$. Moreover, when $\mathcal{L}_K$ reduces to the fractional laplacian operator $-(-\Delta)^s$, $p=\frac{N+2s}{N-2s}$, $\frac{1}{2} (\frac{N+2s}{N-2s}) < q < 1$, $N > 6s$, $\lambda =1$, we find $\mu^*>0$ such that for any $\mu\in(0,\mu^*)$, there exists at least one sign changing solution.

#### Article information

Source
Differential Integral Equations, Volume 30, Number 5/6 (2017), 387-422.

Dates
First available in Project Euclid: 18 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.die/1489802419

Mathematical Reviews number (MathSciNet)
MR3626581

Zentralblatt MATH identifier
06738554

#### Citation

Bhakta, Mousomi; Mukherjee, Debangana. Multiplicity results and sign changing solutions of non-local equations with concave-convex nonlinearities. Differential Integral Equations 30 (2017), no. 5/6, 387--422. https://projecteuclid.org/euclid.die/1489802419