Differential and Integral Equations

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

Mousomi Bhakta and Debangana Mukherjee

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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

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

First available in Project Euclid: 18 March 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35S15: Boundary value problems for pseudodifferential operators 35J20: Variational methods for second-order elliptic equations 49J35: Minimax problems 47G20: Integro-differential operators [See also 34K30, 35R09, 35R10, 45Jxx, 45Kxx] 45G05: Singular nonlinear integral equations


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

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