American Institute of Mathematical Sciences

September  2016, 15(5): 1671-1688. doi: 10.3934/cpaa.2016008

Multiple positive solutions of fractional elliptic equations involving concave and convex nonlinearities in $R^N$

 1 School of Mathematics and Computer Science, Wuhan Polytechnic University, Wuhan, 430023, China

Received  September 2015 Revised  April 2016 Published  July 2016

In this paper, we investigate the existence of positive solutions of the following equation \begin{eqnarray} (-\Delta)^s v +\lambda v=f(x) v^{p-1}+h(x)v^{q-1}, \ x\in R^N,\\ v\in H^s(R^N), \end{eqnarray} where $1\leq q< 2 < p < 2_s^*=\frac{2N}{N-2s}$, $0 < s < 1$, $N>2s$ and $\lambda>0$ is a parameter. Since the concave and convex nonlinearities are involved, the variational functional of the equation has different properties. Via variational method, we show that the equation admits a positive ground state solution for all $\lambda>0$ strictly larger than a threshold value. Moreover, under certain conditions on $f$ and for sufficiently large $\lambda>0$, we also prove that there are at least $k+1$ ($k$ is a positive integer) positive solutions of the equation.
Citation: Qingfang Wang. Multiple positive solutions of fractional elliptic equations involving concave and convex nonlinearities in $R^N$. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1671-1688. doi: 10.3934/cpaa.2016008
References:
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References:
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