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Periodic solutions for a superlinear fractional problem without the Ambrosetti-Rabinowitz condition

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  • The purpose of this paper is to study $T$-periodic solutions to

    $\left\{ \begin{array}{*{35}{l}} [{{(-{{\Delta }_{x}}+{{m}^{2}})}^{s}}-{{m}^{2s}}]u=f(x,u) & \text{ in }{{(0,T)}^{N}} \\ u(x+T{{e}_{i}})=u(x) & \text{for all }x\text{ }\in {{\mathbb{R}}^{N}},i=1,\ldots ,N \\\end{array} \right. \tag{1}$

    where $s∈ (0,1)$, $N> 2s$, $T>0$, $m> 0$ and $f(x,u)$ is a continuous function, $T$ -periodic in $x$ and satisfying a suitable growth assumption weaker than the Ambrosetti-Rabinowitz condition.

    The nonlocal operator $(-Δ_{x}+m^{2})^{s}$ can be realized as the Dirichlet to Neumann map for a degenerate elliptic problem posed on the half-cylinder $\mathcal{S}_{T}=(0,T)^{N}× (0,∞)$. By using a variant of the Linking Theorem, we show that the extended problem in $\mathcal{S}_{T}$ admits a nontrivial solution $v(x,ξ)$ which is $T$ -periodic in $x$. Moreover, by a procedure of limit as $m\to 0$, we prove the existence of a nontrivial solution to (1) with $m=0$.

    Mathematics Subject Classification: Primary: 49J35, 35A15, 35R11; Secondary: 35J60, 35S05.


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