Concave-convex critical problems for the spectral fractional laplacian with mixed boundary conditions

Abstract

In this work we study the existence of solutions to the following critical fractional problem with concave-convex nonlinearities, $$\begin{aligned} \left\{ \begin{array}{lll} (-\varDelta )^su=\lambda u^q+u^{2_s^*-1}, &{} u>0 &{} \hbox { in } \varOmega , \\ u=0&{} &{} \hbox { on } \varSigma _{\mathcal {D}}, \\ \displaystyle \frac{\partial u}{\partial \nu }=0&{} &{} \hbox { on } \varSigma _{\mathcal {N}}, \end{array} \right. \end{aligned}$$with \(\varOmega \subset {\mathbb {R}}^N\), \(N>2s\), a smooth bounded domain, \(\frac{1}{2}<s<1\), \(0<q<2_s^*-1\), \(q\ne 1\), being \(2_s^*=\frac{2N}{N-2s}\) the critical fractional Sobolev exponent, \(\lambda >0\), \(\nu \) is the outwards normal to \(\partial \varOmega \); \(\varSigma _{{\mathcal {D}}}\), \(\varSigma _{{\mathcal {N}}}\) are smooth \((N-1)\)-dimensional submanifolds of \(\partial \varOmega \) such that \(\varSigma _{{\mathcal {D}}}\cup \varSigma _{{\mathcal {N}}}=\partial \varOmega \), \(\varSigma _{{\mathcal {D}}}\cap \varSigma _{{\mathcal {N}}}=\emptyset \), and \(\varSigma _{{\mathcal {D}}}\cap {\overline{\varSigma }}_{{\mathcal {N}}}=\varGamma \) is a smooth \((N-2)\)-dimensional submanifold of \(\partial \varOmega \). In particular, we will prove that, for the sublinear case \(0<q<1\), there exists at least two solutions for every \(0<\lambda <\varLambda \) for certain \(\varLambda \in {\mathbb {R}}\) while, for the superlinear case \(1<q<2_s^*-1\), we will prove that there exists at least one solution for every \(\lambda >0\). We will also prove that solutions are bounded.