We study a nonlinear elliptic boundary value problem defined on a smooth bounded domain involving the fractional Laplace operator and a concave-convex term, together with mixed Dirichlet-Neumann boundary conditions.
We consider radial solutions of the slightly subcritical problem (Formula presented.) either on (Formula presented.) ((Formula presented.)) or in a ball (Formula presented.) satisfying Dirichlet or Neumann boundary conditions. In particular, we provide sharp rates and constants describing the asymptotic behavior (as (Formula presented.)) of all local minima and maxima of (Formula presented.) and of the value of the derivative (Formula presented.) at the zeros of the solution.
We analyze the behavior of the eigenvalues of the following non local mixed
problem $\left\ \beginarrayrcll (-\Delta)^s u &=& \lambda_1(D) \ u
&\inn\Omega,\\ u&=&0&\inn D,\\ \mathcalN_su&=&0&\inn N. \endarray\right $
Our goal is to construct different sequences of problems by modifying the
configuration of the sets $D$ and $N$, and to provide sufficient and necessary
conditions on the size and the location of these sets in order to obtain
sequences of eigenvalues that in the limit recover the eigenvalues of the
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