Perturbations of a critical fractional equation
With the aim to transfer the known results for nonlinear perturbed problems with the usual Laplacian the authors consider the critical problem with fractional Laplacian NEWLINE\[NEWLINE (-\triangle)^{\frac{\alpha}{2}}u=|u|^{p-2}u+f(x)\quad \text{in}\quad \Omega\subset \mathbb{R}^n, \; u=o\quad \text{on}\quad \partial \Omega, \eqno{(P)} NEWLINE\]NEWLINE where \(0<\alpha <2, p=\frac{2N}{N-\alpha}, N>\alpha, f\in H^{\frac{\alpha}{2}}(\Omega)\) and is small in the following sense \(\int\limits_{\Omega}f\varphi<c(\alpha,N)\|\varphi\|_{H_0^{\frac{\alpha}{2}}}^{(N+\alpha)/\alpha}\) for all \(\varphi \in H_0^{\frac{\alpha}{2}}(\Omega)\) with \(\|\varphi\|_p=1,c(\alpha, N)=\frac{2\alpha}{N-\alpha}\left(\frac{N-\alpha}{N+\alpha}\right)^{\frac{N+\alpha}{2\alpha}}\). The main results of the article are the followingNEWLINENEWLINE{Theorem 1.} Under the made assumptions the problem (P) has at least two solutions and when \(f>0\) a.e. in \(\Omega\), then these solutions are nonnegative a.e. in \(\Omega\).NEWLINENEWLINE{Theorem 2.} Let the strict inequality in Theorem 1 be replaced by the nonstrict condition NEWLINE\[NEWLINE \int\limits_{\Omega}f\varphi\leq c(\alpha,N)\|\varphi\|_{H_0^{\frac{\alpha}{2}}}^{(N+\alpha)/\alpha}\quad \text{for}\quad \forall \varphi \in H_0^{\frac{\alpha}{2}}(\Omega),\quad\|\varphi\|_p=1. NEWLINE\]NEWLINE Then the problem (P) has at least one solution and when \(f\) is nonnegative a.e. in \(\Omega\), then this solution is nonnegative a.e. in \(\Omega\).
scientific article; zbMATH DE number 6369629