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Variational methods for non-local operators of elliptic type. (English) Zbl 1303.35121
Summary: We study the existence of non-trivial solutions for equations driven by a non-local integrodifferential operator \({\mathcal L}_K\) with homogeneous Dirichlet boundary conditions. More precisely, we consider the problem \[ \begin{aligned} {\mathcal L}_K u+\lambda u+ f(x,u)=0\quad &\text{in }\Omega,\\ u= 0\quad &\text{in }\mathbb{R}^n\setminus\Omega,\end{aligned} \] where \(\lambda\) is a real parameter and the nonlinear term \(f\) satisfies superlinear and subcritical growth conditions at zero and at infinity. This equation has a variational nature, and so its solutions can be found as critical points of the energy functional \({\mathcal J}_\lambda\) associated to the problem. Here we get such critical points using both the Mountain Pass Theorem and the Linking Theorem, respectively when \(\lambda<\lambda_1\) and \(\lambda\geq \lambda_1\), where \(\lambda_1\) denotes the first eigenvalue of the operator \(-{\mathcal L}_K\).
As a particular case, we derive an existence theorem for the following equation driven by the fractional Laplacian \[ \begin{aligned} (-\Delta)^su- \lambda u= f(x,u)\quad &\text{in }\Omega,\\ u= 0\quad &\text{in }\mathbb{R}^n\setminus\Omega.\end{aligned} \] Thus, the results presented here may be seen as the extension of some classical nonlinear analysis theorems to the case of fractional operators.

35R11 Fractional partial differential equations
35A15 Variational methods applied to PDEs
35A16 Topological and monotonicity methods applied to PDEs
35R09 Integro-partial differential equations
45K05 Integro-partial differential equations
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