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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.

MSC:
 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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