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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 ${\Cal L}_K$ with homogeneous Dirichlet boundary conditions. More precisely, we consider the problem $$\align {\Cal L}_K u+\lambda u+ f(x,u)=0\quad &\text{in }\Omega,\\ u= 0\quad &\text{in }\bbfR^n\setminus\Omega,\endalign$$ 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 ${\Cal J}_\lambda$ 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 $\lambda\ge \lambda_1$, where $\lambda_1$ denotes the first eigenvalue of the operator $-{\Cal L}_K$. As a particular case, we derive an existence theorem for the following equation driven by the fractional Laplacian $$\align (-\Delta)^su- \lambda u= f(x,u)\quad &\text{in }\Omega,\\ u= 0\quad &\text{in }\bbfR^n\setminus\Omega.\endalign$$ Thus, the results presented here may be seen as the extension of some classical nonlinear analysis theorems to the case of fractional operators.

MSC:
35R11Fractional partial differential equations
35A15Variational methods (PDE)
35A16Topological and monotonicity methods (PDE)
35R09Integro-partial differential equations
45K05Integro-partial differential equations
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