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Elliptic problem driven by different types of nonlinearities. (English) Zbl 1486.35418

Summary: In this paper we establish the existence and multiplicity of nontrivial solutions to the following problem: \[ (-\Delta)^{\frac{1}{2}}u+u+\bigl(\ln \vert \cdot \vert \ast \vert u \vert^2\bigr)=f(u)+\mu \vert u \vert^{- \gamma -1}u,\quad \text{in }\mathbb{R}, \] where \(\mu >0, (\ast)\) is the convolution operation between two functions, \(0<\gamma <1\), \(f\) is a function with a certain type of growth. We prove the existence of a nontrivial solution at a certain mountain pass level and another ground state solution when the nonlinearity \(f\) is of exponential critical growth.

MSC:

35R11 Fractional partial differential equations
35J61 Semilinear elliptic equations
35J75 Singular elliptic equations
35R09 Integro-partial differential equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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