## Ground state solutions for Bessel fractional equations with irregular nonlinearities.(English)Zbl 1403.35281

Summary: We consider the semilinear fractional equation $(I-\Delta)^s u=a(x)|u|^{p-2}u\,\,\text{in}\,\,\mathbb R^N,$ where $$N\geq 3$$, $$0<s<1$$, $$2<p<2N/(N-2s)$$ and $$a$$ is a bounded weight function. Without assuming that $$a$$ has an asymptotic profile at infinity, we prove the existence of a ground state solution.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35S05 Pseudodifferential operators as generalizations of partial differential operators 35J61 Semilinear elliptic equations 35R11 Fractional partial differential equations

### Keywords:

Bessel fractional operator; fractional Laplacian
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### References:

 [1] N. Ackermann,J. Chagoya;Ground states for irregular and indefinite superlinear Schr¨odinger equations, J. Differential Equations, 261 (2016), 5180–5201. EJDE-2018/CONF/25GROUND STATE SOLUTIONS233 · Zbl 1347.35114 [2] M. M. Fall, V. Felli; Unique continuation properties for relativistic Schr¨odinger operators with a singular potential, Discrete Contin. Dyn. Syst., 35 (2015), no. 12, 5827–5867. · Zbl 1336.35356 [3] P. Felmer, A. Quaas, J. Tan; Positive solutions of the nonlinear Schr¨odinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237–1262. · Zbl 1290.35308 [4] P. Felmer, I. Vergara; Scalar field equation with non-local diffusion, Nonlinear Differ. Eq. Appl., 22 (2015), 1411–1428. · Zbl 1326.35136 [5] P. H. Rabinowitz; On a class of nonlinear Schr¨odinger equations, Z. Angew. Math. Phys., 43 (1992), 270–291. · Zbl 0763.35087 [6] W. Rudin; Functional analysis, second edition, McGraw-Hill Inc., New York, 1991. · Zbl 0867.46001 [7] S. Secchi; Ground state solutions for nonlinear fractional Schr¨odinger equations in RN, Journal of Mathematical Physics, 54 031501 (2013). [8] S. Secchi; Concave-convex nonlinearities for some nonlinear fractional equations involving the Bessel operator, Complex Variables and Elliptic Equations, 62 (2017), 654-669. · Zbl 1365.35033 [9] S. Secchi; On some nonlinear fractional equations involving the Bessel operator, J. Dynam. Differential Equations, 29 (2017), no. 3, 1173–1193. · Zbl 1377.35107 [10] S. Secchi; On a generalized pseudorelativistic Schr¨odinger equation with supercritical growth, arXiv:1708.03479. [11] E. M. Stein; Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J. 1970. Simone Secchi Dipartimento di Matematica e Applicazioni, Universit‘a degli Studi di Milano Bicocca, Italy E-mail address: simone.secchi@unimib.it 1. Introduction2. Variational setting3
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