Trudinger-Moser Inequalities in Fractional Sobolev-Slobodeckij Spaces and Multiplicity of Weak Solutions to the Fractional-Laplacian Equation

In line with the Trudinger-Moser inequality in the fractional Sobolev-Slobodeckij space due to [S. Iula, A note on the Moser-Trudinger inequality in Sobolev-Slobodeckil spaces in dimension one, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28 (2017), no. 4, 871-88/1 and [E. Parini and B. Ruf, On the Moser-Trudinger inequality in fractional Sobolev-Slobodeckij spaces, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018), no. 2, 315 3191, we establish a new version of the Trudinger Moser inequality in w(s,p) (R-N),. Define parallel to u parallel to(1,tau) = ([u](Ws,p(RN))(p) + tau parallel to u parallel to(p)(p))(1/p) for any tau > 0. There holds sup(u is an element of Ws,p(RN),parallel to u parallel to 1,tau <= 1) integral(RN) Phi(N,S)(alpha vertical bar u vertical bar(N/N-S)) < +infinity where s is an element of(0,1), sp = N, alpha is an element of[0, alpha(*)) and Phi(N,s)(t) = e(t) - Sigma(jp-2)(i=0) t(j)/j(i). Applying this result, we establish sufficient conditions for the existence of weak solutions to the following quasilinear nonhomogeneous fractional-Laplacian equation: (-Delta)(p)(s)u(x) + V(x)vertical bar u(x)vertical bar(p-2)u(x) = f(x,u) +epsilon h(x) in R-N, where V(x) has a positive lower bound, f(x, t) behaves like e(alpha vertical bar t vertical bar N/(N-S)) , h is an element of(W-s,W- p(R-N))* and epsilon > 0. Moreover, we also derive a weak solution with negative energy.