12-Laplacian problems with exponential nonlinearity

Abstract

By exploiting a suitable Trudinger-Moser inequality for fractional Sobolev spaces, we obtain existence and multiplicity of solutions for a class of one-dimensional nonlocal equations with fractional diffusion and nonlinearity at exponential growth.2010 Mathematics Subject Classification. 34K37, 34B10, 46E30.

Ground states of nonlocal scalar field equations with Trudinger-Moser critical nonlinearity

“…In this paper, we use the minimization technique over the Nehari manifold in order to get ground state solutions. We adopt some arguments from [4] combined with those used in [10,23].…”

“…For fractional problems in bounded domains of R N with N > 2s involving critical nonlinearities we cite [5,9,22,30] and [18] for the whole space with vanishing potentials. In [21] the authors investigated properties of the ground state solutions of (−∆) s u + u = u p in R. Recently, in [23], nonlocal problems defined in bounded intervals of the real line involving the square root of the Laplacian and exponential nonlinearities were investigated, using a version of the Trudinger-Moser inequality due to Ozawa [28]. As it was remarked in [23] the nonlinear problem involving exponential growth with fractional diffusion (−∆) s requires s = 1/2 and N = 1.…”

“…In [21] the authors investigated properties of the ground state solutions of (−∆) s u + u = u p in R. Recently, in [23], nonlocal problems defined in bounded intervals of the real line involving the square root of the Laplacian and exponential nonlinearities were investigated, using a version of the Trudinger-Moser inequality due to Ozawa [28]. As it was remarked in [23] the nonlinear problem involving exponential growth with fractional diffusion (−∆) s requires s = 1/2 and N = 1. In [19] some nonlocal problems in R with vanishing potential, thus providing compactifying effects, are considered.…”

Three nontrivial solutions for nonlinear fractional Laplacian equations

“…So, we can prove existence and multiplicity of such solutions by applying to ϕ several abstract results of critical point theory, such as minimax principles (see [32]) and Morse theory (see [13]). Some results of this type can be found, for instance, in [6,14,17,25,27,29,38].…”

Nontrivial Solution for the Fractional p-Laplacian Equations via Perturbation Methods

“…This type of problem arises in many different applications, such as continuum mechanics, phase transition phenomena, population dynamics, and game theory, as they are the typical outcome of stochastically stabilization of Lévy processes; see [1][2][3][4][5][6][7][8][9] and the references therein. The literature on nonlocal operators and their applications is very interesting and quite large; we refer the interested reader to [4,[10][11][12][13][14][15][16][17][18][19][20][21] and the references therein. For the basic properties of fractional Sobolev spaces, we refer the interested reader to [22,23].…”