Ground state and nodal solutions for some elliptic equations involving the fractional Laplacian operator and Trudinger-Moser nonlinearity

Abstract

In this work, we study the existence of ground state and least energy nodal solutions for four classes of problems involving the fractional Laplacian operator with nonlinearities that may have critical exponential growth in the sense of the Trudinguer-Moser inequality. We prove that ground state solutions have a defined signal and we show that the least energy nodal level is greater than twice the ground state level. The first problem is defined in an open bounded interval of R and the second one is defined in the whole real line, both involving the 1/2−Laplacian operator. The third problem, also with the 1/2−Laplacian operator and defined in an open bounded interval, is of Kirchhoff-fractional type with Kirchhoff function of the form mb(t) = a + bt, with a, b > 0. We show the existence of a least energy nodal solution, a nonnegative solution and a nonpositive solution, each of which has minimum energy between the solutions with defined signal. In this case, we also study the asymptotic behavior of nodal solutions, when b → 0+. The last problem addressed is defined in a bounded domain Ω ⊂ R N , N ≥ 2, with Lipschitz boundary ∂Ω and involves the fractional N/s−Laplacian operator, s ∈ (0, 1). In this case, we also found a least energy nodal solution and nontrivial nonnegative and nonpositive solutions, which have minimum energy between the solutions with de ned signal. The main tools used in this study are: Trundiguer-Moser type inequalities, variational methods, deformation lemma and degree theory.