In this paper we study the existence of solutions for fractional Schrödinger equations of the formwhere V is a potential bounded and the nonlinear term f (x, u) has the critical exponential growth. We prove the existence of at least one weak solution by combining the mountain-pass theorem with the Trudinger-Moser inequality and a version of a result due to Lions for critical growth in R.
We establish the existence and multiplicity of weak solutions for a class of nonlocal equations involving the fractional Laplacian operator, nonlinearities with critical exponential growth, and potentials that may change sign. The proofs of our existence results rely on minimization methods and the mountain pass theorem.
In this paper, we discuss the existence of bound and ground state solutions for a class of fractional Kirchhoff equations defined on the whole real line.The equation involves a nonlinear term with critical exponential growth in the Trudinger-Moser sense. We deal with periodic and asymptotically periodic potential, which may change sign. We handle with the lack of compactness because of the unboundedness of the domain and the critical behavior of the nonlinearity. The main theorems are stated without the well-known Ambrosetti-Rabinowitz condition at infinity. KEYWORDS ground state, fractional Kirchhoff-Schrödinger equation, Trudinger-Moser inequality, variational methods R 2is the so-called Gagliardo seminorm of the function u. We study the existence of bound and ground solutions for Equation (1). It is worthwhile to mention that a solution u of finite energy for (1) is called a bound state solution. It is well known that u ≠ 0 is called ground state solution if admits the smallest energy among the all nontrivial bound states of (1). This class of equations imposes several difficulties. The first one is the "lack of compactness" inherited by the nature 806
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