Abstract
Here, a nonlocal nonlinear operator known as the fractional (p,q)-Laplacian is considered. The existence of a mountain pass solution is proved via critical point theory and variational methods. To this aim, the well-known theorem on the construction of the critical set of functionals with a weak compactness condition is applied.
Highlights
The quasilinear operator (p, q)-Laplacian has been used to model steady-state solutions of reaction–diffusion problems arising in biophysics, in plasma physics and in the study of chemical reactions
4 A mountain pass type solution Here, we study the existence of a mountain pass type solution of (1.2)
In order to apply mountain pass theorem, we show that the functional Iλ satisfies in the Palais– Smale compactness condition and has a particular geometric structure
Summary
The quasilinear operator (p, q)-Laplacian has been used to model steady-state solutions of reaction–diffusion problems arising in biophysics, in plasma physics and in the study of chemical reactions. The differential operator p + q is known as the (p, q)-Laplacian operator, if p = q, where j, j > 1 denotes the j-Laplacian and is defined by ju := div(|∇u|j–2∇u) It is not homogeneous, some technical difficulties arise in applying the usual methods of the theory of elliptic equations, for further details see [1, 6, 11, 12]. Marano et al [27] studied the existence of solutions for the nonlinear elliptic problem of (p, q)-Laplacian type. In [13] existence and multiplicity results for fractional (p, q)-Laplacian type equations in RN have been studied. Bhakta et al [7] studied the existence of infinitely many nontrivial weak solutions of the following fractional (p, q)-Laplacian equation:. We study a quasilinear problem, that is, a fractional (p, q)-Laplacian elliptic problem as
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