Nehari manifoldsign-changing weight functions35J6035J5035R11In this paper, we consider a fractional p-Laplacian system with both concave–convex nonlinearities and sign-changing weight functions in bounded domains: ( Δ ) p s u = λ f ( x ) | u | q 2 u + 2 αα + β h ( x ) ...
x ∈ R~N. f has the subcritical growth but higher than Γ(x)|u|~(q-2)u; however, the nonlinearity f(x,u)-Γ(x)|u|~(q-2)u may change sign. If V is coercive, we investigate the existence of ground state solutions for p-Laplacian equation.Huxiao LuoShengjun LiWenfeng He...
In this paper, we consider a fractional p-Laplacian system with both concave-convex nonlinearities and sign-changing weight functions in bounded domains. With the help of the Nehari\\ manifold, we prove that the system has at least two nontrivial solutions when the pair of the parameters (\\...
Yu X.H.: The Nehari manifold for elliptic equation involving the square root of the Laplacian. J. Differ. Equ. 252 (2), 1283–1308 (2012)X. He, M. Squassina, W. Zou, The Nehari manifold for fractional systems involving critical nonlinearities, Comm. Pure Applied Anal. 15 (2016), ...
(1.2), we recall that fractional Laplacian operators are the infinitesimal generators of Lévy stable diffusion processes. They have application in several areas such as anomalous diffusion of plasmas, probability, finance, and population dynamics. For more details on the application background, we ...
Specif i cally, weprove that the knowledge of the local source–to–solution map for the fractionalLaplacian, given on an arbitrary small open nonempty a priori known subset ofa smooth closed connected Riemannian manifold, determines the Riemannianmanifold up to an isometry. This can be viewed as...
Since the fractional Laplacian operator is a nonlocal one, it is difficult to use the methods for local operator directly. For instance, the ground state for −Δ decays exponentially at infinity. In contrast, the ground state for (1.7) decays algebraically at infinity. So, the research of...
So let {ej}∞j=1 be complete or- thonormal system of eigenfunctions of Laplacian and let PN be orthoprojector on the first N eigen- functions ej. And let ξ0N := PN ξ0 be initial data to approximate solution uN , that is (3.8) ∂t2uN + γ(−∆x)α∂tuN − ∆xuN ...
Yamabe problem with an isolated singularity $(-\Delta)^\gamma w= c_{n, {\gamma}}w^{\frac{n+2\gamma}{n-2\gamma}}, w>0 \ \mbox{in} \ \R^n \backslash \{0\}$ We follow a variational approach, in which the key is the computation of the fractional Laplacian in polar ...
Before proving the main results, we denote the Nehari manifold, the critical set, the least energy, and the set of least energy solutions of I_{\varepsilon} as follows: \begin{aligned}& \mathscr{N}_{\varepsilon}:=\bigl\{ u\in H^{s}\setminus\{0 \}:I_{\varepsilon}'(u)u=0...