Sentence examples for many radial solutions from inspiring English sources

Berestycki-Lions [1, 2] advocated it for the first time; they obtained the existence of infinitely many radial solutions of the autonomous equation.

If Q is a radial function such that (Q_(0)=Q_ infty)=0), then there exist infinitely many radial solutions to problem ((mathscr{P}^{Q}_{0})).

If ζ is a radial function on ∂ Ω, then system (5) has infinitely many radial solutions ( u i, ϕ i ) ∈ H 0, r 1 × H r 1 with ∥ ∇ u i ∥ 2 → + ∞ and { ϕ i } bounded in L ∞.

About the existence of infinitely many radial solutions, Jin and Wu in [15] proved the result by applying a fountain theorem for (N=2, 3), (V x) equiv1) and (f x,u)) is subcritical, superlinear at the origin and 4-superlinear at infinity.

For certain f ( | x |, u ), we are interested in the functional on a group invariant subspace, and we obtain the existence of infinitely many radial solutions and non-radial solutions of the equation, which extends the result of (Bartsch and Willem in J. Funct. Anal. 117:447-460, 1993) to the space W 1, p ( R N ).

According to the information I have, for Kirchhoff-type problems in R N, the results are seldom, in [29] Jin and Wu obtained three existence results of infinitely many radial solutions for Kirchhoff-type problems in R N, and in [30] Ji established the existence of infinitely many radially symmetric solutions of Kirchhoff-type p ( x ) -Laplacian equations in R n.

In [11, 12], Wu considered the nonlinear polyharmonic systems and equation of the type (1.1), respectively, and obtained some sufficient conditions for the existence of infinitely many radial positive entire solutions with the prescribed asymptotic behavior at infinity.

In [5], the existence of infinitely many sign-changing solutions of (1.1) with p = 2 has been obtained when N ≥ 4, λ > 0 and Ω is a ball, while it has been shown in [6] that (1.1) with p = 2 has infinitely many sign-changing radial solutions when N ≥ 7, λ > 0 and Ω also is a ball.

In the present paper, we aim to find the existence of infinitely many radial and non-radial solutions of problem (1.1), and extend the result of [5] to the space W 1, p ( R N ).

Later, Bartsch and De Figueiredo [9] proved that the system admits infinitely many radial as well as non-radial solutions.

In [3], [4], the authors proved the existence of infinitely many pairs of high energy radial solutions when 2 < p < 5, and also obtained some existence results for 1 < p ≤ 2. Sun [5] studied the existence of infinitely many solutions when p ∈ ( 0, 1 ).