Abstract
By a change of variables with cut-off functions, we study the existence and the asymptotic behavior of positive solutions for a general quasilinear Schrödinger equation which arises from plasma physics. We extend the results of (Adv. Nonlinear Stud. 18(1):131-150, 2017) from alpha=1 to alpha>frac{1}{2}. Especially, we can consider the exponent p in (2,2^{*}) for all Ngeq3.
Highlights
1 Introduction In this paper, we study the existence and asymptotic behavior of positive solutions for the following general quasilinear elliptic equation:
The existence and boundedness of solution was obtained by the critical point theory when p ∈ (2, 2∗) for N ≥ 4 or p ∈ (2, 4)
Theorem 1.1 Assume V (x) = μ > 0, Eq (1) has a positive solution uγ satisfying: (i) uγ is spherically symmetric and uγ decreases with respect to |x|; (ii) uγ ∈ C2(RN ); (iii) uγ together with its derivatives up to order 2 have exponential decay at infinity |Dαuγ | ≤ Ce–δ|x|, x ∈ RN, for some C, δ > 0 and |α| ≤ 2
Summary
We study the existence and asymptotic behavior of positive solutions for the following general quasilinear elliptic equation:. Using the change of variables introduced in [19] and the cut-off function technique in [5], the authors reduced Eq (1) to a semilinear elliptic equation. We can get the asymptotic properties of the solution of (1) with the use of techniques in [1, 3, 20]. Theorem 1.1 Assume V (x) = μ > 0, Eq (1) has a positive solution uγ satisfying: (i) uγ is spherically symmetric and uγ decreases with respect to |x|; (ii) uγ ∈ C2(RN ); (iii) uγ together with its derivatives up to order 2 have exponential decay at infinity |Dαuγ | ≤ Ce–δ|x|, x ∈ RN , for some C, δ > 0 and |α| ≤ 2.
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