Abstract
This paper is devoted to the 4-superlinear Schrödinger–Kirchhoff equation − a + b ∫ ℝ 3 ∇ u 2 d x Δ u + V x u = f x , u , in ℝ 3 , where a > 0 , b ≥ 0 . The potential V here is indefinite so that the Schrödinger operator − Δ + V possesses a finite-dimensional negative space. By using the Morse theory, we obtain nontrivial solutions for this problem.
Highlights
Introduction and Main ResultsR3 where a > 0, b ≥ 0 are constants. This equation arises when we look for stationary solutions of the equation ρ
Introduction and Main ResultsIn this work, we consider the Schrödinger–Kirchhoff type equation of the form ð− a + b j∇uj2dx Δu + VðxÞu = f ðx, uÞ, x ∈ R3, ð1ÞR3 where a > 0, b ≥ 0 are constants
This equation arises when we look for stationary solutions of the equation ρ
Summary
R3 where a > 0, b ≥ 0 are constants. This equation arises when we look for stationary solutions of the equation ρ. Such an indefinite situation was studied in [16] To overcome these difficulties and the difficulty that the Sobolev embedding H1ðR3Þ°L2ðR3Þ is not compact, it is assumed in Journal of Function Spaces [16] that. (ii) Note that in (f2), we have not required that the limit (8) holds uniformly (iii) In order to produce critical points of Φ, eventually, we will encounter the compactness problem. For this issue, we assume assumption (f4). Having established the (PS) condition, the proof of Theorem 3 is quite similar to that of ([16], Theorem 3); we omit it here
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