Abstract
In this paper, we study the Kirchhoff-type equation: − a + b ∫ ℝ 3 ∇ u 2 d x Δ u + V x u = Q x f u , in ℝ 3 , where a , b > 0 , f ∈ C 1 ℝ 3 , ℝ , and V , Q ∈ C 1 ℝ 3 , ℝ + . V x and Q x are vanishing at infinity. With the aid of the quantitative deformation lemma and constraint variational method, we prove the existence of a sign-changing solution u to the above equation. Moreover, we obtain that the sign-changing solution u has exactly two nodal domains. Our results can be seen as an improvement of the previous literature.
Highlights
Consider the Kirchhoff-type equation ð− a + b j∇uj2dx Δu + VðxÞu = QðxÞf ðuÞ, x ∈ R3, R3ð1Þ where a, b > 0, V, Q : R3 → R+, and f : R3 → R are continuous functions
V and Q are vanishing at infinity
Similar to [1], we say ðV, QÞ ∈ Q, if QðxÞ and VðxÞ satisfy the following conditions: (VQ1) VðxÞ, QðxÞ > 0 for any x ∈ R3 and Q ∈ L∞ðR3Þ (VQ2) If fAngn ⊂ R3 is a sequence of Borel sets such that their Lebesgue measures ∣An ∣ ≤R for all n ∈ N and some R > 0, ð lim
Summary
Ð1Þ where a, b > 0, V, Q : R3 → R+, and f : R3 → R are continuous functions. V and Q are vanishing at infinity. Li et al [8] investigated the sign-changing solution to the following problem by using the constraint variational method. They supposed that f ðx, tÞ satisfies the following conditions: (f1) f ðx, tÞ = oðtÞ as t → 0 uniformly in x ∈ R3 (f2) f ðx, tÞ ∈ CðR3 × R, RÞ and ∣f ðx, tÞ ∣ ≤cð∣t∣+jtjq−1Þ for some q ∈ 1⁄22, 2∗Þ, where c > 0 and 2∗ = 2N/ðN − 2Þ (f3) f ðx, tÞ/t4 → ∞ as t → ∞ uniformly in x ∈ R3 (f4) f ðx, tÞ/t3 is an increasing function of t ∈ R \ f0g.
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