Abstract
Abstract In this paper, we study following Kirchhoff type equation: − a + b ∫ Ω | ∇ u | 2 d x Δ u = f ( u ) + h in Ω , u = 0 on ∂ Ω . $$\begin{array}{} \left\{ \begin{array}{lll} -\left(a+b\int_{{\it\Omega}}|\nabla u|^2 \mathrm{d}x \right){\it\Delta} u=f(u)+h~~&\mbox{in}~~{\it\Omega}, \\ u=0~~&\mbox{on}~~ \partial{\it\Omega}. \end{array} \right. \end{array}$$ We consider first the case that Ω ⊂ ℝ3 is a bounded domain. Existence of at least one or two positive solutions for above equation is obtained by using the monotonicity trick. Nonexistence criterion is also established by virtue of the corresponding Pohožaev identity. In particular, we show nonexistence properties for the 3-sublinear case as well as the critical case. Under general assumption on the nonlinearity, existence result is also established for the whole space case that Ω = ℝ3 by using property of the Pohožaev identity and some delicate analysis.
Highlights
Introduction and main resultsThis paper is concerned with following Kirchho type equation:− a + b Ω |∇u| dx ∆u = f (u) + h in Ω, (1.1) u=on ∂Ω, where Ω ⊂ R is a bounded domain or Ω = R, ≤ h ∈ L (Ω) and f ∈ C(R, R)
Existence of at least one or two positive solutions for above equation is obtained by using the monotonicity trick
Nonexistence criterion is established by virtue of the corresponding Pohožaev identity
Summary
This paper is concerned with following Kirchho type equation:. on ∂Ω, where Ω ⊂ R is a bounded domain or Ω = R , ≤ h ∈ L (Ω) and f ∈ C(R, R). This paper is concerned with following Kirchho type equation:. On ∂Ω, where Ω ⊂ R is a bounded domain or Ω = R , ≤ h ∈ L (Ω) and f ∈ C(R, R). Problem like (1.1) is associated with the stationary analogue of the wave equation arising in the study of string or membrane vibrations: utt − a + b |∇u| dx ∆u = f (x, u). Such type equation proposed rst by Kirchho [1] is used to describe the transversal oscillations of a stretched string. More mathematical and physical background and applications of such problems can be found in [1, 4,5,6] and the references therein
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