Existence, multiplicity and nonexistence results for Kirchhoff type equations

Abstract

In this paper, we study following Kirchhoff type equation: $$\begin{array}{} \left\{ \begin{array}{lll} -\left(a+b\int_{{\it\Omega}}|\nabla u|^2 \mathrm{d}x \right){\it\Delta} u=f(u)+h~~&\mbox{in}~~{\it\Omega}, \\ u=0~~&\mbox{on}~~ \partial{\it\Omega}. \end{array} \right. \end{array}$$ 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.

Initial boundary value problem of a class of mixed pseudo-parabolic Kirchhoff equations

“…For the non-local source case, we refer to some very recent related references, e.g. [24] for the threshold results of the global existence and non-existence for the signchanging weak solutions of thin film equation; [14] and [8] for the Kirchhoff type problem with non-local source 1 |Ω| Ω |u| q−1 udx; [15] for the finite time blow-up of solutions with non-positive initial energy J(u 0 ) and non-local source 1 |Ω| Ω |u| q dx with q > 1; [18] for the well-posedness of pseudo-parabolic equation with singular potential term at three initial energy levels, the logarithmic nonlinearity in [7] and the non-local source 1 |Ω| Ω |u| q−1 udx in [30]. As far as we know, there are few research works concerned with the sign-changing solutions for the mixed pseudoparabolic Kirchhoff equation.…”

Global existence and blowup in infinite time for a fourth order wave equation with damping and logarithmic strain terms

“…For all j, the solution γ h j ∈ C 2 [0, T ] of ( 21) is global and unique, which shows that there exists a unique v h satisfying (19) and solving the problem (20). By the Poincaré inequality, there holds…”

“…We know that the problem ( 16)-( 18) admits a local solution by taking the limit in (20) which satisfies (34). Hence, we obtain the solution v of ( 16)- (18).…”

“…Such special results will inevitably require new estimation techniques and improved potential well methods. An often indispensable tool is the elliptic theory [20,22,35] to establish the potential well theory. The potential well theory is a powerful tool for studying the dependence of the well-posedness of solution to the partial differential equations on the initial data, which can effectively deal with wave equations [28,45], the pseudo-parabolic equations with non-local source [37], polynomial source [41], singular potential [25], and the coupled parabolic systems [40].…”

Ground state solutions for planar periodic Kirchhoff type equation with critical exponential growth

“…For more mathematical and physical background on Kirchhoff type problems, we refer the readers to previous works. [2][3][4][5][6][7][8][9] After Lions 10 proposed an abstract functional analysis framework to Kirchhoff Equation (1.2), based on variational methods, a number of important results of the existence and multiplicity of solutions for problem (1.2) have been established when 𝑓 satisfies various conditions, here we just cite previous studies for the subcritical growth case [11][12][13][14][15][16][17][18][19][20] :…”