Multiple solutions for Kirchhoff‐type problems with variable exponent and nonhomogeneous Neumann conditions

Abstract

The existence of at least three weak solutions for a class of differential equations with ( )-Kirchhoff-type and subject to small perturbations of nonhomogeneous Neumann conditions is established under suitable assumptions. Our technical approach is based on variational methods. In addition, an example to illustrate our results is given. K E Y W O R D SVariable exponent Sobolev spaces, ( )-Kirchhoff-type problems, multiple solutions, variational methods M S C ( 2 0 1 0 ) 35J20, 35J60326

Multiple solutions for a class of bi-nonlocal problems with nonlinear Neumann boundary conditions

“…In [12], Guo et al developed the results of Fan [11] for the p(x)-Kirchhoff type problem with Neumann nonlinear boundary condition. Some further results on Kirchhoff type problems with Neumann nonlinear boundary condition can be found in [14,16,20,21], in which the authors studied the existence and multiplicity of solutions for the problem by using the Nehari manifold and fibering maps, Ekeland variational principle or the variational principles due to Bonanno et al [3,4]. Inspired by the papers mentioned above, in this note we study the existence of solutions for bi-nonlocal problem (1.1) with Neumann nonlinear boundary condition.…”

A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions with Positive Initial Energy

“…They obtained a nontrivial weak solution by using the Mountain Pass theorem. For more results in Kirchhoff type equations with variable-exponents nonlinearities we refer the reader to [18,19,20,21,22,23] and references therein. Motivated by the aforementioned works, in the present paper, we study a r(x)− Kirchhoff type equation with variableexponent nonlinearities.…”

Multiple Solutions for a Class of New p(x)-Kirchhoff Problem without the Ambrosetti-Rabinowitz Conditions

“…where Ω is a smooth bounded domain in R N , a ≥ b > 0, p ∈ C(Ω) with 1 < p(x) < N, λ > 0 is a real number, and g : Ω × R → R is a Carathéodory function whose potential satisfies some conditions which will be stated later on. The Kirchhoff type equations involving variable exponent growth conditions have been a very interesting topic in recent years, and we have seen the publication of a great number of manuscripts dealing with this subject (see, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] and references therein). Problems of this type arise in mathematical models of various physical and biological phenomena.…”