Multiplicity of Solutions for Kirchhoff-Type Problem with Two-Superlinear Potentials

Existence of nonnegative nontrivial solutions for Kirchhoff type problems with variable exponent

“…Moreover, Equation (1.2) is related to the stationary analogue equation, that is, { u tt − (a + b∫ Ω |∇u| 2 dx)Δu = 𝑓 (x, u), in Ω, u = 0, on 𝜕Ω, (1.3) which was first proposed by Kirchhoff [1] in 1883 as an extension of the classical D'Alembert wave equation for free vibrations of elastic strings. The problem (1.2) has been studied by many authors [2][3][4][5][6][7][8][9][10][11][12][13][14]. Many solvability conditions on the nonlinearity 𝑓 near zero and infinity for the problem (1.2) have been considered, such as the superlinear case [8] and asymptotical linear case [11].…”

“…$$ which was first proposed by Kirchhoff [1] in 1883 as an extension of the classical D'Alembert wave equation for free vibrations of elastic strings. The problem () has been studied by many authors [2–14]. Many solvability conditions on the nonlinearity $$$ f $$$ near zero and infinity for the problem () have been considered, such as the superlinear case [8] and asymptotical linear case [11].…”

Sign-changing solutions for Kirchhoff type problems with variable exponent

“…This causes some mathematical difficulties which make the study of (1.3) particularly interesting. In the past few years, The problem (1.3) has been studied by many authors, for example [1,4,5,6,7,13,14,15,16,17,20,21,22]. Many solvability conditions on the nonlinearity f near zero and infinity for the problem (1.3) have been considered, such as the superlinear case [14] and asymptotical linear case [17].…”

Sign-Changing Solutions for Superlinear Kirchhoff Type Problem via the Nehari Method

“…Secondly, we prove that there exists a constant 1 R such that if In order to prove the main result, we need the following lemmas. However, the proofs of them are standard and similar to Lemmas 3.1 -3.4 of our recent paper [10], so we omit their proofs. Note that in [10], we only proved the existence of sign-changing solution for (1.1) when b is sufficiently small, and the number of nodal domains is not obtained there.…”

“…However, the proofs of them are standard and similar to Lemmas 3.1 -3.4 of our recent paper [10], so we omit their proofs. Note that in [10], we only proved the existence of sign-changing solution for (1.1) when b is sufficiently small, and the number of nodal domains is not obtained there. By contrast, here for any 0 b > , a sign-changing solution is obtained, and it has exactly two nodal domains.…”