Abstract
The paper focuses on the modified Kirchhoff equation \begin{align*} -\left(a+b\int_{\mathbb{R}^N}|\nabla u|^2dx\right)\Delta u-u\Delta (u^2)+V(x)u=\lambda f(u), \quad x\in \mathbb{R}^N, \end{align*} where $a,b>0$, $V(x)\in C(\mathbb{R}^N,\mathbb{R})$ and $\lambda<1$ is a positive parameter. We just assume that the nonlinearity $f(t)$ is continuous and superlinear in a neighborhood of $t = 0$ and at infinity. By applying the perturbation method and using the cutoff function, we get existence and multiplicity of nontrivial solutions to the revised equation. Then we use the Moser iteration to obtain existence and multiplicity of nontrivial solutions to the above original Kirchhoff equation. Moreover, the nonlinearity f(t) may be supercritical.
Highlights
In this paper, we are devoted to studying the following modified Kirchhoff equation: (1.1)where a, b > 0, V(x) ∈ C(RN, R), λ < 1 is a positive parameter and f is continuous in R
In 2015, Wu [20] studied the existence of infinitely many small energy solutions for equation (1.5) by applying Clark’s Theorem to a perturbation functional
By Lemma 2.3 and Mountain Pass Theorem, we know that the equation (2.1) has a positive weak solution
Summary
− a + b |∇u|2dx ∆u − u∆(u2) + V(x)u = λ f (u), x ∈ RN, RN where a, b > 0, V(x) ∈ C(RN, R) and λ < 1 is a positive parameter. We just assume that the nonlinearity f (t) is continuous and superlinear in a neighborhood of t = 0 and at infinity. By applying the perturbation method and using the cutoff function, we get existence and multiplicity of nontrivial solutions to the revised equation. We use the Moser iteration to obtain existence and multiplicity of nontrivial solutions to the above original Kirchhoff equation. The nonlinearity f (t) may be supercritical.
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