Existence,multiplicity, and nonexistence of solutions for a ss="a-plus-plus">p-Kirchhoff elliptic equation on <span class="a-plus-plus inline-equation id-i-eq1"> <span class="a-plus-plus equation-source format-t-e-x" xmlns:search="http://marklogic.com/appservices/search">\(\mathbb{R}^{N}\)span> span>

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• 作者：Lihua Liu ; Chunxiu Zhou
• 关键词：p ; Kirchhoff elliptic equation ; bounded potential ; variational methods ; mountain pass lemma
• 刊名：Boundary Value Problems
• 出版年：2017
• 出版时间：December 2017
• 年：2017
• 卷：2017
• 期：1
• 全文大小：1495KB
• 刊物主题：Difference and Functional Equations; Ordinary Differential Equations; Partial Differential Equations; Analysis; Approximations and Expansions; Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1687-2770
• 卷排序：2017

文摘

In this paper, we study the multiplicity of solutions for the following nonhomogeneous p-Kirchhoff elliptic equation: $$ \biggl(a+\lambda \biggl( \int_{\mathbb{R}^{N}} \bigl(|\nabla{u}|^{p}+|u|^{p} \bigr)\,dx \biggr)^{m} \biggr) \bigl(-\Delta_{p}u+|u|^{p-2}u \bigr) =f(u)+h(x),\quad x\in\mathbb{R}^{N}, $$ (0.1) with \(a,\lambda,m>0\) and \(1< p< N\). By variational methods we prove that problem (0.1) admits at least two solutions under appropriate assumptions on \(f(u)\) and \(h(x)\). The main difficulty to overcome is the lack of an a priori bound for Palais-Smale sequence. Motivated by Jeanjean (Proc. R. Soc. Edinb., Sect. A 129:787-809, 1999), we use a cut-off functional to obtain a bounded (PS) sequence. Also, if \(f(u)=|u|^{q-2}u\), \(p< q<\min\{p(m+1), p^{*}=\frac{pN}{N-p}\}\), and \(h(x)=0\), then we prove that problem (0.1) has at least one nontrivial solution for any \(\lambda\in(0, \lambda^{*}]\) and has no nontrivial weak solutions for any \(\lambda\in(\lambda^{*}, +\infty)\).Keywordsp-Kirchhoff elliptic equationbounded potentialvariational methodsmountain pass lemma1 IntroductionIn this paper, we are interested in the multiplicity of solutions to the following nonhomogeneous p-Kirchhoff elliptic problem: $$ \biggl(a+\lambda \biggl( \int_{\mathbb{R}^{N}} \bigl(\vert \nabla{u} \vert ^{p}+\vert u \vert ^{p} \bigr)\,dx \biggr)^{m} \biggr) \bigl(- \Delta_{p}u+\vert u\vert ^{p-2}u \bigr) =f(u)+h(x),\quad x\in \mathbb{R}^{N}, $$ (1.1) where \(\Delta_{p}u=\operatorname {div}(\vert \nabla{u} \vert ^{p-2}\nabla{u})\) is the p-Laplacian operator, and the nontrivial function \(h(x)\) can be seen as a perturbation term. Problem (1.1) is a generalization of the model introduced by Kirchhoff [2]. More precisely, Kirchhoff proposed the model given by the equation $$ \rho_{tt}- \biggl(\frac{P_{0}}{h}+\frac{E}{2L} \int^{L}_{0}u^{2}_{x}\,dx \biggr)u_{xx}=0,\quad 0< x< L, t>0, $$ (1.2) which takes into account the changes in length of string produced by transverse vibration. The parameters in (1.2) have the following meaning: L is the length of the string, h is the area of cross-section, E is the Young modulus of material, ρ is the mass density, and \(P_{0}\) is the initial tension.The equation $$ \rho_{tt}-M \bigl(\Vert \nabla u\Vert ^{2}_{2} \bigr)\Delta u=f(x,u), \quad x\in\Omega, t>0, $$ (1.3) generalizes equation (1.2), where \(M: \mathbb{R}^{+}\to\mathbb{R}\) is a given function, Ω is a domain of \(\mathbb{R}^{N}\). The stationary counterpart of (1.3) is the Kirchhoff-type elliptic equation $$ -M \bigl(\Vert \nabla u\Vert ^{2}_{2} \bigr) \Delta u=f(x,u), \quad x\in\Omega, t>0. $$ (1.4) Some classical and interesting results on Kirchhoff-type elliptic equations can be found, for example, in [3–<span class="CitationRef">9].Particularly, Li et al. [10] considered the Kirchhoff-type problem $$ \biggl(a+\lambda \biggl( \int_{\mathbb{R}^{N}} \bigl(\vert \nabla{u} \vert ^{2}+b \vert u\vert ^{2} \bigr)\,dx \biggr) \biggr) (-\Delta u+bu) =f(u),\quad x \in \mathbb{R}^{N}, $$ (1.5) where \(N \geq3\), with constants \(a, b > 0\) and \(\lambda\geq0\) under the following assumptions: \((H_{1})\)\(f\in C(\mathbb{R}^{+},\mathbb{R}^{+})\), \(\vert f(t)\vert \leq C(1+t^{q-1})\) for all \(t\in\mathbb{R}^{+}=[0,+\infty)\) and some \(q\in(2,2^{*})\), where \(2^{*}=\frac{2N}{N-2}\) for \(N\geq3\);