Existence of solutions for a p(x)-Kirchhoff-type equation

By Guowei Dai, R. Hao. 2009

195 citations recorded by the source corpus.

This paper is concerned with the existence and multiplicity of solutions to a class of p(x)-Kirchhoff-type problem with Dirichlet boundary data. By means of a direct variational approach and the theory of the variable exponent Sobolev spaces, we establish conditions ensuring the existence and multiplicity of solutions for the problem.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5e14e7961f64146b675e7e48ceacee0915125cf6

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Existence and multiplicity of the solutions of the p(x)–Kirchhoff type equation via genus theory (2011, methodology): “In [12], the authors studied the problem (P) and showed the weak and nontrivial weak solutions by the help of the Ambrosetti– Rabinowitz’s condition and Mountain Pass theorem when f : R!R satisfies the Carathéodory condition and M.”
Three solutions for a nonlocal Dirichlet boundary value problem involving the p(x)-Laplacian (2013, methodology): “In [33], by a direct variational approach, we establish conditions ensuring the existence and multiplicity of solutions for the following problem:”
Multiplicity of solutions for p ( x ) -polyharmonic elliptic Kirchhoff equations (2011, methodology): “4) when L = 1 andM is strictly positive have already been given in [16] via variational methods, while in [17] the authors use a three critical point theorem due to Ricceri to solve the problem.”
Overview of differential equations with non-standard growth (2010, background): “After the completion of this paper, we learned that Liu, Qiu and Zhang [174, 177, 179] have partially generalized these to equations with ∆p(·) on the left hand side.”
Multiplicity results for a class of $p(x)$-Kirchhoff type equations with combined nonlinearities (2012, methodology): “Also in [8], the authors considered the case when f verifies the condition f(x, t) ≥ C|t|, t→ 0, (1.”
Existence and multiplicity results for a newp(x)-Kirchhoff problem (2019, background): “Some important and interesting results can be found, for example, in [4, 5, 3, 2, 7, 10, 8, 11, 12, 13, 15, 18, 21, 19, 22, 23, 24, 25, 29, 30, 31, 32, 33, 40] and references therein.”
Soliton solutions to Kirchhoff type problems involving the critical growth in RN (2013, methodology): “In [17], by means of a direct variational method, the authors proved the existence and multiplicity of solutions to a class of p-Kirchhoff-type problem with Dirichlet boundary data.”
Multiple solutions for a p(x)-Kirchhoff-type equation with sign-changing nonlinearities (2013, methodology): “Motivated by the papers [11,12], the properties of p(x)-Kirchhoff Laplace operators in [15] and the ideas introduced in [21,22], the goal of this article is to study the multiplicity of weak solutions for problem (1.”