The Solvability of p-Kirchhoff Type Problems with Critical Exponent
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Authors: H. BENCHIRA, A. MATALLAH, A. BENAISSA, S. G. GEORGIEV AND K. ZENNIR
DOI: 10.46793/KgJMat2609.1445B
Abstract:
The article aims to study new and current problems in the theory of nonclassical partial differential equations and their applications, proving the existence and nonexistence of solutions to p-Kirchhoff type problems with critical exponent of Sobolev in ℝn, which are of great interest in the study of mathematical physics equations. We show the existence of a local minimizer with negative/positive energy by using variational methods. More precisely, we considered a minimization of Eλ constrained in a neighborhood of zero using the Ekeland variational principle, then we found the first critical point of Eλ which achieves the local minimum of Eλ whose level is negative; next around the zero point, using the mountain pass theorem, we also obtained a critical point whose level is positive. In addition, we studied the case of λ = 0, where there is no non-trivial solution using the contradiction principle. We also established infinite solutions and discuss the different cases.
Keywords:
Variational methods, Kirchhoff equations, critical Sobolev exponent, Lyapunov dunctions, nonlinear equations.
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