Critical quasilinear Schroedinger equations with electromagnetic fields

TL;DR

The paper studies the existence of nontrivial weak solutions to the nonlinear electromagnetic Schrödinger equation with critical growth in $\mathbb{R}^N$, driven by the magnetic $p$-Laplacian for $1<p<N$ and with Sobolev exponent $p^*=Np/(N-p)$. Using a variational approach and the Mountain Pass Theorem in the complex-valued space $W^{1,p}_{A,V}$, it overcomes the double lack of compactness via Lions concentration-compactness and a specialized complex inequality, while addressing gauge invariance. The main contributions include establishing Mountain Pass geometry for the associated energy functional $J_A$, proving the existence of a $(PS)_c$ sequence below a critical threshold $c_P$, and confirming convergence to a nontrivial critical point, aided by a tailored test function family and energy estimates. This work extends the theory of quasilinear PDEs with magnetic fields to critical nonlinearities, offering a foundational existence result with potential implications for nonlinear quantum models and magnetic materials.

Abstract

The p-Laplace operator in the entire N-dimensional Euclidean space, subject to external electromagnetic potentials, is investigated. In the general case 1<p<N, the existence of at least one solution of mountain pass type to a weighted critical equation is proved. Our technique relies on variational methods and faces a twofold difficulty: double lack of compactness, which requires concentration compactness arguments; and a complex quasilinear framework, which entails appropriate inequalities.

Theorems & Definitions (19)

• Theorem 1
• Proposition 1: LionsRev1
• Proposition 2: BTW
• Theorem 2: AR
• Lemma 1
• Remark 1
• Definition 1
• Lemma 2
• proof
• Lemma 3