[Paper Review] Existence of Nontrivial Solutions for p-Laplacian Equations in {R}^{N}

Chungen Lıu, Youquan Zheng|arXiv (Cornell University)|Nov 24, 2010

Nonlinear Partial Differential Equations22 references4 citations

TL;DR

This paper establishes the existence of nontrivial weak solutions for p-Laplacian equations in R^N with sign-changing potentials and subcritical p-superlinear nonlinearities, using the cohomological linking method on cones. The key result proves that such solutions exist for all real values of the parameter λ, even without periodicity assumptions.

ABSTRACT

In this paper, we consider a p-Laplacian equation in {R}^{N}with sign-changing potential and subcritical p-superlinear nonlinearity. By using the cohomological linking method for cones developed by Degiovanni and Lancelotti in 2007, an existence result is obtained. We also give a result on the existence of periodic solutions for one-dimensional $p$-Laplacian equations which can be proved by the same method.

Motivation & Objective

• To establish the existence of nontrivial weak solutions for p-Laplacian equations in R^N with sign-changing potentials.
• To extend existing results beyond constant-sign potentials and periodic settings.
• To prove existence under weaker growth and coercivity conditions than the Ambrosetti-Rabinowitz condition.
• To demonstrate that the cohomological linking method applies to p-Laplacian problems with sign-changing potentials.
• To provide a parallel existence result for periodic solutions in one dimension using the same method.

Proposed method

• Utilizes the cohomological linking method for cones developed by Degiovanni and Lancelotti (2007) to find critical points of the energy functional.
• Defines the energy functional Φ(u) = (1/p)∫|∇u|^p + |u|^p dx − (λ/p)∫V(x)|u|^p dx − ∫F(x,u) dx on the Sobolev space W^{1,p}(R^N).
• Imposes conditions (B), (V), and (f1)–(f4) on the potential U(x) = b(x) − λV(x) and nonlinearity f(x,t), including p-superlinear growth and a generalized convexity condition (f4).
• Establishes the Cerami condition by proving boundedness of Palais-Smale sequences through contradiction arguments involving energy estimates.
• Applies linking theorems via cones C_− and C_+ defined by spectral projections related to eigenvalues of the linearized operator.
• Adapts the method to the one-dimensional periodic case by defining a suitable Banach space W^{1,p}(S^1) and using periodic eigenvalue structure.

Experimental results

Research questions

• RQ1 Can nontrivial solutions be guaranteed for p-Laplacian equations in R^N when the potential changes sign and no periodicity is assumed?
• RQ2 Does the cohomological linking method on cones remain effective for p-Laplacian problems with sign-changing potentials and p-superlinear nonlinearities?
• RQ3 Can the classical Ambrosetti-Rabinowitz condition be weakened while still ensuring existence of solutions?
• RQ4 Is the existence result robust across all real values of the parameter λ, including λ = 0?
• RQ5 Can the same method be applied to derive existence of periodic solutions in the one-dimensional case?

Key findings

• The problem (1.1) admits at least one nontrivial weak solution for every λ ∈ R under conditions (B), (V), and (f1)–(f4).
• The condition inf b(x) > −∞ is sufficient; the requirement b(x) ≥ b₀ > 0 can be relaxed by shifting b and V appropriately.
• The generalized convexity condition (f4) replaces the Ambrosetti-Rabinowitz condition (1.3), allowing weaker growth assumptions.
• The Cerami condition is verified for the energy functional, ensuring convergence of bounded Palais-Smale sequences.
• The method applies verbatim to the one-dimensional periodic p-Laplacian equation (5.22), yielding a nontrivial solution for all λ ∈ R.
• The result holds even when V⁺(x) ≡ 0 or λ < λ₁, by adjusting the cone structure to C₋ = {0} and C₊ = W^{1,p}(S¹).