[Paper Review] Ground states of elliptic problems involving non homogeneous operators

Giovany M. Figueiredo, Humberto Ramos Quoirin|arXiv (Cornell University)|Mar 25, 2015

Nonlinear Partial Differential Equations24 references3 citations

TL;DR

This paper establishes the existence of ground states for elliptic problems with nonhomogeneous operators using a generalized Nehari manifold method, demonstrating that the method applies even when the principal part of the functional is not homogeneous. The key contribution is extending variational methods to nonhomogeneous settings, proving existence of at least one nontrivial non-negative ground state and infinitely many pairs of solutions for three classes of boundary value problems involving variable exponent and nonhomogeneous operators.

ABSTRACT

We investigate the existence of ground states for functionals with nonhomogenous principal part. Roughly speaking, we show that the Nehari manifold method requires no homogeinity on the principal part of a functional. This result is motivated by some elliptic problems involving nonhomogeneous operators. As an application, we prove the existence of a ground state and infinitely many solutions for three classes of boundary value problems.

Motivation & Objective

• To extend the Nehari manifold method beyond homogeneous settings to functionals with nonhomogeneous principal parts.
• To establish existence of ground states for elliptic problems involving nonhomogeneous operators, such as those with variable exponent or nonstandard growth.
• To provide a unified variational framework applicable to problems with nonhomogeneous operators, including p-Laplacian-type and variable exponent operators.
• To prove the existence of at least one nontrivial non-negative ground state and infinitely many pairs of solutions for three classes of boundary value problems.
• To generalize classical results on ground states in variational settings to nonhomogeneous and non-regular growth conditions.

Proposed method

• The paper employs the Nehari manifold method on a functional Φ = I₀ - I, where I₀ is a nonhomogeneous, uniformly convex, and C¹ functional on a Banach space X, and I is a C¹ functional with specific asymptotic and monotonicity properties.
• It establishes that the functional Φ satisfies geometric conditions (A2) and (A3): for each w ∈ S (the unit sphere), the map t ↦ Φ(tw) has a unique maximum at t_w > 0, and t_w is uniformly bounded away from zero and bounded on compact subsets.
• The method relies on proving that the Nehari manifold 𝒩 is a C¹ submanifold of X ​\ {0}, and that minimizing Φ over 𝒩 yields a critical point of Φ.
• The analysis uses weak lower semicontinuity of Φ and its derivative, and the compactness of the embedding D₀¹,𝐩(Ω) ↪ Lσ(Ω) for σ ∈ [1, p*), which allows convergence arguments in Palais-Smale sequences.
• It applies a generalized Poincaré-type inequality and uses the fact that the functional I₀ satisfies a homogeneity-like lower bound: I₀(tu) ≥ C t^{p₁} ||u||^{p_N} for t > 1 and ||u|| ≤ 1.
• The proof leverages the monotonicity of s ↦ I′(su)/s^{p-1} and the uniform convergence of I(su)/s^p → ∞ as s → ∞ on weakly compact sets, ensuring the geometry required for minimization on the Nehari manifold.

Experimental results

Research questions

• RQ1 Can the Nehari manifold method be extended to functionals with nonhomogeneous principal parts, without requiring homogeneity of the energy functional?
• RQ2 Does the existence of a ground state solution persist for elliptic problems involving nonhomogeneous operators such as variable exponent p(x)-Laplacians or nonstandard growth operators?
• RQ3 Under what conditions on the nonhomogeneous functional I₀ and the nonlinearity f does the functional Φ = I₀ - I admit a minimizer on the Nehari manifold that yields a critical point?
• RQ4 Can the method be applied to prove existence of infinitely many solutions when the functional is even, even in the absence of homogeneity?
• RQ5 What structural assumptions on the nonlinearity f and the operator I₀ are sufficient to ensure the geometric properties (A2) and (A3) required for the Nehari manifold approach?

Key findings

• The Nehari manifold method is extended to nonhomogeneous functionals, proving that ground states exist even when the principal part of the functional is not positively homogeneous.
• For the problem −div(A(|∇u|^{p−2})∇u) = f(u) in Ω with u ∈ W₀¹,p(Ω), a nontrivial non-negative ground state solution exists under suitable growth and monotonicity conditions on f.
• The existence of infinitely many pairs of solutions is established when the functional is even, via the application of the symmetric mountain pass theorem on the Nehari manifold.
• The functional Φ satisfies the geometric conditions (A2) and (A3) even when I₀ is nonhomogeneous, provided it satisfies a lower bound of the form I₀(tu) ≥ C t^{p₁} ||u||^{p_N} for t > 1 and ||u|| ≤ 1.
• The paper proves that the Nehari manifold is a C¹ submanifold and that minimization of Φ over 𝒩 yields a critical point, which is a ground state solution.
• The convergence of Palais-Smale sequences is established via weak lower semicontinuity and a generalized Poincaré inequality, ensuring that weak convergence implies strong convergence in the energy space.