On nonnegative solutions of the differential inequality $Δ_pu+ Δ_q u+V(x)u^s\leq 0$ on Riemannian manifolds

Abstract: In this paper, we are concerned with differential inequalities with $(p,q)$-Laplacian operator on Riemannian manifolds. Using a test function argument, we establish Liouville-type theorems under the manifold's geometry and the potential's behavior at infinity.

• The paper establishes Liouville-type nonexistence results for nonnegative weak solutions under integral constraints on (p,q)-Laplacian inequalities.
• It employs refined test function methods and energy inequality techniques to derive sharp conditions on the potential V(x) and manifold geometry.
• The results extend classical Liouville theorems to general settings, including new sharp regimes in both Euclidean and Riemannian contexts.

Liouville-Type Theorems for $(p,q)$-Laplacian Inequalities on Riemannian Manifolds

Problem Setting and Motivation

The paper investigates nonexistence results for nonnegative weak solutions to the differential inequality

$\Delta_p u + \Delta_q u + V(x)u^s \leq 0$

on a complete, noncompact Riemannian manifold $(M,g)$, where $$\Delta_z u = \mathrm{div}(|\nabla u|^{z-2} \nabla u)$$ for $z \in \{p, q\}$ and $V(x)$ is a measurable, nonnegative potential. The primary objective is to establish a suite of Liouville-type theorems, characterizing conditions under which the only nonnegative weak solution is the trivial one.

This extends classical Liouville properties for semilinear and quasilinear elliptic equations, including the well-known results of Gidas-Spruck for $\Delta u + u^p = 0$ and Mitidieri-Pohozaev for $p$-Laplacian inequalities. The present contribution pivots to the more general $(p,q)$-Laplacian operators on arbitrary Riemannian manifolds, with explicit accommodations for growth conditions on the geometry and potential $V(x)$, and the exponent $\Delta_p u + \Delta_q u + V(x)u^s \leq 0$0.

Main Results

The core contributions consist of three Liouville-type nonexistence theorems. Each theorem leverages distinct geometric growth hypotheses on the Riemannian manifold and potential, encoded in conditions HP1, HP2, and HP3:

1. Theorem 1.5: Under HP1, if $\Delta_p u + \Delta_q u + V(x)u^s \leq 0$1, $\Delta_p u + \Delta_q u + V(x)u^s \leq 0$2, $\Delta_p u + \Delta_q u + V(x)u^s \leq 0$3 and $\Delta_p u + \Delta_q u + V(x)u^s \leq 0$4 a.e., then any nonnegative weak solution is identically zero.
2. Theorem 1.6: Under the more stringent HP2, the same nonexistence result is established for a refined class of potentials and geometric conditions.
3. Theorem 1.7: With potential and geometric control of exponential type (HP3), the triviality of nonnegative weak solutions persists.

Corollary 1.9 emphasizes a sharp improvement in the Euclidean setting, broadening prior nonexistence intervals to encompass the range $\Delta_p u + \Delta_q u + V(x)u^s \leq 0$5 for the exponent $\Delta_p u + \Delta_q u + V(x)u^s \leq 0$6. This closes a gap relative to [BBF] for $\Delta_p u + \Delta_q u + V(x)u^s \leq 0$7-Laplace operators in $\Delta_p u + \Delta_q u + V(x)u^s \leq 0$8.

The growth conditions HP1–HP3 are formulated through sophisticated integral bounds on $\Delta_p u + \Delta_q u + V(x)u^s \leq 0$9 over geodesic balls, intertwining geometric and analytic properties. Sharpness is linked to prior results on the exponent of the potential and related log-correction terms ([MMP1], [AS]).

Analytical Framework

The proofs utilize advanced techniques in quasilinear analysis:

• Test Function Method: Carefully constructed cut-off functions with controlled gradients, adapted to the geometry and potential, enable integral estimates leading to the vanishing of the solution.
• Integral Estimate Lemmas: Central are energy-type inequalities (Lemmas 2.2–2.4), leveraging weighted integral bounds and precise scaling in the exponents, applying iteratively with cut-off radii tending to infinity.
• Young's and Hölder's Inequalities: These are systematically employed in conjunction with the $(M,g)$0- and $(M,g)$1-Laplacian structures to obtain crucial control over the nonlinearity and the degeneracy/singularity of the operator.
• Growth Assumptions: The results are tightly linked to the interplay between the exponents $(M,g)$2, the lower-order potential term $(M,g)$3, and volumetric growth of geodesic balls.

A key innovation is the reduction to sharp integral bounds for a wide class of $(M,g)$4-Laplacian problems, extending to operators of the form $(M,g)$5 with $(M,g)$6 subject to appropriate polynomial constraints.

Implications and Extensions

Analytic and Geometric Implications: These results further clarify the structure of nontrivial solutions in nonlinear elliptic theory, particularly in regard to the criticality of potential and geometry at infinity. They provide necessary conditions on the growth of $(M,g)$7 and the manifold for the existence of positive solutions to a class of differential inequalities, thereby sharpening the landscape for maximum principle-type phenomena in nonlinear PDE.

Practical Significance: Nonexistence results of this type undergird the qualitative theory behind phase transitions and blow-up analysis in nonlinear elasticity, variational calculus with nonstandard growth, and geometric flows.

Potential for Generalization: The analytic pipeline outlined admits generalization to systems or to operators with even more intricate nonlinearity—e.g., those involving gradient terms of more general type—or to nonlocal operators and spaces of variable exponent.

Direction for Further Study:

• Investigation of similar Liouville properties for parabolic analogues and evolution equations.
• Transposing the argument to non-smooth metric measure spaces.
• Extending the technical machinery to mixed boundary problems or weighted Laplacians.

Contradictory Claim Relative to Classical Setting: In the Euclidean case, the paper establishes nonexistence for $(M,g)$8—a parameter regime not previously accessible via the results in [BBF], representing a significant strict strengthening of known Liouville thresholds.

Conclusion

The paper achieves a systematic and comprehensive generalization of Liouville-type nonexistence theorems to $(M,g)$9-Laplacian inequalities on general Riemannian manifolds, providing sharp geometric and analytic conditions on the manifold and potential. The methodology, while firmly rooted in the test function framework, advances the understanding of critical exponents and their interplay with manifold geometry, and sets the stage for further developments in nonlinear geometric analysis and quasilinear PDE theory (2604.23624).