On fractional Schrödinger equations in (\mathbb{R}^N) without the Ambrosetti-Rabinowitz condition

Abstract: In this note we prove the existence of radially symmetric solutions for a class of fractional Schrödinger equation in (\mathbb{R}^N) of the form {equation*} \slap u + V(x) u = g(u), {equation*} where the nonlinearity $g$ does not satisfy the usual Ambrosetti-Rabinowitz condition. Our approach is variational in nature, and leans on a Pohozaev identity for the fractional laplacian.

• The paper proves the existence of nontrivial radially symmetric solutions to fractional Schrödinger equations without the AR condition.
• It employs variational methods, penalization arguments, and the Pohozaev identity to handle nonlinearities lacking traditional superlinear growth.
• Results emphasize the limitations of using the Pohozaev manifold for constrained minimization, suggesting new strategies for nonlocal PDEs.

Existence Theory for Fractional Schrödinger Equations Without the Ambrosetti-Rabinowitz Condition

Introduction and Background

This paper investigates the existence of solutions to a class of fractional Schrödinger equations in $\mathbb{R}^N$, given by

$u + V(x)u = g(u)$

where the nonlinearity $g$ does not satisfy the Ambrosetti-Rabinowitz (AR) condition. This study is motivated by the burgeoning interest in nonlocal PDEs, particularly those involving the fractional Laplacian. Such equations are fundamental in modeling diverse phenomena, including phase transitions and anomalous diffusion, due to their inherent nonlocality.

The primary novelty is the removal of the AR condition, which is typically imposed to establish the boundedness of Palais–Smale sequences and ensure the applicability of variational methods. The present work achieves existence and some non-existence results using alternative compactness arguments and the Pohozaev identity for the fractional Laplacian.

Analytical Foundations: Fractional Laplacians and Function Spaces

The operator under consideration is the fractional Laplacian, defined via the Fourier multiplier $|\xi|^{2s}$ for $s\in(0,1)$. The study takes place in the fractional Sobolev (Hilbert) space $H^s(\mathbb{R}^N)$, equipped with its natural norm. The paper provides a thorough account of function space embeddings, relying on the established regularity theory (cf. [Di Nezza et al., 2012], [Cabre, 2010]):

• When $sp<N$, $H^s$ embeds locally compactly into $L^q$ for $2 < q < 2^*$
• For $s>1/2$, the Strauss compactness lemma for radially symmetric functions applies, circumventing issues of mass escaping to infinity

These properties undergird the variational analysis, especially in the absence of the AR condition.

Main Results: Existence and Regularity

The existence proof is variational, seeking critical points of the Euler-Lagrange functional

$$I(u) = \frac{1}{2}\int_{\mathbb{R}^N} |\xi|^{2s} |\hat{u}(\xi)|^2 \, d\xi + \frac{1}{2} \int_{\mathbb{R}^N} V(x) |u(x)|^2 \, dx - \int_{\mathbb{R}^N} G(u(x))\, dx$$

with $G(s) = \int_0^s g(t)\,dt$.

The main theorem asserts: For $1/2 < s < 1$, under general growth conditions on $g$ that do not require AR, there exists a nontrivial radially symmetric solution in $H^s(\mathbb{R}^N)$.

A broader result is established for $0 < s < 1$ if $g$ grows subcritically at infinity, specifically if $|g(t) - mt| \leq C|t|^{q-1}$ for large $|t|$ with $q < 2^*$. Notably, the nonlinearity $g$ need not be superlinear at infinity—a significant relaxation compared to classical approaches.

Regularity theory for weak solutions is built upon advances for the fractional Laplacian, ensuring that solutions inherit sufficient regularity to invoke the Pohozaev identity.

The Pohozaev Identity

A key analytic element is the derivation and application of the Pohozaev identity for the fractional problem. For any (suitably regular) solution, the identity reads:

$$\frac{N-2s}{2}\int_{\mathbb{R}^N} |\xi|^{2s} |\hat{u}(\xi)|^2 \, d\xi + \frac{N}{2} \int_{\mathbb{R}^N} V(x)|u(x)|^2 dx + \frac{1}{2}\int_{\mathbb{R}^N} \langle\nabla V(x), x\rangle |u(x)|^2 dx = N \int_{\mathbb{R}^N} G(u(x)) dx$$

This identity is central for characterizing ground states and understanding variational constraints.

Non-criticality of the Pohozaev Manifold and Failure of Constrained Minimization

The classical procedure (in the local or AR case) of minimizing the energy functional on the Pohozaev manifold often produces ground state solutions. However, for the present fractional problem without AR, b = inf_{u\in\mathcal{P}} I(u) is not a critical level of $I$ under natural conditions on $V$. This is proved by constructing suitable test functions, involving rescalings and translations, and showing that minimizers on the Pohozaev set cannot, in general, be Palais–Smale sequences.

This result highlights the limitation of simply enforcing the Pohozaev constraint as a substitute for AR-type superlinearity. The potential $V(x)$—especially its behavior at infinity—plays a delicate role, and the structure of the fractional Laplacian further complicates variational methods.

Methodological Contributions

The analysis leverages:

• Penalization arguments: splitting $g$ into bounded and unbounded parts to establish boundedness of critical sequences,
• Radial symmetry and decay properties: capitalizing on compact embeddings for radially symmetric $H^s$ functions, and
• A careful adaptation of minimax/the mountain-pass framework: following Azzollini et al. and Jeanjean, but overcoming noncompactness without AR by working in symmetric subspaces.

Technical details include refined regularity estimates (inspired by Caffarelli–Silvestre extension methods) and compactness theorems tailored for nonlocal operators.

Implications and Future Perspectives

Practically, the results offer existence theorems for nonlinear equations driven by the fractional Laplacian under minimal growth controls. The study broadens the applicable range of nonlinearities for which nonlocal field theories in mathematical physics can be rigorously analyzed.

Theoretically, the demonstration that the Pohozaev manifold does not generally yield ground states in this setting draws a sharp distinction between the variational structure of local and nonlocal problems. This emphasizes that for nonlocal operators, new techniques and constraints may be necessary, and that variational approaches must be carefully tailored to the peculiarities of fractional diffusion.

Looking forward, further advances may include:

• Sharpening compactness criteria for nonlocal operators in non-symmetric and non-radial settings,
• Extending these results to systems, quasilinear variants, or equations on bounded domains,
• Deriving multiplicity and qualitative properties of solutions in the absence of AR.

Conclusion

This paper significantly advances the existence theory for fractional Schrödinger equations by removing the need for the Ambrosetti-Rabinowitz condition, and by demonstrating both the successes and the limits of existing variational techniques in the nonlocal context. The results provided open the way to a more flexible analysis of nonlinear fractional PDEs, while also highlighting the nontrivial role of symmetry and the limitations of projection onto natural constraint sets such as the Pohozaev manifold (1210.0755).