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Open research questions in Nonlinear Partial Differential Equations

88 unresolved questions extracted from the limitations and future-work sections of 831 Nonlinear Partial Differential Equations papers in our library. Each links back to the study that raised it.

What the literature leaves open

  • Future research could focus on extending the results of the paper to more general domains and potentials. Future research could also focus on applying the methods of the paper to other non-local equations.

    Multi-peak solutions for the fractional Schrödinger equation with Dirichlet datum · 2026 · DOI

  • The paper identifies a gap in the prior literature on the study of multi-peak solutions for the fractional Schrödinger equation. The paper aims to fill this gap by establishing the existence of multi-peak solutions.

    Multi-peak solutions for the fractional Schrödinger equation with Dirichlet datum · 2026 · DOI

  • The paper identifies a gap in the existing literature by considering the case with non-convex constitutive stress relations and non-degenerate Lax’s shock. The paper also addresses the lack of results for large-amplitude shocks and space-periodic perturbations.

    Stability of large-amplitude viscous shock under periodic perturbation for 1-d viscoelasticity with non-convex constitutive relations · 2026 · DOI

  • The paper assumes certain conditions on the parameters μ, κ, and q. The analysis is limited to the case of a ground state normalized solution. The authors do not provide numerical simulations or experimental results.

    Multiple Normalized Solutions for the Quasilinear Sobolev Critical Schrödinger-Poisson System · 2026 · DOI

  • Future research could focus on numerical simulations and experimental results. The study of related systems and phenomena could benefit from the paper's findings. Further analysis of the quasilinear Sobolev critical Schrödinger-Poisson system could lead to new insights.

    Multiple Normalized Solutions for the Quasilinear Sobolev Critical Schrödinger-Poisson System · 2026 · DOI

  • The Grushin operator is degenerate on the subspace {0} × Rn. The paper needs to establish the Palais-Smale compactness condition for the functional Jγ.

    Bifurcation and multiplicity results for critical Grushin-Choquard problems · 2026 · DOI

  • The presence of sign-changing prescribed functions makes the mean field equation challenging to solve. The lack of sufficient conditions for the existence of solutions in critical cases is a significant challenge.

    Critical mean field equations for equilibrium turbulence with sign-changing prescribed functions · 2026 · DOI

  • The 1-Laplacian operator introduces analytical challenges due to its degenerate and non-smooth nature. The lack of compactness is a challenge that is overcome using the Vitali Convergence Theorem.

    Quasicritical elliptic equation involving 1-Laplacian operator with Robin and Neumann Boundary conditions · 2026 · DOI

  • The nonlocal nature of the Hamilton-Jacobi equations. The need for a generalized notion of Mather measure. The challenge of characterizing the constant c.

    The vanishing discount problem for nonlocal Hamilton–Jacobi equations · 2026 · DOI

  • The paper identifies a gap in the existing literature regarding boundary pointwise regularity results for weak solutions to the Poisson problem on uniform domains. The authors seek to address this gap using a novel definition of weak solutions and compactness methods.

    Boundary pointwise regularity for the Poisson problem on uniform domain · 2026 · DOI

  • There is a gap in the existing literature regarding higher regularity properties of solutions to fully nonlinear elliptic equations. The existing literature does not provide a complete understanding of the regularity properties of solutions to fully nonlinear elliptic equations.

    Higher regularity of solutions to fully nonlinear elliptic equations · 2026 · DOI

  • The paper suggests that future research could focus on extending the estimates to more general classes of equations. The paper notes that the estimates could be used to study the regularity properties of solutions of quasilinear elliptic equations in divergence form with rapidly oscillating periodic coefficients. The paper suggests that the estimates could be applied to various applications, including homogenization of p-Laplace type equations.

    Regularity for monotone operators and applications to homogenization of p-Laplace type equations · 2026 · DOI

  • The paper identifies a gap in the existing literature, where estimates are typically derived under strong monotonicity conditions. The paper notes that a weaker condition is needed to obtain Calderon-Zygmund type estimates for quasilinear elliptic equations in divergence form with rapidly oscillating periodic coefficients.

    Regularity for monotone operators and applications to homogenization of p-Laplace type equations · 2026 · DOI

  • To study the behavior of solutions to nonlinear elliptic equations on the hyperbolic space. To generalize the result to other types of equations or spaces.

    Conformally invariant equations with negative critical exponents on the three dimensional hyperbolic space · 2026 · DOI

  • The lack of symmetry results for positive entire solutions to fourth-order equations on the hyperbolic space. The need to generalize previous results on the Euclidean space to the hyperbolic space.

    Conformally invariant equations with negative critical exponents on the three dimensional hyperbolic space · 2026 · DOI

  • The gap involves analyzing regularly varying functions at zero and infinity. The paper aims to establish the existence of positive solutions.

    A p-Laplacian problem with slightly subcritical regularly varying nonlinearity · 2026 · DOI

  • The results are limited to interior critical points. The estimates are limited to fully nonlinear elliptic equations.

    Higher regularity of solutions to fully nonlinear elliptic equations · 2026 · DOI

  • The Concentration-Compactness Principle has not been established for Musielak-Orlicz spaces - The existence of weak solutions to quasilinear equations with critical nonlinear terms is not well understood

    The concentration–compactness principle for Musielak–Orlicz spaces and applications · 2026 · DOI

  • There is a lack of analysis for the truncated Laplacian in punctured balls. The literature lacks a characterization of the solutions in terms of their asymptotic behavior near the origin.

    Radial solutions of truncated Laplacian equations in punctured balls · 2026 · DOI

  • The lack of compactness in R 3 and the presence of competing nonlinear terms require a refined concentration-compactness analysis. The presence of the variable mass term and mixed nonlinearities requires a more delicate analysis.

    Normalized Solutions and Numerical Approximation for a Critical p-Laplacian Schrödinger–Bopp–Podolsky System with Variable Mass and Logarithmic Nonlinearity · 2026 · DOI

  • The paper identifies a gap in the existing literature on anisotropic equations. The problem of establishing existence results for weak and entropy solutions is addressed.

    Regularity of weak and entropy solution for a $$p_i(x)-$$elliptic problems with lower order terms · 2026 · DOI

  • Future research could consider problems with supercritical growth. The paper provides a new tool for analyzing regularity in partial differential equations, which could be applied to other problems.

    Global boundedness for generalized Schrödinger-type double phase problems in $${{\mathbb {R}}}^N$$ and applications to supercritical double phase problems · 2026 · DOI

  • There is a gap in the literature regarding global boundedness results for weak solutions to generalized Schrödinger-type double phase problems. The paper fills this gap by establishing global boundedness results.

    Global boundedness for generalized Schrödinger-type double phase problems in $${{\mathbb {R}}}^N$$ and applications to supercritical double phase problems · 2026 · DOI

  • Future research should focus on extending the results of the paper to more general domains. Future research should focus on studying the critical points of solutions of elliptic equations in non-simply-connected domains with non-smooth boundaries.

    Critical Points of Solutions of Elliptic Equations in Divergence Form in Planar Non-simply-connected Domains with Smooth or Nonsmooth Boundary · 2026 · DOI

  • The existing literature on the critical points of solutions of elliptic equations is limited to simply connected domains. There is a need for a comprehensive study of the critical points of solutions of elliptic equations in non-simply-connected domains.

    Critical Points of Solutions of Elliptic Equations in Divergence Form in Planar Non-simply-connected Domains with Smooth or Nonsmooth Boundary · 2026 · DOI

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88 open questions have been extracted from the limitations and future-work passages of 831 Nonlinear Partial Differential Equations papers in our library. Each one below links back to the study that raised it, so you can read the original claim in context.

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