Open research questions in Nonlinear Partial Differential Equations

What the literature leaves open

• The lack of compactness becomes more delicate in Klein-Gordon-Maxwell systems involving critical Sobolev exponents. The study of systems with sign-changing or unbounded potentials under even weaker compactness assumptions.

  On the Existence of Infinitely Many Solutions for Nonlinear Klein–Gordon–Maxwell Systems Under Weak Nonlinearity Conditions · 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

• 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

• 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 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 paper identifies a gap in prior work as the lack of a detailed analysis of the bijection between normalized critical points of Rα and solutions of the p-Laplace equations for different relations between p, q, r. The authors argue that prior work has not provided a comprehensive study of the properties of Rα.

  On Rayleigh quotients connected to p-Laplace equations with polynomial nonlinearities · 2026 · DOI

• The lack of regularity results for the free boundary of the minimizers is a gap in the literature. The paper identifies the need for a comprehensive study of the regularity of the free boundary in all possible cases.

  On a Multiphase Vectorial Bernoulli Free Boundary Problem · 2026

• The study of sign-changing bubble tower solutions for non-local elliptic problems involving almost critical non-power non-linearity is limited. Prior work has focused on Laplacian elliptic operators.

  Sign-changing bubble tower solutions of a non-local elliptic problem involving almost critical non-power non-linearity · 2026 · DOI

• There is a gap in the literature regarding the asymptotic behavior of solutions to the semilinear heat equation in exterior domains. Prior work has not considered the case where the domain is unbounded.

  On the decay of mass with respect to an invariant measure for semilinear heat equations in exterior domains · 2026 · DOI

• The need for a classification of solutions to the one-dimensional one-phase Stefan problem based on the size of the initial function. The lack of understanding of the behavior of solutions at the blow-up time.

  A one-dimensional Stefan problem for the heat equation with a nonlinear boundary condition of blow-up type · 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 paper identifies a gap in the literature, as the critical Grushin-Choquard problem has not been previously studied. The paper generalizes the classical results of the Hardy-Littlewood-Sobolev inequality to the setting of the Grushin operator.

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

• 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

• The study is limited to the case of fractional Laplacian, - The results are obtained for a specific range of values of n and s, - The study does not consider the case of unbounded…

  Critical Fractional Problems with Weights: Existence, Minimization, and Pohozaev Obstructions · 2026 · DOI

• Study of the case of unbounded domains, - Extension of the results to other types of operators, - Investigation of the behavior of minimizers for different values…

  Critical Fractional Problems with Weights: Existence, Minimization, and Pohozaev Obstructions · 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

• To study the behavior of solutions to the heat equation in more general types of domains. To consider the case where p ≤ 1. To study the behavior of solutions to other types of partial differential equations.

  On the decay of mass with respect to an invariant measure for semilinear heat equations in exterior domains · 2026 · DOI

• Extension to more complex problems and models. Application to fields such as materials science and engineering. Further analysis of the behavior of solutions at the blow-up time.

  A one-dimensional Stefan problem for the heat equation with a nonlinear boundary condition of blow-up type · 2026 · DOI

• Further development of the Galerkin approximation theory for more general systems. Application of the theory to specific physical and geometric problems.

  The Galerkin method for quasilinear elliptic systems with form-bounded nonlinearities · 2026 · DOI

Most-cited papers in Nonlinear Partial Differential Equations

• Recent developments in problems with nonstandard growth and nonuniform ellipticity · Journal of Mathematical Analysis and Applications · 2021 · 239 citations
• Superlinear indefinite elliptic problems and nonlinear Liouville theorems · Topological Methods in Nonlinear Analysis · 1994 · 169 citations
• On the existence of sign changing solutions for semilinear Dirichlet problems · Topological Methods in Nonlinear Analysis · 1996 · 109 citations
• Quasilinear elliptic equations with critical exponents · Topological Methods in Nonlinear Analysis · 1996 · 90 citations
• A note on additional properties of sign changing solutions to superlinear elliptic equations · Topological Methods in Nonlinear Analysis · 2003 · 88 citations
• Nontrivial critical groups in $p$-Laplacian problems via the Yang index · Topological Methods in Nonlinear Analysis · 2003 · 86 citations
• A three critical points theorem and its applications to the ordinary Dirichlet problem · Topological Methods in Nonlinear Analysis · 2003 · 58 citations
• Simplified Samuelson Conditions for Cubic and Quartic Equations* · Manchester School · 1973 · 56 citations
• Multiplicity of solutions for quasilinear elliptic equations · Topological Methods in Nonlinear Analysis · 1995 · 48 citations
• Infinitely many solutions of superlinear fourth order boundary value problems · Topological Methods in Nonlinear Analysis · 2002 · 45 citations

Most recent work

• Fujita-type results for the semilinear heat equations driven by mixed local-nonlocal operators · Journal of Differential Equations · 2026
• Normalized solutions for the nonlinear Schrödinger equation with potential: the purely Sobolev critical case · Calculus of Variations and Partial Differential Equations · 2026
• Optimal stability of Hardy–Littlewood–Sobolev and Sobolev inequalities of arbitrary orders with dimension-dependent constants · Mathematische Annalen · 2026
• Describing smooth small-data solutions to a quasilinear hyperbolic-parabolic system by W1, energy analysis · Nonlinear Analysis: Real World Applications · 2026
• Multi-bubble solutions for the Brezis-Nirenberg problem in four dimensions · Calculus of Variations and Partial Differential Equations · 2026
• Extremals for Poincaré-Sobolev sharp constants in Steiner symmetric sets · Nonlinear Analysis · 2026
• Parabolic De Giorgi classes with doubly nonlinear, nonstandard growth: local boundedness under exact integrability assumptions · Journal of Differential Equations · 2026
• Blow-up and global existence for semilinear parabolic equations on infinite graphs · Calculus of Variations and Partial Differential Equations · 2026
• Critical spacetime crystals in continuous dimensions · SciPost Physics · 2026
• On a semilinear heat equation on infinite graphs I: blow-up for large initial data · Journal of Differential Equations · 2026

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