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Given a real Banach space X, we show that the Nehari manifold method can be applied to functionals which are C 1 in X \ { 0 } . In particular we deal with functionals that can be unbounded near 0, and prove the existence of a ground state and infinitely many critical points for such functionals. These results are then applied to three classes of problems: the prescribed energy problem for a family of functionals depending on a parameter, problems involving the affinep-Laplacian operator, and degenerate Kirchhoff type problems. [ABSTRACT FROM AUTHOR]

Our paper is devoted to studying the existence results of distributional solutions in the anisotropic Sobolev space with variable exponents and zero boundary $ \mathring {W}^{1,\overrightarrow {p}(\cdot)}(\Omega,\mathbb {R}^m) $ W ˚ 1 , p → (⋅) (Ω , R m) for a family of boundary value problems, which is a variable exponent anisotropic nonlinear elliptic system with $ L^{\overrightarrow {p'}(\cdot)} $ L p ′ → (⋅) data for the datum f (i.e. $ \in L^{\overrightarrow {p'}(\cdot)}(\Omega,\mathbb {R}^m) $ ∈ L p ′ → (⋅) (Ω , R m)), and has a sum of two Carathéodory nonlinearities, one of which includes the solution u and the other its partial derivatives $ \partial _iu,\,i=1,\ldots,N $ ∂ i u , i = 1 , ... , N. [ABSTRACT FROM AUTHOR]

We consider the steady Navier–Stokes equations with mixed boundary conditions, where a regularized directional do-nothing (RDDN) condition is defined on the Neumann boundary portion. An auxiliary Stokes reference flow, which also works as a lifting of the inhomogeneous Dirichlet boundary values, is used to define the RDDN condition. Our aim is to study the minimization of a velocity tracking cost functional with controls localized on a part of the boundary. We prove the existence of a solution for this optimal control problem and derive the corresponding first order necessary optimality conditions in terms of dual variables. All results are obtained under appropriate assumptions on the size of the data and the controls, which, however, are less restrictive when compared with the case of the classical do-nothing outflow condition. This is further confirmed by the numerical examples presented, which include scenarios where only noisy data is available. [ABSTRACT FROM AUTHOR]

We consider an eigenvalue problem for the nonautonomous double phase differential operator. We show that there exist two positive numbers 0 < λ ^ ≤ λ ∗ depending only on the data of the equation such that every λ ≥ λ ∗ is an eigenvalue, while 0 < λ < λ ^ is not an eigenvalue. It is an open problem if λ ^ = λ ∗ and also if λ ^ < λ ∗ , then what can be said about the interval [ λ ^ , λ ∗) , does it contain any eigenvalues? [ABSTRACT FROM AUTHOR]

On a compact three-dimensional Riemannian manifold with boundary, we prove the compactness of the full set of conformal metrics with positive constant scalar curvature and constant mean curvature on the boundary. This involves a blow-up analysis of a Yamabe equation with critical Sobolev exponents both in the interior and on the boundary. [ABSTRACT FROM AUTHOR]

We set up a general framework tailor-made to solve complement value problems governed by symmetric nonlinear nonlocal integro-differential p-Lévy operators. A prototypical example of integro-differential p-Lévy operators is the well-known fractional p-Laplace operator. Our main focus is on nonlinear integro-differential equations in the presence of Dirichlet, Neumann and Robin conditions and we show well-posedness results. Several results are new even for the fractional p-Laplace operator but we develop the approach for general translation-invariant nonlocal operators. We also bridge the gap from nonlocal to local, by showing that solutions to the local Dirichlet and Neumann boundary value problems associated with p-Laplacian are strong limits of the nonlocal ones. [ABSTRACT FROM AUTHOR]

The present study is related to the existence and the asymptotic behavior of the solution of a nonlinear elliptic Steklov problem imposed on a nanostructure depending on the thickness parameter ε\varepsilon (nano-scale), distributed on the boundary of the domain when the parameter ε\varepsilon goes to 0, under some appropriate conditions on the data involved in the problem. We use epi-convergence method in order to establish the limit behavior by characterizing the weak limits of the energies for the solutions. An intermediate step in the proof provides a homogenization result for the considered structure.

We establish a supercritical Trudinger–Moser type inequality for the k-Hessian operator on the space of the k-admissible radially symmetric functions Φ 0 , rad k (B) , where B is the unit ball in R N . We also prove the existence of extremal functions for this new supercritical inequality. [ABSTRACT FROM AUTHOR]

In this article, we study the uniqueness and stability of nonnegative solutions for a class of semilinear elliptic problems in a ball, when the nonlinearity has more than one zero, negative at the origin and concave. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish the existence of infinitely many solutions for a class of anisotropic p → -Laplacian problems with Dirichlet boundary conditions, depending on two real parameters. These problems have a variational structure and some critical point theorems are applied. We also give an example to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In this present paper, we are first of all interested in some continuity and compactness results for the space ψ-fractional 핊 풜 α , β ; ψ ⁢ ( Λ ) {\mathbb{S}^{\alpha,\beta;\psi}_{\mathcal{A}}(\Lambda)} . In this sense, we investigate the existence of at least two solutions with constant signs using truncation arguments and comparison methods of a new class of fractional differential equations with m ⁢ ( ξ ) {m(\xi)} -Laplacian with double phase. [ABSTRACT FROM AUTHOR]

We study the existence of conformal metrics on noncompact Riemannian manifolds with noncompact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature on the boundary. These metrics are constructed on smooth manifolds obtained by removing d-dimensional submanifolds from certain n-dimensional compact spaces locally modelled on generalized solid cones. We prove the existence of such metrics if and only if d > n - 2 2 . Our main theorem is inspired by the classical results by Aviles–McOwen and Loewner–Nirenberg, known in the literature as the "singular Yamabe problem". [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of conformal metrics of the disk with prescribed Gaussian and geodesic curvatures. Being more specific, given nonnegative smooth functions K : D ¯ → R and h : ∂ D → R , we consider the problem of finding a conformal metric realizing K and h as Gaussian and geodesic curvatures, respectively. This is the natural analogue of the classical Nirenberg problem posed on the disk. As we shall see, both curvatures play a role in the existence of solutions. Indeed we are able to give existence results under conditions that involve K and H, where H denotes the harmonic extension of h. The proof is based on the computation of the Leray-Schauder degree of the associated operator. [ABSTRACT FROM AUTHOR]

In this paper, we use the variational method to study some singular Steklov-type problem with variable exponents. More precisely, we use the min-max method in order to prove the existence of a solution to such a problem. [ABSTRACT FROM AUTHOR]

The paper is concerned with Lane–Emden and Brezis–Nirenberg problems involving the affine p-Laplace nonlocal operator Δ p 풜 , which has been introduced in [J. Haddad, C. H. Jiménez and M. Montenegro, From affine Poincaré inequalities to affine spectral inequalities, Adv. Math. 386 2021, Article ID 107808] driven by the affine L p energy ℰ p , Ω from convex geometry due to [E. Lutwak, D. Yang and G. Zhang, Sharp affine L p Sobolev inequalities, J. Differential Geom. 62 2002, 1, 17–38]. We are particularly interested in the existence and nonexistence of positive C 1 solutions of least energy type. Part of the main difficulties are caused by the absence of convexity of ℰ p , Ω and by the comparison ℰ p , Ω ⁢ (u) ≤ ∥ u ∥ W 0 1 , p ⁢ (Ω) generally strict. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of a double phase problem with variable exponents and Dirichlet boundary condition. Based on an abstract critical point theorem, we establish existence results under very general assumptions on the nonlinear term, such as a subcritical growth and a superlinear condition. In particular, we prove the existence of two bounded weak solutions with opposite energy sign and we state some special cases in which they turn out to be nonnegative. [ABSTRACT FROM AUTHOR]

In this article, we study a p-Laplacian Schrödinger-Poisson system involving a parameter q≠0q\ne 0 in bounded domains. By using the Nehari manifold and the fibering method, we obtain the non-existence and multiplicity of nontrivial solutions. On one hand, there exists q*>0{q}^{* }\gt 0 such that only the trivial solution is admitted for q∈(q*,+∞).q\in \left({q}^{* },+\infty ). On the other hand, there are two positive solutions existing for q∈(0,q0*+ε)q\in \left(0,{q}_{0}^{* }+\varepsilon ), where ε>0\varepsilon \gt 0 and q0*+ε

In this paper we extend the Banach spaces-based fully mixed approach recently developed for the coupled Stokes and Poisson–Nernst–Planck equations, to cover the coupled Navier–Stokes and Poisson–Nernst–Planck equations. In addition to the velocity and pressure of the fluid, the velocity gradient and the Bernoulli-type stress tensor are added as further unknowns. Similarly, fully mixed formulations for the Poisson and Nernst–Planck sub-problems are achieved by considering, alongside the electrostatic potential and the concentration of ionized particles, the electric current field and total ionic fluxes as new mixed variables. As a consequence, two saddle-point problems, one of them non-linear, and both involving nonlinear source terms depending on the other unknowns, along with a perturbed saddle-point problem that is in turn further perturbed by a bilinear form depending on the remaining unknowns, constitute the resulting variational formulation of the whole coupled system. Fixed-point strategies are then employed to prove, under smallness assumptions on the data, the well-posedness of the continuous and associated Galerkin schemes, the latter for arbitrary finite element subspaces under suitable stability assumptions. The main theoretical tools utilized include the Babuška–Brezzi and Banach–Nečas–Babuška theories in Banach spaces, an abstract result for perturbed saddle-point problems (also in Banach spaces), and the classical Banach and Brouwer fixed-point theorems. Strang-type lemmas are then applied to establish a priori error estimates. Next, specific finite element subspaces (defined by Raviart–Thomas elements of order k ≥ 0 and piecewise polynomials of degree ≤ k ) are shown to satisfy the required hypotheses, and this yields specific convergence rates. Finally, several numerical results are reported, confirming the theoretical findings and illustrating the good performance of the method. [ABSTRACT FROM AUTHOR]

In this paper, we study a double phase problem with both variable exponents. Such problem has a reaction consisting of a Carathéodory perturbation defined only locally and of a critical term. The presence of the critical term does not permit to use results of the critical point theory for the corresponding energy functional. Consequently, using suitable cut-off functions and truncation techniques we focus on an auxiliary coercive problem on which, differently from our main problem, we can act with variational tools. In this way, we are able to produce a sequence of sign-changing solutions to our main problem converging to 0 in L ∞ and in the Musielak–Orlicz Sobolev space. [ABSTRACT FROM AUTHOR]

The present work deals with theory of critical points to the energy functional on W 0 1 , p (Ω) defined by Φ A (u) = 1 p E p , Ω p (u) - ∫ Ω F (x , u) d x , where E p , Ω p stands for the affine p-energy introduced for p > 1 by Lutwak et al. (J Differ Geom 62:17–38, 2002). Its development is inspired in the study of solutions of elliptic equations involving the affine p-Laplace non-local operator Δ p A introduced recently by Haddad et al. (Adv Math 386:107808, 2021). New results on regularity of E p , Ω p and (S +) compactness property associated to Δ p A are established. The latter is fundamental on the discussion of Palais–Smale compactness for Φ A when affine mountain-pass and coercive geometries are considered. The mountain-pass case is quite intricate, being addressed by means of asymptotic analysis as ε → 0 of corresponding critical points of the one-parameter perturbation Φ A ε (u) = Φ A (u) + ε ‖ u ‖ W 0 1 , p (Ω) . [ABSTRACT FROM AUTHOR]

This paper investigates the existence, nonexistence, and qualitative properties of p-harmonic functions in the upper half-space R + N (N ≥ 3) satisfying nonlinear boundary conditions for 1 < p < N . Moreover, the symmetry of positive solutions is shown by using the method of moving planes. [ABSTRACT FROM AUTHOR]

The main result of this paper is to investigate the existence of a solution of a class of fractional problems involving the operator p-Laplacian with periodic potential and supercritical growth via the Mountain Pass theorem-Cerami version. [ABSTRACT FROM AUTHOR]

This article is devoted to the existence of optimal controls in various control problems associated with a novel nonlinear interface problem on an unbounded domain with non-monotone set-valued transmission conditions. This interface problem involves a nonlinear monotone partial differential equation in the interior domain and the Laplacian in the exterior domain. Such a scalar interface problem models non-monotone frictional contact of elastic infinite media. The variational formulation of the interface problem leads to a hemivariational inequality (HVI), which, however, lives on the unbounded domain, and thus cannot be analyzed in a reflexive Banach space setting. Boundary integral methods lead to another HVI that is amenable to functional analytic methods using standard Sobolev spaces on the interior domain and Sobolev spaces of fractional order on the coupling boundary. Broadening the scope of this article, we consider extended real-valued HVIs augmented by convex extended real-valued functions. Under a smallness hypothesis, we provide existence and uniqueness results; moreover, we establish a stability result with respect to the extended real-valued function as a parameter. Based on the latter stability result, we prove the existence of optimal controls for four kinds of optimal control problems: distributed control on the bounded domain, boundary control, simultaneous boundary-distributed control governed by the interface problem, and control of the obstacle driven by a related bilateral obstacle interface problem.

In this article, we study the uniqueness of positive symmetric solutions of the following mean curvature problem in Euclidean space: (P)u′1+∣u′∣2′+h(x)f(u)=0,−1

In an open, bounded Lipschitz polygon Ω ⊂ R 2 , we establish weighted analytic regularity for a semilinear, elliptic PDE with analytic nonlinearity and subject to a source term f which is analytic in Ω . The boundary conditions on each edge of ∂ Ω are either homogeneous Dirichlet or homogeneous Neumann BCs. The presently established weighted analytic regularity of solutions implies exponential convergence of various approximation schemes: hp-finite elements, reduced order models via Kolmogorov n-widths of solution sets in H 1 (Ω) , quantized tensor formats and certain deep neural networks. [ABSTRACT FROM AUTHOR]

We prove a general result characterizing a specific class of Serrin domains as supports of unbounded and periodic constant mean curvature graphs. We apply this result to prove the existence of a family of unbounded periodic constant mean curvature graphs, each supported by a Serrin domain and intersecting its boundary orthogonally, up to a translation. We also show that the underlying Serrin domains are calibrable and Cheeger in a suitable sense, and they solve the 1-Laplacian equation. [ABSTRACT FROM AUTHOR]

We establish some embedding results for weighted Sobolev spaces. As an application, we obtain one nonzero solution for the equation - div (| ∇ u | N - 2 ∇ u) + V (x) | u | N - 2 u = λ Q (x) f (u) , in R N , where V , Q are nonnegative potentials, λ > 0 is a large parameter and f has critical growth in the Trudinger–Moser sense. [ABSTRACT FROM AUTHOR]

We study a nonlinear generalization of a free boundary problem that arises in the context of thermal insulation. We consider two open sets Ω ⊆ A , and we search for an optimal A in order to minimize a nonlinear energy functional, whose minimizers u satisfy the following conditions: Δ p u = 0 inside A \ Ω , u = 1 in Ω , and a nonlinear Robin-like boundary (p, q)-condition on the free boundary ∂ A . We study the variational formulation of the problem in SBV , and we prove that, under suitable conditions on the exponents p and q, a minimizer exists and its jump set satisfies uniform density estimates. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the Kirchhoff-type variable-order fractional Laplacian problems involving critical exponents and logarithmic nonlinearity. By using the constraint variational method, we show the existence of one least energy sign-changing solution. Moreover, we show that this energy is strictly larger than twice the ground energy. [ABSTRACT FROM AUTHOR]

The Miura ori is a very classical origami pattern used in numerous applications in engineering. A study of the shapes that surfaces using this pattern can assume is still lacking. A constrained nonlinear partial differential equation (PDE) that models the possible shapes that a periodic Miura tessellation can take in the homogenization limit has been established recently and solved only in specific cases. In this paper, the existence and uniqueness of a solution to the unconstrained PDE is proved for general Dirichlet boundary conditions. Then an H 2 -conforming discretization is introduced to approximate the solution of the PDE coupled to a Newton method to solve the associated discrete problem. A convergence proof for the method is given as well as a convergence rate. Finally, numerical experiments show the robustness of the method and that nontrivial shapes can be achieved using periodic Miura tessellations. [ABSTRACT FROM AUTHOR]

In this paper, a new class of Sobolev spaces with kernel function satisfying a Lévy-integrability-type condition on compact Riemannian manifolds is presented. We establish the properties of separability, reflexivity, and completeness. An embedding result is also proved. As an application, we prove the existence of solutions for a nonlocal elliptic problem involving the fractional p (· , ·) -Laplacian operator. As one of the main tools, topological degree theory is applied. [ABSTRACT FROM AUTHOR]

In this paper, we propose a gradient type term for the k-Hessian equation that extends for k > 1 the classical quadratic gradient term associated with the Laplace equation. We prove that such as gradient term is invariant by the Kazdan–Kramer change of variables. As applications, we ensure the existence of solutions for a new class of k-Hessian equation in the sublinear and superlinear cases for Sobolev type growth. The threshold for existence is obtained in some particular cases. In addition, for the Trudinger–Moser type growth regime, we also prove the existence of solutions under either subcritical or critical conditions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of nontrivial weak solutions for a class of p(x)-Laplacian problem under Steklov boundary condition. The variable exponent theory of generalized Lebesgue-Sobolev spaces and the concentration-compactness principle for weighted variable exponent spaces are used for this purpose. [ABSTRACT FROM AUTHOR]

We prove the existence of the first eigenvalue and an associated eigenfunction with Dirichlet condition for the complex Monge–Ampère operator on a bounded strongly pseudoconvex domain in C n . We show that the eigenfunction is plurisubharmonic, smooth with bounded Laplacian in Ω and boundary values 0. Moreover it is unique up to a positive multiplicative constant. To this end, we follow the strategy used by P. L. Lions in the real case. However, we have to prove a new theorem on the existence of solutions for some special complex degenerate Monge–Ampère equations. This requires establishing new a priori estimates of the gradient and Laplacian of such solutions using methods and results of Caffarelli et al. (Commun Pure Appl Math 38(2):209–252, 1985) and Guan (Commun Anal Geom 6(4):687–703, 1998). Finally we provide a Pluripotential variational approach to the problem and using our new existence theorem, we prove a Rayleigh quotient type formula for the first eigenvalue of the complex Monge–Ampère operator. [ABSTRACT FROM AUTHOR]

We consider a nonlinear Neumann problem, with periodic oscillation in the elliptic operator and on the boundary condition. Our focus is on problems posed in half-spaces, but with general normal directions that may not be parallel to the directions of periodicity. As the frequency of the oscillation grows, quantitative homogenization results are derived. When the homogenized operator is rotation-invariant, we prove the Hölder continuity of the homogenized boundary data. While we follow the outline of Choi and Kim (Homogenization for nonlinear PDEs in general domains with oscillatory Neumann boundary data, Journal de Mathématiques Pures et Appliquées 102 (2014), no. 2, 419–448), new challenges arise due to the presence of tangential derivatives on the boundary condition in our problem. In addition, we improve and optimize the rate of convergence within our approach. Our results appear to be new even for the linear oblique problem.

We consider a parametric (with two parameters μ,λ>0\mu ,\lambda \gt 0) Dirichlet problem driven by the double phase differential operator and a reaction which has the competing effect of a singular term and of a superlinear perturbation. We prove a bifurcation-type result in the parameter λ>0\lambda \gt 0, when the other parameter μ>0\mu \gt 0 is large.

We are concerned with the Neumann problem with Minkowski-curvature operator (P) { − div (∇ u (x) 1 − | ∇ u (x) | 2 ) = g (u (x)) in Ω , u (x) > 0 in Ω , u → 0 as | x | → ∞ , ∂ u ∂ ν = − σ on ∂ Ω , where Ω is an exterior domain in R N and ∂ ∂ ν denotes the normal interior derivative on ∂ Ω. We have shown the existence of radial positive solutions of (P) when g satisfies some suitable assumptions. The proof of our main result is based upon the shooting method. [ABSTRACT FROM AUTHOR]

In this present paper, we investigate a new class of singular double phase p-Laplacian equation problems with a ψ -Hilfer fractional operator combined from a parametric term. Motivated by the fibering method using the Nehari manifold, we discuss the existence of at least two weak solutions to such problems when the parameter is small enough. Before attacking the main contribution, we discuss some results involving the energy functional and the Nehari manifold. [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of two kinds of new double phase problems with mixed boundary conditions and multivalued convection terms, which are, exactly, a double phase inclusion problem with Dirichlet–Neumann–Dirichlet–Neumann boundary conditions (DNDN, for short) and a double phase inclusion problem with Dirichlet–Neumann–Neumann–Neumann boundary conditions (DNNN, for short), respectively. On the one hand, we examine the nonemptiness, boundedness and closedness of solution sets to (DNDN) and (DNNN), respectively, by employing the theory of nonsmooth analysis and a surjectivity theorem for multivalued mappings which is formulated by the sum of a multivalued maximal monotone operator and a multivalued bounded pseudomonotone mapping. On the other hand, we explore a significant result on asymptotic behavior of solution set to (DNNN) which reveals that the solution set of (DNDN) can be approached by the solution set of (DNNN) in the sense of Kuratowski, when a parameter tends to infinity. [ABSTRACT FROM AUTHOR]

In this paper we present some existence and multiplicity results for a class of anisotropic p⃗-Laplacian problems with Dirichlet boundary conditions. In particular, the existence of three solutions is pointed out. The approach is based on variational methods and our main tool is a three critical point theorem. [ABSTRACT FROM AUTHOR]

Considering the second boundary value problem of the Lagrangian mean curvature equation, we obtain the existence and uniqueness of the smooth uniformly convex solution, which generalizes the Brendle–Warren's theorem about minimal Lagrangian diffeomorphism in Euclidean metric space. [ABSTRACT FROM AUTHOR]

We prove two Hölder regularity results for solutions of generated Jacobian equations. First, that under the A3 condition and the assumption of nonnegative L p valued data solutions are C 1 , α for an α that is sharp. Then, under the additional assumption of positive Dini continuous data, we prove a C 2 estimate. Thus the equation is uniformly elliptic and when the data is Hölder continuous solutions are in C 2 , α . [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of distributional solutions to certain anisotropic nonlinear weighted elliptic equations with variable exponents, where the weight function belongs to the anisotropic Sobolev space with variable exponents and zero boundary. The functional setting involves anisotropic variable exponents Lebesgue–Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this paper we study a Dirichlet double phase problem with a parametric superlinear right-hand side that has subcritical growth. Under very general assumptions on the data, we prove the existence of at least two nontrivial bounded weak solutions to such problem by using variational methods and critical point theory. In contrast to other works we do not need to suppose the Ambrosetti–Rabinowitz condition. [ABSTRACT FROM AUTHOR]