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This paper continues the study of explicit asymptotic formulas for standing coastal trapped waves, focusing on the spectral properties of the operator ⟨ ∇ , D (x) ∇ ⟩ , which is the spatial component of the wave operator with a degenerating wave propagation velocity. We aim to construct spectral series—pairs of asymptotic eigenvalues and formal asymptotic eigenfunctions—corresponding to the high-frequency regime, where the eigenvalue is ω → ∞ . Extending earlier results, this study addresses the nearly integrable case, providing a more detailed asymptotic behavior of eigenfunctions. Depending on their domain of localization, these eigenfunctions can be expressed in terms of Airy functions and their derivatives or Bessel functions. In addition, we introduce a canonical operator with violated (imprecisely satisfied) quantization conditions. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of degenerate elliptic equations with rapidly oscillating coefficients in periodically perforated domains, which arises in the study of spectrum problems for uniformly elliptic equations in perforated domains. We establish a quantitative convergence rate and obtain the uniform weighted Lipschitz and W1,p estimates. [ABSTRACT FROM AUTHOR]

The Dirichlet problem of elliptic equation from two phase problems, b (u) − div (| ∇ u) | p (x) − 2 ∇ u + μ (x) | ∇ u | q (x) − 2 ∇ u) = f (x) in Ω, u = 0 on ∂Ω is considered, where Ω is a bounded domain with a smooth boundary in R N , b is a continuous and nondecreasing function. By the theory of the weighted variable Sobolev space, the existence and uniqueness of an entropy solution for L 1 -data f are proved. [ABSTRACT FROM AUTHOR]

We study the numerical solutions for a class of quasilinear Schrödinger equations arising from the self-channeling of high-power ultra short lasers in matter, which are associated with energy functionals with nonlinear principle parts so that the classical algorithm cannot directly be used. By the method of variable replacement to transform the quasilinear Schrödinger equation into an semilinear elliptic equation, the numerical mountain pass algorithm is then applied. Some numerical experiments are also performed, including zero potential, nonzero constant potential and singular problem. [ABSTRACT FROM AUTHOR]

With the help of De Giorgi's method, the embedding of the $ \mathfrak {B}_{m(x)}(\Omega,M,\gamma,\gamma _1,\delta,\frac {1}{q}) $ B m (x) (Ω , M , γ , γ 1 , δ , 1 q) class into $ C^{0,\alpha }(\Omega) $ C 0 , α (Ω) is proved. This result can be applied to the $ C^{0,\alpha } $ C 0 , α continuity to some elliptic equations with variable exponent. [ABSTRACT FROM AUTHOR]

We exploit the existence and non-existence of positive solutions to the eigenvalue problem driven by the nonhomogeneous fractional $ p\& q $ p &q Laplacian operator with indefinite weights \[ \left(-\Delta_p\right)^{\alpha}u + \left(-\Delta_q\right)^{\beta}u = \lambda\left[a \left|u\right|^{p-2}u + b \left|u\right|^{q-2}u \right]\quad{\rm in}\ \Omega, \] (− Δ p) α u + (− Δ q) β u = λ [ a | u | p − 2 u + b | u | q − 2 u ] in Ω , where $ \Omega \subseteq \mathbb {R}^N $ Ω ⊆ R N is a smooth bounded domain that has been extended by zero. We further show the existence of a continuous family of eigenvalues in the case $ \Omega =\mathbb {R}^N $ Ω = R N and $ b\equiv 0 $ b ≡ 0 a.e. Our approach relies strongly on variational Analysis, in which the Mountain pass theorem plays the key role. Due to the lack of spatial compactness and the embedding $ \mathcal {W}^{\alpha, p}\left (\mathbb {R}^N\right) \hookrightarrow \mathcal {W}^{\beta, q}\left (\mathbb {R}^N\right) $ W α , p (R N) ↪ W β , q (R N) in $ \mathbb {R}^N $ R N , we employ the concentration-compactness principle of P.L. Lions [The concentration-compactness principle in the calculus of variations. The limit case. II, Rev Mat Iberoamericana. 1985;1(2):45–121]. to overcome the difficulty. Our paper can be considered as a counterpart to the important works [Alves et al. Existence, multiplicity and concentration for a class of fractional $ p\& q $ p &q Laplacian problems in $ \Bbb R^N $ R N , Commun Pure Appl Anal, 2019;18(4):2009–2045], [Benci et al. An eigenvalue problem for a quasilinear elliptic field equation. J Differ Equ, 2002;184(2):299–320], [Bobkov et al. On positive solutions for $ (p,q) $ (p , q) -Laplace equations with two parameters, Calc Var Partial Differ Equ, 2015;54(3):3277–3301], [Colasuonno and Squassina. Eigenvalues for double phase variational integrals, Ann Mat Pura Appl (4), 2016;195(6):1917–1956], [Papageorgiou et al. Positive solutions for nonlinear Neumann problems with singular terms and convection, J Math Pures Appl (9), 2020;136:1–21], [Papageorgiou et al. Ground state and nodal solutions for a class of double phase problems, Z Angew Math Phys, 2020;71:1–15], and may have further applications to deal with other problems. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the existence of ground state solutions for the p (x) -Laplacian equation { − Δ p (x) u + λ | u | p (x) − 2 u = f (x , u) + h (x) in Ω , u = 0 , on ∂ Ω. Using the constraint variational method, quantitative deformation lemma, and strong maximum principle, we proved that the above problem admits three ground state solutions, especially speaking, one solution is sign-changing, one is positive, and one is negative. Our results improve on those existing in the literature. [ABSTRACT FROM AUTHOR]

In this work, we study a class of degenerate Dirichlet problems, whose prototype is { − div (∇ u (1 + | u |) γ + c (x) | u | θ − 1 u log β (1 + | u |)) = f in Ω , u = 0 on ∂ Ω , where Ω is a bounded open subset of R N . 0 < γ < 1 , 0 < θ ≤ 1 and 0 ≤ β < 1 . We prove existence of bounded solutions when f and c belong to suitable Lebesgue spaces. Moreover, we investegate the existence of renormalized solutions when the function f belongs only to L 1 (Ω) . [ABSTRACT FROM AUTHOR]

In this article we prove existence, uniqueness and regularity for the singular equation { | ∇ u | α (F (D 2 u) + h (x) ⋅ ∇ u) + c (x) | u | α u + p (x) u − γ = 0 in Ω u > 0 in Ω , u = 0 on ∂ Ω when p is some continuous and positive function, c and h are continuous, α > − 1 and F is Fully nonlinear elliptic. Some conditions on the first eigenvalue for the operator − | ∇ u | α (F (D 2 u) + h (x) ⋅ ∇ u) − c (x) | u | α u are required. The results generalize the well known results of Lazer and McKenna. [ABSTRACT FROM AUTHOR]

In this paper, we prove existence and regularity results for solutions of some nonlinear Dirichlet problems for an elliptic equation defined by a degenerate coercive operator and a singular right-hand side (1) { − div (a (x , u , ∇ u)) = f u γ in Ω , u = 0 on ∂ Ω , where Ω is bounded open subset of I R N (N ≥ 2) , γ > 0 and f is a non-negative function that belongs to some Lebesgue space. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of a class of p (x) -Kirchhoff equation involving concave-convex nonlinearities. The main tools used are the perturbation technique, variational method, and a priori estimation. [ABSTRACT FROM AUTHOR]

This paper considers a class of degenerate quasilinear elliptic equations with discontinuous nonlinearities. The existence of positive weak solutions and S-solutions is discussed using variational methods. The results assert that the (λ , a) -space of the parameters involved is divided into three regions - no solution, at least one S-solution, and at least two weak solutions (one is S-solution among them), in each region respectively. The regions are separated by a continuous, nondecreasing curve and line segment. Further, there exists an S-solution at each point on the separating curve. [ABSTRACT FROM AUTHOR]

We consider equations involving a combination of local and nonlocal degenerate p-Laplace operators. The main contribution of the paper is almost Lipschitz regularity for the homogeneous equation and Hölder continuity with an explicit Hölder exponent in the general case. For certain parameters, our results also imply Hölder continuity of the gradient. In addition, we establish existence, uniqueness and local boundedness. The approach is based on an iteration in the spirit of Moser combined with an approximation method. [ABSTRACT FROM AUTHOR]

We consider the Dirichlet problem for partial trace operators which include the smallest and the largest eigenvalue of the Hessian matrix. It is related to two-player zero-sum differential games. No Lipschitz regularity result is known for the solutions, to our knowledge. If some eigenvalue is missing, such operators are nonlinear, degenerate, non-uniformly elliptic, neither convex nor concave. Here we prove an interior Lipschitz estimate under a non-standard assumption: that the solution exists in a larger, unbounded domain, and vanishes at infinity. In other words, we need a condition coming from far away. We also provide existence results showing that this condition is satisfied for a large class of solutions. On the occasion, we also extend a few qualitative properties of solutions, known for uniformly elliptic operators, to partial trace operators.

Existence of radially symmetric positive solutions for a general class of quasi linear elliptic equations in the zero mass situation is proven. Our results include ground-state solutions for both power-type and exponential-type critical growths in the whole space and the class concerned includes, as particular cases, the Laplace, p-Laplace and k-Hessian operators in radial form. [ABSTRACT FROM AUTHOR]

We study the existence and regularity results for non-linear elliptic equation with degenerate coercivity and a singular gradient lower order term. The model problems is { - d i v (b (x) | ∇ u | p - 2 ∇ u (1 + | u |) γ ) + | ∇ u | p | u | θ = f , i n Ω , u = 0 , o n ∂ Ω , swhere Ω is a bounded open subset in ℝN, 1 ≤ θ < 2, p > 2 and γ > 0. We will show that, even if the lower order term is singular, we obtain existence and regularity of positive solution, under various assumptions on the summability of the source f. [ABSTRACT FROM AUTHOR]

In this paper, the authors will apply De Giorgi-Nash-Moser iteration to establish boundary Hölder estimates for a class of degenerate elliptic equations in piecewise C2-smooth domains. [ABSTRACT FROM AUTHOR]

We prove higher-order fractional Sobolev regularity for fully nonlinear, uniformly elliptic equations in the presence of unbounded source terms. More precisely, we show the existence of a universal number 0 < ε < 1 , depending only on ellipticity constants and dimension, such that if u is a viscosity solution of F (D 2 u) = f (x) ∈ L p , then, u ∈ W 1 + ε , p , with appropriate estimates. Our strategy suggests a sort of fractional feature of fully nonlinear diffusion processes, as what we actually show is that F (D 2 u) ∈ L p ⇒ (− Δ) θ u ∈ L p , for a universal constant 1 2 < θ < 1. We believe our techniques are flexible and can be adapted to various models and contexts. [ABSTRACT FROM AUTHOR]

In this paper we obtain the interior Hölder regularity of the gradient for the elliptic p (x) -Laplacian equation with the logarithmic function d i v A ∇ u ⋅ ∇ u p (x) − 2 2 ln e + A ∇ u ⋅ ∇ u 1 / 2 A ∇ u = d i v | f | p (x) − 2 ln e + f f under some proper assumptions on the Hölder continuous functions p , f and A. This extends previous results obtained by Giannetti and Passarelli di Napoli [Regularity results for a new class of functionals with non-standard growth conditions. J. Differ Equ. 2013;254(3):1280–1305], where A is the identity matrix and hence does not depend on x. [ABSTRACT FROM AUTHOR]

In a bounded domain Ω ⊂ ℝN, we consider the Dirichlet boundary-value problem for an elliptic equation with a non-Lipschitz nonlinearity of the form Δ u = λ u − u α − 1 u , λ ∈ ℝ , 0 < α < 1. The problem of the existence of a solution of the ground-state-type with compact support is examined. [ABSTRACT FROM AUTHOR]

We prove the concavity of classical solutions to a wide class of degenerate elliptic differential equations on strictly convex domains of the unit sphere. The proof employs a suitable two-point maximum principle, a technique which originates in works of Korevaar, Kawohl and Kennington for equations on Euclidean domains. We emphasize that no differentiability of the differential operator is needed, but only some monotonicity and concavity properties. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the oscillation of solutions of the nonlinear degenerate elliptic equation in the Heisenberg group H n . We first derive a critical inequality in H n . Based on it, we establish a Picone-type differential inequality and a Sturm-type comparison principle. Then we obtain an oscillation theorem. Our result generalizes the related conclusions for the nonlinear elliptic equations in R n . [ABSTRACT FROM AUTHOR]

We consider a class of equations in divergence form with a singular/degenerate weight − div (| y | a A (x , y) ∇ u) = | y | a f (x , y) or div (| y | a F (x , y)). Under suitable regularity assumptions for the matrix A and f (resp. F) we prove Hölder continuity of solutions which are even in y ∈ R , and possibly of their derivatives up to order two or more (Schauder estimates). In addition, we show stability of the C 0 , α and C 1 , α a priori bounds for approximating problems in the form − div ( (ε 2 + y 2) a / 2 A (x , y) ∇ u) = (ε 2 + y 2) a / 2 f (x , y) or div ( (ε 2 + y 2) a / 2 F (x , y)) as ε → 0. Finally, we derive C 0 , α and C 1 , α bounds for inhomogenous Neumann boundary problems as well. Our method is based upon blow-up and appropriate Liouville type theorems. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence and regularity of solutions of the homogeneous Dirichlet initial-boundary value problem for a class of degenerate elliptic equations with lower order terms. The results we obtained here, extend some existing ones of [2, 9, 11] in some sense. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity of weak solutions to the boundary value problem − G α u = f (x , y , u) + g (x , y , u) in Ω , u = 0 on ∂ Ω , where Ω is a bounded domain with smooth boundary in R N (N ≥ 2) , α ∈ N , f (x , y , ξ) is odd in ξ and g (x , y , ξ) is a perturbation term. Under some growth conditions on f and g, we show that there are infinitely many weak solutions to the problem. Here we do not require that f satisfies the Ambrosetti-Rabinowitz (AR) condition. The conditions on f and g are relatively weak and our result is new even in the case α = 0 , i.e. for the classical Laplace equation with the Dirichlet boundary condition. [ABSTRACT FROM AUTHOR]

We prove the local Lipschitz continuity and the higher differentiability of local minimizers of functionals of the form 𝔽 ⁢ (u , Ω) = ∫ Ω (F ⁢ (x , D ⁢ u ⁢ (x)) + f ⁢ (x) ⋅ u ⁢ (x) ) ⁢ d ⁢ x \mathbb{F}(u,\Omega)=\int_{\Omega}(F(x,Du(x))+f(x)\cdot u(x))\mathop{}\!dx with non-autonomous integrand F ⁢ (x , ξ) {F(x,\xi)} which is degenerate convex with respect to the gradient variable. The main novelty here is that the results are obtained assuming that the partial map x ↦ D ξ ⁢ F ⁢ (x , ξ) {x\mapsto D_{\xi}F(x,\xi)} has weak derivative in the almost critical Zygmund class L n ⁢ log α ⁡ L {L^{n}\log^{\alpha}L} and the datum f is assumed to belong to the same Zygmund class. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of multiple solutions for the quasilinear equation - div ⁡ 𝐀 ⁢ (x , ∇ ⁡ u) + | u | α ⁢ (x) - 2 ⁢ u = f ⁢ (x , u) in ℝ N , {-\operatorname{div}\mathbf{A}(x,\nabla u)+|u|^{\alpha(x)-2}u=f(x,u)\quad\text% {in ${\mathbb{R}^{N}}$,}} which involves a general variable exponent elliptic operator 𝐀 {\mathbf{A}} in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has various types of behavior like | ξ | q ⁢ (x) - 2 ⁢ ξ {|\xi|^{q(x)-2}\xi} for small | ξ | {|\xi|} and like | ξ | p ⁢ (x) - 2 ⁢ ξ {|\xi|^{p(x)-2}\xi} for large | ξ | {|\xi|} , where 1 < α ⁢ (⋅) ≤ p ⁢ (⋅) < q ⁢ (⋅) < N {1<\alpha(\,\cdot\,)\leq p(\,\cdot\,)

We establish the interior Hölder continuity for locally bounded solutions, and the Harnack inequality for non-negative continuous solutions to a class of anisotropic elliptic equations with bounded and measurable coefficients, whose prototype equation is u xx + Δ q , y u = 0 locally in R × R N - 1 , for q < 2 , via ideas and tools originating from the regularity theory for degenerate and singular parabolic equations. [ABSTRACT FROM AUTHOR]

In this paper we study the strong maximum principle for equations of the form F [ u ] = H (u , | D u |) where F is either a fully nonlinear elliptic operator or is the p-Laplace operator. We give sufficient conditions on H to ensure that the strong maximum principle (SMP) holds. The condition is also necessary for SMP to hold for the the equation F [ u ] = g (| D u |). [ABSTRACT FROM AUTHOR]

Integrals of the Calculus of Variations with p , q {p,q} -growth may have not smooth minimizers, not even bounded, for general p , q {p,q} exponents. In this paper we consider the scalar case, which contrary to the vector-valued one allows us not to impose structure conditions on the integrand f ⁢ (x , ξ) {f(x,\xi)} with dependence on the modulus of the gradient, i.e. f ⁢ (x , ξ) = g ⁢ (x , | ξ |) {f(x,\xi)=g(x,|\xi|)}. Without imposing structure conditions, we prove that if q p {\frac{q}{p}} is sufficiently close to 1, then every minimizer is locally Lipschitz-continuous. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of a class of nonlinear anisotropic elliptic equations with degenerate coercivity, lower order term and L 1 datum in appropriate anisotropic variable exponent Sobolev space. We obtain the existence of the entropy solutions via strong convergence of the truncations and we derive some properties of these solutions. [ABSTRACT FROM AUTHOR]

We are concerned with the following p (x) -Laplacian equations in R N − △ p (x) u + | u | p (x) − 2 u = f (x , u) in R N. The nonlinearity is superlinear but does not satisfy the Ambrosetti-Rabinowitz type condition. Our main difficulty is that the weak limit of (PS) sequence is not always the weak solution of this problem. To overcome this difficulty, by adding potential term and using mountain pass theorem, we get the weak solution u λ of perturbation equations. First, we prove that u λ ⇀ u as λ → 0 . Second, by using vanishing lemma, we get that u is a nontrivial solution of the original problem. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of viscosity solutions to the Gelfand problem for the 1-homogeneous p-Laplacian in a bounded domain Ω⊂ℝN{\Omega\subset\mathbb{R}^{N}}, that is, we deal with

We have investigated the behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities in bounded and unbounded domains. We found exponents of the solution’s decreasing rate near the boundary singularities.

We study two obstacle problems involving the p-Laplace operator in domains with n-th pre-fractal and fractal boundary. We perform asymptotic analysis for p ∞ and n ∞. [ABSTRACT FROM AUTHOR]

We consider a function U satisfying a degenerate elliptic equation on R+N+1:= (0, +∞) × RN with mixed Dirichlet-Neumann boundary conditions. The Neumann condition is prescribed on a bounded domain Ω ⊂ RN of class C1,1, whereas the Dirichlet data is on the exterior of Ω. We prove H¨older regularity estimates of … , where dΩ is a distance function defined as dΩ(z) := dist(z, RN \ Ω), for z ∈ R+N+1. The degenerate elliptic equation arises from the Caffarelli-Silvestre extension of the Dirichlet problem for the fractional Laplacian. Our proof relies on compactness and blow-up analysis arguments. [ABSTRACT FROM AUTHOR]

In this paper, we study the differentiability of solutions on the boundary for equations of type Lλu=∂2u∂x2+|x|2λ∂2u∂y2=f(x,y), where λ is an arbitrary positive number. By introducing a proper metric that is related to the elliptic operator Lλ, we prove the differentiability on the boundary when some well-posed boundary conditions are satisfied. The main diffculty is the construction of new barrier functions in this article. [ABSTRACT FROM AUTHOR]

We consider the quasilinear degenerate elliptic equation with rough and singular coefficients of the form divA(x,u,∇u)=B(x,u,∇u)in an open set Ω⊂Rn, where the quadratic form associated with the principle part of this equation allows vanishing and the coefficients in structural conditions on A(x,u,∇u) and B(x,u,∇u) require no smoothness but belong to some Stummel-Kato class. We prove a Fefferman-Phong type inequality related to the Stummel-Kato class and then an embedding inequality. Based on these inequalities, the local boundedness and Harnack’s inequality of the weak solutions are derived. As applications, the continuity and Hölder continuity for the nonnegative weak solutions are given. [ABSTRACT FROM AUTHOR]

We study the behaviour near the boundary angular or conical point of weak solutions to the Robin problem for elliptic quasi-linear second-order equation with thep-Laplacian. [ABSTRACT FROM PUBLISHER]

We investigate the existence of solutions of the Kirchhoff elliptic equations with nonlinearity in $R^{N}$ . Using the ideas developed in Orlicz spaces and the technique of variation principle, we prove that there are at least three solutions in Orlicz-Sobolev spaces. [ABSTRACT FROM AUTHOR]

We define a homogeneous De Giorgi class of order p ⩾ 2 which suits two classes of evolution equations of the type μ ( x , t ) u t + A u = 0 and ( μ ( x , t ) u ) t + A u = 0 , where μ can be positive, null and negative and A a suitable elliptic operator. For functions belonging to this class we prove a local boundedness result. [ABSTRACT FROM AUTHOR]

Let,is a bounded domain withboundarysuch thatandwhereis the fractional critical Sobolev exponent. Letbe a Carathéodory function withand there existssatisfyingif,otherwise, such thatfor alland for someConsider the associated functionaldefined aswhereTheorem 1.1 proves that ifis a local minimum ofIin the-topology, then it is also a local minimum in-topology. This result is useful for proving multiple solutions to the associated Euler–Lagrange equation (P) defined below. Theorem 1.1 given in the present paper can be also extended to more general quasilinear elliptic equations. [ABSTRACT FROM AUTHOR]

The diffuse domain method for partial differential equations on complicated geometries recently received strong attention in particular from practitioners, but many fundamental issues in the analysis are still widely open. In this paper, we study the diffuse domain method for approximating second order elliptic boundary value problems posed on bounded domains and show convergence and rates of the approximations generated by the diffuse domain method to the solution of the original second order problem when complemented by Robin, Dirichlet or Neumann conditions. The main idea of the diffuse domain method is to relax these boundary conditions by introducing a family of phase-field functions such that the variational integrals of the original problem are replaced by a weighted average of integrals of perturbed domains. From a functional analytic point of view, the phase-field functions naturally lead to weighted Sobolev spaces for which we present trace and embedding results as well as various types of Poincaré inequalities with constants independent of the domain perturbations. Our convergence analysis is carried out in such spaces as well, but allows to draw conclusions also about unweighted norms applied to restrictions on the original domain. Our convergence results are supported by numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, motivated by recent works on the study of the equations which model the electrostatic MEMS devices, we study the quasilinear elliptic equation involving a singular nonlinearity { − ( r α | u ′ ( r ) | β u ′ ( r ) ) ′ = λ r γ f ( r ) ( 1 − u ( r ) ) 2 , r ∈ ( 0 , 1 ) , 0 ≤ u ( r ) < 1 , r ∈ ( 0 , 1 ) , u ′ ( 0 ) = u ( 1 ) = 0 . According to the choice of the parameters α , β and γ , the differential operator which we are dealing with corresponds to the radial form of the Laplacian, the p -Laplacian and the k -Hessian. In this work we present conditions over which we can assert regularity for solutions, including the case λ = λ ∗ , where λ ∗ is a critical value for the existence of solutions. Moreover, we prove that whenever the critical solution is regular, there exists another solution of mountain pass type for λ close to the critical one. In addition, we use the Shooting Method to prove uniqueness of solutions for λ in a neighborhood of 0 . [ABSTRACT FROM AUTHOR]