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We investigate Bernoulli free boundary problems prescribing infinite jump conditions. The mathematical set-up leads to the analysis of non-differentiable minimization problems of the form ∫ ∇ u · (A (x) ∇ u) + φ (x) 1 { u > 0 } d x → min , where A(x) is an elliptic matrix with bounded, measurable coefficients and φ is not necessarily locally bounded. We prove universal Hölder continuity of minimizers for the one- and two-phase problems. Sharp regularity estimates along the free boundary are also obtained. Furthermore, we perform a thorough analysis of the geometry of the free boundary around a point ξ of infinite jump, ξ ∈ φ - 1 (∞) . We show that it is determined by the blow-up rate of φ near ξ and we obtain an analytical description of such cusp geometries. [ABSTRACT FROM AUTHOR]

We present new regularity result in the study of elliptic equations with mixed boundary conditions. We obtain small higher integrability of the gradient (Meyers property). The result is new for both linear and nonlinear equations with degenerate coefficients with a new sharp Maz’ya-type capacity condition on the part with Dirichlet boundary condition, while prior research was limited to uniformly elliptic weights. [ABSTRACT FROM AUTHOR]

The present paper is concerned by the study of a nonlinear elliptic equation which contains a Hardy potential, lower order term, singular term and L m (·) data. Our approach is based on approximating the initial problem with a non-singular problem that is well-posed. We then establish the necessary estimates to pass to the limit. [ABSTRACT FROM AUTHOR]

We present two Liouville-type results for solutions to anisotropic elliptic equations that have a growth of power 2 along the first ss coordinate directions and of power pp, with 10p+\left(N-s)\left(p-2)\gt 0 of non-degeneration, which is a purely parabolic shade. Then we break free from this constraint at the price of assuming the solution to be bounded also from above.

In this paper, by using the variational methods, we consider the existence of nontrivial solitary wave for a generalized Kadomtsev–Petviashvili equation with a constant potential and a periodic potential in $ \mathbb {R}^{2} $ R 2 , respectively. In our problem, the nonlinear term f is only continuous, which allows to consider larger classes of nonlinearities than usually assumed. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Dirichlet problem in a bounded open set Ω ⊂ R n ( n ⩾ 2 ) for a class of second-order nonlinear elliptic equations with right-hand side f in L 1 (Ω) . We study the summability of entropy and weak solutions of this problem under the stronger assumption that f G (| f |) ∈ L 1 (Ω) , where G is a nonnegative increasing continuous function on [ 0 , + ∞) . We show how the summability of the solutions depends on the function G. Our conditions on G imply that L 1 + ε (Ω) ⊂ K G ⊂ L 1 (Ω) for every ε > 0 , where K G is the set of all measurable functions v on Ω such that v G (| v |) ∈ L 1 (Ω) . [ABSTRACT FROM AUTHOR]

We prove partial regularity of solutions u of the nonlinear quasimonotone system div A x , u , D u + B x , u , D u = 0 under natural polynomial growth of its coefficient functions A and B. We propose a new direct method based on an L p estimate with low exponent p > 1 for a linear elliptic system with constant coefficient. [ABSTRACT FROM AUTHOR]

We consider the functional ℱ(u)≔∫Ωf(x,Du(x))dx,{\mathcal{ {\mathcal F} }}\left(u):= \mathop{\int }\limits_{\Omega }f\left(x,Du\left(x)){\rm{d}}x, where f(x,z)f\left(x,z) satisfies a (p,q)\left(p,q)-growth condition with respect to zz and can be approximated by means of a suitable sequence of functions. We consider BR⋐Ω{B}_{R}\hspace{0.33em}\Subset \hspace{0.33em}\Omega and the spaces X=W1,p(BR,RN)andY=W1,p(BR,RN)∩Wloc1,q(BR,RN).X={W}^{1,p}\left({B}_{R},{{\mathbb{R}}}^{N})\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}Y={W}^{1,p}\left({B}_{R},{{\mathbb{R}}}^{N})\cap {W}_{\hspace{0.1em}\text{loc}\hspace{0.1em}}^{1,q}\left({B}_{R},{{\mathbb{R}}}^{N}). We prove that the lower semicontinuous envelope of ℱ∣Y{\mathcal{ {\mathcal F} }}{| }_{Y} coincides with ℱ{\mathcal{ {\mathcal F} }} or, in other words, that the Lavrentiev term is equal to zero for any admissible function u∈W1,p(BR,RN)u\in {W}^{1,p}\left({B}_{R},{{\mathbb{R}}}^{N}). We perform the approximations by means of functions preserving the values on the boundary of BR{B}_{R}.

We establish an Amann-Zehnder-type result for resonance systems of quasilinear elliptic equations with homogeneous Dirichlet boundary conditions, involving nonlinearities growing asymptotically (p,q)\left(p,q)-linear at infinity. The proof relies on a cohomological linking in a product Banach space where the properties of cones of the sublevels are missing, differently from the single quasilinear equation. We also perform critical group computations of the energy functional at the origin, in spite of the lack of its C2{C}^{2} regularity, to exclude that the found mini-max solution is trivial. Finally, we furnish a local condition which guarantees that the found solution is not semi-trivial.

We consider viscosity solutions to two-phase free boundary problems for the p(x)-Laplacian with non-zero right hand side. We prove that flat free boundaries are C 1 , γ . No assumption on the Lipschitz continuity of solutions is made. These regularity results are the first ones in literature for two-phase free boundary problems for the p(x)-Laplacian and also for two-phase problems for singular/degenerate operators with non-zero right hand side. They are new even when p (x) ≡ p , i.e., for the p-Laplacian. The fact that our results hold for merely viscosity solutions allows a wide applicability. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear discontinuous sub-elliptic system with drift on the Heisenberg group, where the coefficients in system are discontinuous and satisfy the vanishing mean oscillation condition and growth conditions with the growth index $ 2 2 < p < ∞ , and the non-homogeneous terms satisfy the controllable growth conditions and the natural growth conditions, respectively. The partial Hölder regularity to weak solutions and the partial Morrey regularity to horizontal gradients of weak solutions are proved by the $ \mathcal {A} $ A -harmonic approximation. One of the difficulties in this paper is how to reasonably define the weak solution and avoid the requirement of integrability of Tu. In fact, we used $ T={X_i}{X_{n+i}}-{X_{n+i}}{X_i} $ T = X i X n + i − X n + i X i . [ABSTRACT FROM AUTHOR]

We work with a double-phase energy functional exhibiting log L -perturbed p & q -growth. We look for regularity properties of such functional in the setting of Musielak-Orlicz-Sobolev space, by imposing suitable conditions on the data. We further obtain the existence and uniqueness results for the solution of perturbed Dirichlet double-phase problems. We deal with both the cases of uncontrolled growth and controlled growth. [ABSTRACT FROM AUTHOR]

In this paper, we study the boundary regularity for viscosity solutions of fully nonlinear elliptic equations. We use a unified, simple method to prove that if the domain Ω satisfies the exterior C 1 , Dini condition at x 0 ∈ ∂ Ω , the solution is Lipschitz continuous at x 0 ; if Ω satisfies the interior C 1 , Dini condition at x 0 , the Hopf lemma holds at x 0 . The key idea is that the curved boundaries are regarded as perturbations of a hyperplane. Moreover, we show that the C 1 , Dini conditions are optimal. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and regularity results for some parabolic equations with degenerate coercivity, and a singular right-hand side. The model problem is 0.1 { ∂ u ∂ t − div ( (1 + | ∇ u | − Λ) | ∇ u | p − 2 ∇ u (1 + | u |) θ) = f (e u − 1) γ in Q T , u (x , 0) = 0 on Ω , u = 0 on ∂ Q T , where Ω is a bounded open subset of R N N ≥ 2 , T > 0 , Λ ∈ [ 0 , p − 1) , f is a non-negative function belonging to L m (Q T) , Q T = Ω × (0 , T) , ∂ Q T = ∂ Ω × (0 , T) , 0 ≤ θ < p − 1 + p N + γ (1 + p N) and 0 ≤ γ < p − 1 . [ABSTRACT FROM AUTHOR]

We consider weak solutions (u , π) : Ω → ℝ n × ℝ to stationary ϕ-Navier-Stokes systems of the type − div a (x , E u) + ∇ π + [ D u ] u = f div u = 0 in Ω ⊂ ℝ n , and to the corresponding ϕ-Stokes systems, in which the convective term [Du]u does not appear. In the above system, the function a(x,ξ) depends Hölder continuously on x and satisfies growth conditions with respect to the second variable expressed through a Young function ϕ. The notation E u is used for the symmetric part of the gradient Du. We prove results on the fractional higher differentiability of both the symmetric part of the gradient E u and of the pressure π. [ABSTRACT FROM AUTHOR]

We prove the absence of the Lavrentiev gap for non-autonomous functionals ℱ ⁢ (u) ≔ ∫ Ω f ⁢ (x , D ⁢ u ⁢ (x)) ⁢ 푑 x , where the density f ⁢ (x , z) is α-Hölder continuous with respect to x ∈ Ω ⊂ ℝ n , it satisfies the (p , q) -growth conditions | z | p ⩽ f ⁢ (x , z) ⩽ L ⁢ (1 + | z | q) , where 1 < p < q < p ⁢ (n + α n) , and it can be approximated from below by suitable densities f k . [ABSTRACT FROM AUTHOR]

We study nonlinear elliptic double obstacles problems with an asymptotically regular nonlinearity in a non-smooth domain. Here, we obtain a global Calderón–Zygmund type estimate in the setting of Lorentz spaces by making use of the perturbation method. More precisely, we approximate the solutions of asymptotically regular double obstacles problems to the solutions of regular single obstacle problems when the gradients of solutions are close to infinity. [ABSTRACT FROM AUTHOR]

We obtain sharp local C 1 , α regularity of solutions for singular obstacle problems, Euler-Lagrange equation of which is given by Δ p u = γ (u - φ) γ - 1 in { u > φ } , for 0 < γ < 1 and p ≥ 2 . At the free boundary ∂ { u > φ } , we prove optimal C 1 , τ regularity of solutions, with τ given explicitly in terms of p, γ and smoothness of φ , which is new even in the linear setting. [ABSTRACT FROM AUTHOR]

This article aims to prove a local Calderón–Zygmund estimate for asymptotically regular elliptic double obstacle problems of multi-phase. By approximating the solutions of asymptotically regular double obstacle problems to the solutions of regular single obstacle problems when the gradients of solutions close to infinity, we get that the gradient of the solution is as integrable as both the nonhomogeneous term and the gradients of the associated double obstacles. [ABSTRACT FROM AUTHOR]

In this paper we present new embedding results for Musielak–Orlicz Sobolev spaces of double phase type. Based on the continuous embedding of W 1 , H (Ω) into L H ∗ (Ω) , where H ∗ is the Sobolev conjugate function of H , we present much stronger embeddings as known in the literature. Based on these results, we consider generalized double phase problems involving such new type of growth with Dirichlet and nonlinear boundary condition and prove appropriate boundedness results of corresponding weak solutions based on the De Giorgi iteration along with localization arguments. [ABSTRACT FROM AUTHOR]

In this paper, we are devoted to a global W 1 , γ (⋅) -estimate for elliptic obstacle problem with double phase growth for the borderline setting in nonsmooth domain. More precisely, we consider the variational inequality ∫ Ω 〈 A (x , D u) , D v − D u 〉 d x ≥ ∫ Ω 〈 B (x , F) , D v − D u 〉 d x ∀ v ∈ A 0 (Ω) , whose characteristics of ellipticity and growth switch between a type of polynomial and its logarithm, then we prove a variable exponent Calderón-Zygmund estimate with an implication that H (x , F) , H (x , D ψ) ∈ L γ (x) (Ω) ⇒ H (x , D u) ∈ L γ (x) (Ω) provided that the nonlinearity satisfies a small BMO condition while the boundary of the underlying domain is flat in the sense of Reifenberg. [ABSTRACT FROM AUTHOR]

By applying the De Giorgi-Nash-Moser theory, we obtain an interior Hölder continuity of weak solutions to nonlocal Schrödinger equations given by an integro-differential operator LK as follows; L K u + V u = 0 in Ω , u = g in ℝ n ∖ Ω where V = V+ − V− with V − ∈ L loc 1 (ℝ n) and V + ∈ L loc q (ℝ n) (q > n 2 s > 1 , 0 < s < 1) is a potential such that (V − , V + b , k) belongs to the (A1,A1)-Muckenhoupt class and V + b , k is in the A1-Muckenhoupt class for all k ∈ ℕ (here, V + b , k = V + max { b , 1 / k } / b for a nonnegative bounded function b on ℝ n with V + / b ∈ L loc q (ℝ n) ), g ∈ H s (ℝ n) and Ω is a bounded domain in ℝ n with Lipschitz boundary. In addition, we get the local boundedness of weak subsolutions of the nonlocal Schrödinger equations. In particular, we note that all the above results still work for any nonnegative potential in L loc q (ℝ n) (q > n 2 s > 1 , 0 < s < 1). [ABSTRACT FROM AUTHOR]

Let Ω ⊂ R n be a bounded convex domain with n ≥ 2 . Suppose that A is uniformly elliptic and belongs to W 1 , n when n ≥ 3 or W 1 , q for some q > 2 when n = 2 . For 1 < p < ∞ , we establish a global second-order regularity estimate ‖ D [ | D u | p - 2 D u ] ‖ L 2 (Ω) + ‖ D [ ⟨ A D u , D u ⟩ p - 2 2 A D u ] ‖ L 2 (Ω) ≤ C ‖ f ‖ L 2 (Ω) for the inhomogeneous p-Laplace type equation - div ( ⟨ A D u , D u ⟩ p - 2 2 A D u) = f in Ω with Dirichlet or Neumann homogeneous boundary condition. Similar result was also established for certain bounded Lipschitz domains whose boundary is weakly second-order differentiable and satisfies some smallness assumptions. [ABSTRACT FROM AUTHOR]

We consider a semilinear boundary value problem - Δ u = f (x , u) , in Ω , with Dirichlet boundary conditions, where Ω ⊂ R N with N > 2 , is a bounded smooth domain, and f is a Carathéodory function, superlinear and subcritical at infinity. We provide L ∞ (Ω) a priori estimates for weak solutions in terms of their L 2 ∗ (Ω) -norm, where 2 ∗ = 2 N N - 2 is the critical Sobolev exponent. In particular, our results also apply to f (x , s) = a (x) | s | 2 N / r ∗ - 2 s [ log (e + | s |) ] β , where a ∈ L r (Ω) with N / 2 < r ≤ ∞ , and 2 N / r ∗ : = 2 ∗ 1 - 1 r . Assume N / 2 < r ≤ N . We show that for any ε > 0 there exists a constant C ε > 0 such that for any solution u ∈ H 0 1 (Ω) , the following holds: [ log (e + ‖ u ‖ ∞) ] β ≤ C ε (1 + ‖ u ‖ 2 ∗ ) (2 N / r ∗ - 2) (1 + ε) . To establish our results, we do not assume any restrictions on the sign of the solutions, or on the non-linearity. Our approach is based on Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg interpolation inequalities. Finally, we state sufficient conditions for having H 0 1 (Ω) uniform a priori bounds for non-negative solutions, so finally we provide suficient conditions for having L ∞ (Ω) uniform a priori bounds, which holds roughly speaking for superlinear and subcritical non-linearities. [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 and regularity of solutions to quasilinear elliptic equations having quadratic growth with respect to the gradient and two singularities. [ABSTRACT FROM AUTHOR]

Boundary value problems for second-order elliptic equations in divergence form, whose nonlinearity is governed by a convex function of non-necessarily power type, are considered. The global boundedness of their solutions is established under boundary conditions of Dirichlet, or Neumann, or Robin type. A decisive role in the results is played by optimal forms of Orlicz-Sobolev embeddings and boundary trace embeddings, which allow for critical growths of the coefficients. [ABSTRACT FROM AUTHOR]

We consider a singular nonlinear elliptic Dirichlet problems with lower-order terms, where the combined effects of a superlinear growth in the gradient and a singular term allow us to establish some existence and regularity results. The model problem is 0.1 - div (| ∇ u | p - 2 ∇ u) + μ | u | p - 1 u = b (x) | ∇ u | q u θ + f (x) u γ in Ω , u = 0 on ∂ Ω , where Ω is an open and bounded subset of R N , μ ≥ 0 , 0 < θ ≤ 1 , 0 ≤ γ < 1 and f is a nonnegative function that belong to some Lebesgue space. [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.

We consider the following doubly nonlocal nonlinear logistic problem driven by the fractional p-Laplacian (- Δ) p s u = f (x , u) - ∫ Ω | u (y) | r | x - y | α d y | u (x) | r - 2 u (x) in Ω , u = 0 in R N \ Ω. Here Ω ⊂ R N (N ≥ 2) is a bounded domain with C 1 , 1 boundary ∂ Ω , s ∈ (0 , 1) , p ∈ (1 , ∞) are such that p s < N . Also p s , α # ≤ r < ∞ , where p s , α # = (2 N - α) / 2 N . Under suitable and general assumptions on the nonlinearity f, we study the existence, nonexistence, uniqueness, and regularity of weak solutions. As for applications, we treat cases of subdiffusive type logistic Choquard problem. We also consider in the superdiffusive case the Brezis-Nirenberg type problem with logistic Choquard and show the existence of a nontrivial solution for a suitable choice of λ . Finally for a particular choice of f viz. f (x , t) = λ t q - 1 with 1 < p < 2 r < q , we show the existence of at least one energy nodal solution. [ABSTRACT FROM AUTHOR]

In this paper, we establish the pointwise boundary differentiability on Reifenberg domains for viscosity solutions of fully nonlinear elliptic equations which extends the result under the usual C 1 , Dini condition and generalizes the result for linear equations in Huang et al (Manuscr. Math 162(3–4):305–313, 2020). Moreover, our proofs are relatively simple. [ABSTRACT FROM AUTHOR]

P-Laplacian, a priori estimates, 35J60, 35J60, stable solutions, extremal solutions, regularity, 35B65, 35B35 Keywords: p-Laplacian; stable solutions; extremal solutions; regularity; a priori estimates; 35B65; 35B35; 35J60 EN p-Laplacian stable solutions extremal solutions regularity a priori estimates 35B65 35B35 35J60 749 785 37 10/03/22 20221001 NES 221001 1 Introduction Given HT ht , a function HT ht , and a smooth bounded domain HT ht , we consider the elliptic equation involving the I p i -Laplacian HT lt;mo stretchy="false">} ht since here the equation is uniformly elliptic; see Remark 2.2. [Extracted from the article]

In this paper, we prove up to the boundary gradient estimates for viscosity solutions to inhomogeneous nonlinear Free Boundary Problems (FBP) governed by fully nonlinear and quasilinear elliptic equations with unbounded measurable ingredients. Here, we build upon our previous results in [9] to construct Inhomogeneous Pucci Barriers (IPB) for the Pucci extremal equations with unbounded coefficients. Using these barriers, we obtain a version of a boundary growth type lemma for inhomogeneous nonlinear equations that may be of independent interest. In a certain way, this lemma detects the expansion of the level sets of supersolutions from the boundary to the interior. The use of this boundary growth type lemma together with the geometry of IPB bridge the interchanging information between the free boundary condition and Dirichlet boundary data on the free and fixed boundary respectively. This produces an estimate on the trace of solutions to FBP along the fixed boundary. This way, control of such solutions (up to the boundary) by the distance to the negative phase is obtained. Finally, this distance control combined with the PDE boundary gradient estimates render our final result. [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]

A sharp pointwise differential inequality for vectorial second-order partial differential operators, with Uhlenbeck structure, is offered. As a consequence, optimal second-order regularity properties of solutions to nonlinear elliptic systems in domains in R n are derived. Both local and global estimates are established. Minimal assumptions on the boundary of the domain are required for the latter. In the special case of the p-Laplace system, our conclusions broaden the range of the admissible values of the exponent p previously known. [ABSTRACT FROM AUTHOR]

Under the lack of variational structure and nondegeneracy, we investigate three notions of generalized principal eigenvalue for a general infinity Laplacian operator with gradient and homogeneous term. A Harnack inequality is proved to support our analysis. This is a continuation of our first work (Biswas and Vo in Liouville theorems for infinity Laplacian with gradient and KPP type equation. Ann. Sc. Norm. Super. Pisa Cl. Sci. https://doi.org/10.2422/2036-2145.202105%5f050) and a contribution in the development of the theory of generalized principal eigenvalue beside the works (Berestycki et al. in Commun Pure Appl Math 47(1):47–92, 1994; Berestycki and Rossi in JEMS 8:195–215, 2006; Berestycki and Rossi in Commun Pure Appl Math 68(6):1014–1065, 2015; Berestycki et al. in J Math Pures Appl 103:1276–1293, 2015; Nguyen and Vo in Calc Var Partial Differ Equ 58(3):102 2019). We use these notions to characterize the validity of maximum principle and study the existence, nonexistence and uniqueness of positive solutions of Fisher-KPP type equations in the whole space. The sliding method is intrinsically improved for infinity Laplacian to solve the problem. The results are related to the Liouville type results, which will be meticulously explained. [ABSTRACT FROM AUTHOR]

The authors study the continuity estimate of the solutions of almost harmonic maps with the perturbation term f in a critical integrability class (Zygmund class) L n 2 , logqL, n is the dimension with n ≥ 3. They prove that when q > n 2 the solution must be continuous and they can get continuity modulus estimates. As a byproduct of their method, they also study boundary continuity for the almost harmonic maps in high dimension. [ABSTRACT FROM AUTHOR]

We analyze stable weak solutions to the fractional Geľfand problem

We obtain an explicit Hölder regularity result for viscosity solutions of a class of second order fully nonlinear equations led by operators that are neither convex/concave nor uniformly elliptic. [ABSTRACT FROM AUTHOR]

Given certain functions φ , q , h and a nonnegative constant λ , we establish regularities for the free boundary problem J (u) = ∫ Ω (G (| ∇ u |) + q F (u +) + h u + λ χ { u > 0 } ) d x → min , over the set { u ∈ W 1 , G (Ω) : u - φ ∈ W 0 1 , G (Ω) } in the setting of Orlicz spaces, where the functions G and F satisfy the structural conditions of Tolksdorf's type. Moreover, F allows for subcritical exponents. The main results obtained in this paper include: the local C 1 , α - and Log-Lipschitz continuities of minimizers in the subcritical case for λ = 0 and λ ≥ 0 , respectively; the growth rates near the free boundary for non-negative minimizers in the subcritical case for λ ≥ 0 , which give the optimal growth rates of non-negative minimizers for the p-Laplacian problems; the local Lipschitz continuity of non-negative minimizers for λ > 0 under the natural growth condition that F (t) ≲ 1 + G (t) for t ≥ 0 . All the results presented in this paper are new even for the free boundary problems of p-Laplacian. [ABSTRACT FROM AUTHOR]