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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]

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.

In this paper, the authors study the multiplicity of solutions to the weighted p-Laplacian with isolated singularity and diffusion suppressed by convection subject to nonlinear Robin boundary value condition where λ > 0, B ⊂ ℝN (N ≥ 2) is the unit ball centered at the origin, α > 0, p > 1, β ∈ ℝ, γ > −N, g ∈ C([0, 1]) with g(0) > 0, A ∈ ℝ, ρ > 0 and is the unit outward normal. The same problem with diffusion promoted by convection, namely λ ≤ 0, has already been discussed by the last two authors (Song-Yin (2012)), where the existence, nonexistence and classification of singularities for solutions are presented. Completely different from [Song, H. J. and Yin, J. X., Removable isolated singularities of solutions to the weighted p-Laplacian with singular convection, Math. Meth. Appl. Sci., 35, 2012, 1089–1100], in the present case λ > 0, namely the diffusion is suppressed by the convection, non-singular solutions are not only existent but also may be infinite which vary according only to the values of solutions at the isolated singular point. At the same time, the singular solutions may exist only if the diffusion dominates the convection. [ABSTRACT FROM AUTHOR]

Let Ω⊂ℝN{\Omega\subset\mathbb{R}^{N}} (N≥3{N\geq 3}) be a C2{C^{2}} bounded domain, and let δ be the distance to ∂⁡Ω{\partial\Omega}. We study equations (E±){(E_{\pm})}, -Lμ⁢u±g⁢(u,|∇⁡u|)=0{-L_{\mu}u\pm g(u,\lvert\nabla u\rvert)=0} in Ω, where Lμ=Δ+μδ2{L_{\mu}=\Delta+\frac{\mu}{\delta^{2}}}, μ∈(0,14]{\mu\in(0,\frac{1}{4}]} and g:ℝ×ℝ+→ℝ+{g\colon\mathbb{R}\times\mathbb{R}_{+}\to\mathbb{R}_{+}} is nondecreasing and locally Lipschitz in its two variables with g⁢(0,0)=0{g(0,0)=0}. We prove that, under some subcritical growth assumption on g, equation (E+){(E_{+})} with boundary condition u=ν{u=\nu} admits a solution for any nonnegative bounded measure on ∂⁡Ω{\partial\Omega}, while equation (E-){(E_{-})} with boundary condition u=ν{u=\nu} admits a solution provided that the total mass of ν is small. Then we analyze the model case g⁢(s,t)=|s|p⁢tq{g(s,t)=\lvert s\rvert^{p}t^{q}} and obtain a uniqueness result, which is even new with μ=0{\mu=0}. We also describe isolated singularities of positive solutions to (E+){(E_{+})} and establish a removability result in terms of Bessel capacities. Various existence results are obtained for (E-){(E_{-})}. Finally, we discuss existence, uniqueness and removability results for (E±){(E_{\pm})} in the case g⁢(s,t)=|s|p+tq{g(s,t)=\lvert s\rvert^{p}+t^{q}}.

In this paper, we establish the existence of at least one positive solution to a singular elliptic equation with zero boundary data and critical Hardy–Sobolev exponent. We show that the existence of the positive solution in high dimension depends on the sign of the mean curvature of the boundary near zero and in the low dimensions, it depends on the mass of the domain. [ABSTRACT FROM AUTHOR]

We consider the problem

Let Ω ⊂ R N ( N ≥ 3 ) be a C 2 bounded domain and F ⊂ ∂ Ω be a C 2 submanifold with dimension 0 ≤ k ≤ N - 2 . Denote δ F = dist (· , F) , V = δ F - 2 and C H (V) the Hardy constant relative to V in Ω . We study positive solutions of equations (LE) - L γ V u = 0 and (NE) - L γ V u + f (u) = 0 in Ω where L γ V = Δ + γ V , γ < C H (V) and f ∈ C (R) is an odd, monotone increasing function. We extend the notion of normalized boundary trace introduced in Marcus and Nguyen (Ann Inst H. Poincaré (C) Non Linear Anal 34:69–88, 2015) and employ it to investigate the linear equation (LE). Using these results we obtain properties of moderate solutions of (NE). Finally we determine a criterion for subcriticality of points on ∂ Ω relative to f and study b.v.p. for (NE). In particular we establish existence and stability results when the data is concentrated on the set of subcritical points. [ABSTRACT FROM AUTHOR]

In this work we study the asymptotic behavior of a class of quasilinear elliptic problems posed in a domain perforated by ε -periodic holes of size ε . The quasilinear equations present a nonlinear singular lower order term f ζ ( u ε ) , where u ε is the solution of the problem at ε -level, ζ is a continuous function singular in zero and f a function whose summability depends on the growth of ζ near its singularity. We prescribe a nonlinear Robin condition on the boundary of the holes contained in Ω and a homogeneous Dirichlet condition on the exterior boundary. The particular case of a Neumann boundary condition on the holes is already new. The main tool in the homogenization process consists in proving a suitable convergence result, which shows that the gradient of u ε behaves like that of the solution of a suitable linear problem associated with a weak cluster point of the sequence { u ε } , as ε → 0 . This allows us not only to pass to the limit in the quasilinear term, but also to study the singular term near its singularity, via an accurate a priori estimate. We also get a corrector result for our problem. The main novelty of this work is that for the first time the unfolding method is used to treat a singular term as f ζ ( u ε ) . This plays an essential role in order to get an almost everywhere convergence of the solution u ε , needed in the study the asymptotic behavior of the problem. [ABSTRACT FROM AUTHOR]

In this paper we prove some existence, uniqueness and regularity results for a weak solution of a quasilinear elliptic nonlinear problem posed in a domain perforated by one or more holes. The nonlinear term is of the form $$f\zeta (u)$$ , where $$\zeta (s)$$ is singular at $$s=0$$ . On the boundary of the holes we impose a nonlinear Robin condition, while on the exterior boundary we prescribe a homogeneous Dirichlet condition. The difficulty here arises in dealing simultaneously with the quasilinear matrix field, the singular datum and the nonlinear Robin condition. To show the existence of a solution we approximate the problem with a sequence of nonsingular problems for which the existence of a solution is proved via the Schauder fixed-point theorem. The main tool when passing to the limit in the approximate problem is to split the integral of the singular term into the sum of two integrals, one on the set where the solution is very close to the singularity and one where it is far from it. To obtain the uniqueness of the solution we need to require additional assumptions on the quasilinear term and a monotonicity property for the singular one. Under suitable stronger hypotheses on the data, we also prove the boundedness of the solution. [ABSTRACT FROM AUTHOR]

In this article, we combine the existing regularity theory, perturbation method and the lower and upper solutions method to study the existence and asymptotic behaviour of positive solution to a boundary value problem for the p-Laplacian operator. More exactly, we study the existence and asymptotic behaviour of the positive solution to a quasi-linear elliptic problem of the form-Δpu=λ a(x)g(u) in D'(Ω), u>0 in Ω, lim x→∂ Ωu(x)=0. Under some conditions on a and g, we show that there is a non-negative number Λ0 such that for all λ∈(0, Λ0], the problem has a solution uλ in the sense of distribution, which is bounded from above by some positive numbers μ(λ). Such estimates and the asymptotic behaviour are important in computer programs when we know an algorithm for determining the solution. [ABSTRACT FROM AUTHOR]