Singular Quasilinear Elliptic Problems Research Articles

Calderón–Zygmund type estimates for singular quasilinear elliptic obstacle problems with measure data

Calderón–Zygmund type estimates for singular quasilinear elliptic obstacle problems with measure data

Existence of two solutions for singular Φ-Laplacian problems

AbstractExistence of two solutions to a parametric singular quasi-linear elliptic problem is proved. The equation is driven by theΦ\Phi-Laplacian operator, and the reaction term can be nonmonotone. The main tools employed are the local minimum theorem and the Mountain pass theorem, together with the truncation technique. GlobalC1,τ{C}^{1,\tau }regularity of solutions is also investigated, chiefly viaa prioriestimates and perturbation techniques.

Existence of solutions for a class of singular quasilinear elliptic equations

ABSTRACT In this paper, we study the existence of positive solutions to a class of singular quasilinear elliptic problems where , the potential and . By Nehari manifold approach and the technique of splitting a minimizing sequence, we prove the existence of positive solutions for the above problem.

Singular quasilinear elliptic problems with changing sign datum: existence and homogenization

We study singular quasilinear elliptic equations whose model is $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\,\Delta u = \lambda u + \mu (x) \frac{|\nabla u|^q}{|u|^{q-1}}+f(x) &{} \quad \hbox {in}\,\, \Omega , \\ u=0 &{}\quad \hbox {on} \,\, \partial \Omega , \end{array}\right. } \end{aligned}$$ where \(\Omega \) is a bounded smooth domain of \({\mathbb {R}}^N\) (\(N\ge 3\)), \(\lambda \in {\mathbb {R}}\), \(1 \frac{N}{2}\), may change sign. We prove existence of solution and we deal with the homogenization problem posed in a sequence of domains \(\Omega ^\varepsilon \) obtained by removing many small holes from a fixed domain \(\Omega \).

Weighted Calderón-Zygmund Estimates for Weak Solutions of Quasi-Linear Degenerate Elliptic Equations

This paper studies regularity estimates in Sobolev spaces for weak solutions of a class of degenerate and singular quasi-linear elliptic problems of the form div[A(x,u,∇u)] = div[F] with non-homogeneous Dirichlet boundary conditions over bounded non-smooth domains. The coefficients A could be be singular, degenerate or both in x in the sense that they behave like some weight function μ, which is in the A2 class of Muckenhoupt weights. Global and interior weighted W1,p(Ω,ω)-regularity estimates are established for weak solutions of these equations with some other weight function ω. The results obtained are even new for the case μ = 1 because of the dependence on the solution u of A. In case of linear equations, our W1,p-regularity estimates can be viewed as the Sobolev’s counterpart of the Holder’s regularity estimates established by B. Fabes, C. E. Kenig, and R. P. Serapioni.

Three Solutions for a Singular Quasilinear Elliptic Problem

AbstractIn the present paper we deal with a quasilinear problem involving a singular term. By combining truncation techniques with variational methods, we prove the existence of three weak solutions. As far as we know, this is the first contribution in this direction in the high-dimensional case.

Homogenization of singular quasilinear elliptic problems with natural growth in a domain with many small holes

In this paper we consider the homogenization problem for quasilinear elliptic equations with singularities in the gradient, whose model is the following \begin{document}$\begin{equation*}\begin{cases}\displaystyle -Δ u^\varepsilon + \frac{|\nabla u^\varepsilon|^2}{{(u^\varepsilon})^θ} = f (x)& \mbox{in} \; Ω^\varepsilon,\\u^\varepsilon = 0&\mbox{on} \; \partial Ω^\varepsilon,\\\end{cases}\end{equation*}$ \end{document} where Ω is an open bounded set of $\mathbb{R}^N$, $θ ∈ (0,1)$ and $f$ is positive function that belongs to a certain Lebesgue's space. The homogenization of these equations is posed in a sequence of domains $Ω^\varepsilon$ obtained by removing many small holes from a fixed domain Ω. We also give a corrector result.

Bifurcation analysis of a singular elliptic problem modelling the equilibrium of anisotropic continuous media

In this work we obtain an existence result for a class of singular quasilinear elliptic Dirichlet problems on a smooth bounded domain containing the origin. By using a Caffarelli-Kohn-Nirenberg type inequality, a critical point result for differentiable functionals is exploited, in order to prove the existence of a precise open interval of positive eigenvalues for which the treated problem admits at least one nontrivial weak solution. In the case of terms with a sublinear growth near the origin, we deduce the existence of solutions for small positive values of the parameter. Moreover, the corresponding solutions have smaller and smaller energies as the parameter goes to zero.

Singular quasilinear elliptic problems on unbounded domains

We prove the existence of a solution between an ordered pair of sub and supersolutions for singular quasilinear elliptic problems on unbounded domains. Further, we use this result to establish the existence of a positive solution to the problem {−Δpu=λK(x)f(u)in B1c,u=0on ∂B1,u(x)→0as |x|→∞, where B1c={x∈Rn||x|>1}, Δpu=div(|∇u|p−2∇u),1<p<n, λ is a positive parameter, K belongs to a class of functions which satisfy certain decay assumptions and f belongs to a class of (p−1)-subhomogeneous functions which may be singular at the origin, namely lims→0+f(s)=−∞. Our methods can be also applied to establish a similar existence result when the domain is entire Rn.

Existence and multiplicity results for singular quasilinear elliptic equation on unbounded domain

In this paper, we are interested in the existence and multiplicity results of solutions for the singular quasilinear elliptic problem with concave–convex nonlinearities (0.1) where is an unbounded exterior domain with smooth boundary ∂Ω, 1 < p < N,0 ≤ a < (N − p) ∕ p,λ > 0,1 < s < p < r < q = pN ∕ (N − pd),d = a + 1 − b,a ≤ b < a + 1. By the variational methods, we prove that problem 0.1 admits a sequence of solutions uk under the appropriate assumptions on the weight functions H(x) and H(x). For the critical case, s = q,h(x) = | x | − bq, we obtain that problem 0.1 has at least a nonnegative solution with p < r < q and a sequence of solutions uk with 1 < r < p < q and J(uk) → 0 as k → ∞ , where J(u) is the energy functional associated to problem 0.1. Copyright © 2013 John Wiley & Sons, Ltd.

Existence of nontrivial solutions for singular quasilinear elliptic equations on [formula omitted

In this paper, we study the existence of weak solutions for the singular quasilinear elliptic problem (0.1){−div(|x|−ap|∇u|p−2∇u)+V(x)|u|p−2u=h(x)|u|s−2u±H(x)|u|r−2u,x∈RN,u(x)→0,as|x|→∞, where 1<p<N,−∞<a<(N−p)/p,1<s,r<p∗=pN/(N−p). Using an approach which strongly relies on the Nehari manifold and the fibering method in combination with related variational methods, we prove that problem (0.1) admits multiple solutions under appropriate assumptions on the weight functions V(x), h(x), and H(x).

Singular quasilinear elliptic problems with indefinite weights and critical potential

The Dirichlet problem for a quasilinear sub-critical inhomogeneous elliptic equation with critical potential and singular coefficients, which has indefinite weights in R N , is studied in this paper. We discuss the corresponding eigenvalue problems by the variational techniques and Picone’s identity, and obtain the existence of non-trivial solutions for the inhomogeneous Dirichlet problem by using Hardy inequality, Mountain Pass Lemma in conjunction with the property of eigenvalues.

Existence of multiple solutions for a singular quasilinear elliptic equation in [formula omitted

In this paper, we are concerned with the existence of multiple positive solutions for the singular quasilinear elliptic problem in R N { − div ( | x | − a p | ∇ u | p − 2 ∇ u ) + g ( x ) | u | p − 2 u = h ( x ) | u | m − 2 u + λ H ( x ) | u | n − 2 u , x ∈ R N u ( x ) > 0 , x ∈ R N . where λ > 0 is a real parameter and 1 < p < N ( N ≥ 3 ) , 1 < n < p < m < p ∗ , 0 ≤ a < ( N − p ) / p , p ∗ = N p / ( N − p d ) , a ≤ b < a + 1 , d = a + 1 − b > 0 . The weight function g ( x ) is a bounded nonnegative function with ‖ g ‖ ∞ > 0 and h ( x ) , H ( x ) are continuous functions which change sign in R N . We prove that there admits at least two positive solutions by using the Nehari manifold and the fibrering maps associated with the Euler functional for this problem.

On a class of critical singular quasilinear elliptic problem with indefinite weights

In this paper, the eigenvalue problem for a class of quasilinear elliptic equations involving critical potential and indefinite weights is investigated. We obtain the simplicity, strict monotonicity and isolation of the first eigenvalue λ 1 . Furthermore, because of the isolation of λ 1 , we prove the existence of the second eigenvalue λ 2 . Then, using the Trudinger–Moser inequality, we obtain the existence of a nontrivial weak solution for a class of quasilinear elliptic equations involving critical singularity and indefinite weights in the case of 0 < λ < λ 1 by the Mountain Pass Lemma, and in the case of λ 1 ≤ λ < λ 2 by the Linking Argument Theorem.

Existence of solutions for a class of singular quasilinear elliptic resonance problems

The paper concerns a resonance problem for a class of singular quasilinear elliptic equations in weighted Sobolev spaces. The equation set studied is one of the most useful sets of Navier–Stokes equations; these describe the motion of viscous fluid substances such as liquids, gases and so on. By using Galerkin-type techniques, the Brouwer fixed point theorem, and a new weighted compact Sobolev-type embedding theorem established by Shapiro, we show the existence of a nontrivial solution.

Existence and regularity of nonnegative solution of a singular quasi-linear anisotropic elliptic boundary value problem with gradient terms

In this paper, we consider the singular quasi-linear anisotropic elliptic boundary value problem (P) { f 1 ( u ) u x x + u y y + g ( u ) | ∇ u | q + f ( u ) = 0 , ( x , y ) ∈ Ω , u | ∂ Ω = 0 , where Ω is a smooth, bounded domain in R 2 ; 0 < q < 2 ; f 1 ( 0 ) = 0 , f 1 ( t ) > 0 ( t ≠ 0 ) , f 1 is a smooth function in ( − ∞ , + ∞ ) and is a non-decreasing function in ( 0 , + ∞ ) ; g ( t ) ≥ 0 , g is a smooth function in ( − ∞ , 0 ) ∪ ( 0 , + ∞ ) and is a non-increasing function in ( 0 , + ∞ ) ; f ( t ) > 0 , f is a smooth function in ( − ∞ , 0 ) ∪ ( 0 , + ∞ ) and is a strictly decreasing function in ( 0 , + ∞ ) . Clearly, this is a boundary degenerate elliptic problem if f 1 ( 0 ) = 0 . We show that the solution of the Dirichlet boundary value problem (P) is smooth in the interior and continuous or Lipschitz continuous up to the degenerate boundary and give the conditions for which gradients of solutions are bounded or unbounded. We believe that these results on regularity of the solution should be very useful.

Exact number of solutions of stationary reaction–diffusion equations

We study both existence and the exact number of positive solutions of the problem ( P λ ) λ | u ′ | p - 2 u ′ ′ + f ( u ) = 0 in ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 , where λ is a positive parameter, p > 1 , the nonlinearity f is positive in (0,1), and f ( 0 ) = f ( 1 ) = 0 . Assuming that f satisfies the condition lim s → 1 - f ( s ) ( 1 - s ) θ = ω > 0 where θ ∈ ( 0 , p - 1 ) , we study its behavior near zero, and we obtain existence and exactness results for positive solutions. We prove the results using the shooting method. We show that there always exist solutions with a flat core for λ sufficiently small. As an application, we prove the existence of a non-negative solution for a class of singular quasilinear elliptic problems in a bounded domain in R N having a flat core in a ball.

Quasilinear elliptic problems with critical exponents and Hardy terms in ℝN

AbstractWe deal with a singular quasilinear elliptic problem, which involves critical Hardy-Sobolev exponents and multiple Hardy terms. Using variational methods and analytic techniques, the existence of ground state solutions to the problem is obtained.

QUASILINEAR ELLIPTIC EQUATIONS OF THE HENON-TYPE: EXISTENCE OF NON-RADIAL SOLUTIONS

In this work, we prove results on existence and multiplicity of non-radial solutions for a class of singular quasilinear elliptic problems of the form [Formula: see text] where B = {x ∈ ℝN: |x| &lt; 1} (N ≥ 3) is a unit open ball centered at the origin, -∞ &lt; a &lt; (N - p)/p, β &gt; 0 and [Formula: see text].

The existence and non-existence of positive solutions to a singular quasilinear elliptic problem in [formula omitted

This paper deals with the existence and nonexistence of entire positive solutions of the quasilinear elliptic equation − Δ p u = ρ a ( x ) g ( u ) + λ b ( x ) f ( u ) in R N , 1 < p < N , N ≥ 3 , with u ( x ) → 0 as | x | → ∞ . Here either g or f (or both of them) are singular at 0 in the sense that g ( t ) , f ( t ) → ∞ as t → 0 . By using a perturbation method which eliminates the singularity on a ball with radius R and then let R tends to infinity, with help of the bounded super-solution of the original problem, we obtain the existence of a weak solution of the problem. The main results of this paper improve the corresponding results of [Dragos-Patru Covei, the existence and asymptotic behavior of a positive solution to a quasilinear elliptic problem in R N , Nonlinear Anal. 69 (2008) 2615–2622; J.V.Goncalves, C.A.Santos, Singular elliptic problems: Existence, non-existence and boundary behavior, Nonlinear Anal. 66 (2007) 2078–2090].