Quasilinear Dirichlet Problems Research Articles

Existence and nonexistence of positive radial solutions of a quasilinear Dirichlet problem with diffusion

In this paper existence and nonexistence results of positive radial solutions of a Dirichlet m-Laplacian problem with different weights and a diffusion term inside the divergence of the form (a(|x|)+g(u))−γ, with γ>0 and a, g positive functions satisfying natural growth conditions, are proved. Precisely, we obtain a new critical exponent mα,β,γ⁎, which extends the one relative to case with no diffusion and it divides existence from nonexistence of positive radial solutions. The results are obtained via several tools such as a suitable modification of the celebrated blow up technique, Liouville type theorems, a fixed point theorem and a Pohožaev-Pucci-Serrin type identity.

Ordering properties of positive solutions for a class of $ \varphi $-Laplacian quasilinear Dirichlet problems

<abstract><p>We study ordering properties of positive solutions $ u $ for the one-dimensional $ \varphi $-Laplacian quasilinear Dirichlet problem</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left \{\begin{array}{l} -\left (\varphi (u^{ \prime })\right )^{ \prime } = \lambda f (u) , \;\; -L <x <L, \\ u ( -L) = u (L) = 0, \end{array}\right . \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ \lambda, L > 0 $ are two parameters. Assume that $ \varphi \in C (-\kappa, \kappa) \cap C^{2} ((-\kappa, 0) \cup (0, \kappa)) $ is odd for some positive $ \kappa \leq \infty, $ and $ \varphi ^{ \prime } (t) > 0 $ for all $ t \in (-\kappa, 0) \cup (0, \kappa) $ and $ f \in C[0, \eta) $, $ f (0) \geq 0 $, $ f (u) > 0 $ on $ (0, \eta) $ for some positive $ \eta \leq \infty $, where either $ \eta = \infty $, or $ \eta < \infty $ with $ \lim_{u \rightarrow \eta ^{ -}}f (u) = \infty $ or $ \lim_{u \rightarrow \eta ^{ -}}f (u) = 0 $. Some applications are given, including $ f (u) = u^{p} $ ($ p > 0 $)$, $ $ u^{p} +u^{q} $ ($ 0 < p < q < \infty $), $ \frac{1}{(1 -u)^{p}} $ $ (p > 0), $ $ \exp (u), \; \exp \left({\frac{{au}}{{a + u}}} \right) $ ($ a > 0 $)$, $ and $ \frac{1}{(1 -u)^{2}} -\frac{\varepsilon ^{2}}{(1 -u)^{4}} $ ($ \varepsilon \in (0, 1) $).</p></abstract>

Quasilinear Dirichlet Problems with Degenerated p-Laplacian and Convection Term

The paper develops a sub-supersolution approach for quasilinear elliptic equations driven by degenerated p-Laplacian and containing a convection term. The presence of the degenerated operator forces a substantial change to the functional setting of previous works. The existence and location of solutions through a sub-supersolution is established. The abstract result is applied to find nontrivial, nonnegative and bounded solutions.

Quasilinear Dirichlet problems with competing operators and convection

Abstract The paper deals with a quasilinear Dirichlet problem involving a competing (p,q)-Laplacian and a convection term. Due to the lack of ellipticity, monotonicity and variational structure, the known methods to find a weak solution are not applicable. We develop an approximation procedure permitting to establish the existence of solutions in a generalized sense. If in place of competing (p,q)-Laplacian we consider the usual (p,q)-Laplacian, our results ensure the existence of weak solutions.

Uniqueness for a class of singular quasilinear Dirichlet problem

We prove uniqueness of positive solutions for the problem −Δpu=λf(u)uαinΩ,u=0 on∂Ω,where p>1+α,α∈(0,1),Ω is bounded domain in Rn with smooth boundary ∂Ω,f:[0,∞)→(0,∞) is nondecreasing with f(z)∕zα decreasing for z large, and λ is a large parameter.

Existence of solutions for quasilinear Dirichlet problems with gradient terms

In this paper we prove an existence theorem for positive solutions of a nonlinear Dirichlet problem involving the p-Laplacian operator on a smooth bounded domain when a nonlinearity depending on the gradient is considered. Our main theorem extends a previous result by Ruiz in [ 19 ], in which a slight modification of the celebrated blowup technique due to Gidas and Spruck, [ 11 ] and [ 12 ] is introduced.

A Class of Quasilinear Dirichlet Problems with Unbounded Coefficients and Singular Quadratic Lower Order Terms

We study existence and regularity of positive solutions of problems like $$\left\{\begin{array}{cl} - {\rm div}([a(x) + u^q] \nabla u) + b(x)\frac{1}{u^\theta}|\nabla u|^2 = f & {\rm in} \, \Omega,\\ u > 0 &{\rm in} \, \Omega, \\ u = 0 & {\rm on} \, \partial\Omega ,\end{array}\right.$$ depending on the values of q > 0, 0 < θ < 1, and on the summability of the datum f ≥ 0 in Lebesgue spaces.

Asymptotic Behavior of Solutions to Singular Quasilinear Dirichlet Problem with a Convection Term

In this paper, we study the boundary behavior of solution to the singular Dirichlet problem 8 : div(jruj m 2 ru) = b(x)g(u) + jru(x)j q(m 1) ; x2 ; u > 0; x2 ; uj@ = 0; where is a bounded domain with smooth boundary in R N , 2 R;m > 1; 0 < q m=(m 1), lims!0+ g(s) = +1, and b 2 C () , which is non-negative on and may be vanishing on the boundary, mainly, we investigate the exact asymptotic behavior of solution to the above problem.

A quasilinear Dirichlet problem with quadratic growth respect to the gradient and [formula omitted] data

For an open, bounded set Ω⊂RN, measurable bounded functions a(x),b(x) which are strictly positive and p,q>0, we prove the existence of a weak solution of the quasilinear b.v.p {−div[(a(x)+|u|q)∇u(x)]+b(x)u|u|p−1|∇u|2=f(x),x∈Ω;u(x)=0,x∈∂Ω. The datum f is assumed to be in L1(Ω) and does not satisfy any sign assumption.

S. Bernstein’s idea for bounding the gradient of solutions to the quasi-linear Dirichlet problem

S. Bernstein’s idea is outlined to estimate \({\| u_{x} \|_{0,0,\Omega}}\) of a solution u(x) to the quasi-linear elliptic differential equation, especially in the case of n > 2. Up to about 1956, investigations of nonlinear problems have been dealing with the case of n = 2. A large number of methods were developed, but none of them were applicable for n > 2. The maximum-minimum principle had been a powerful tool to find bounds, however in the case of n > 2 this tool is not available for the gradient of the solution. In 1956, Cordes [Math. Ann. 130 (1956), 278–312] gave estimates for the case n > 2. Later, Ladyzhenskaya and Ural’tseva [Linear and Quasilinear Elliptic Equations. Academic Press, New York, 1968] established estimates in the Sobolev space \({W^{2}_{2} (\Omega)}\) . In their line of reasoning they used the idea of S. Bernstein and investigated the transformed function υ(x) with u(x) = ϕ(υ(x)). In this paper, we give a more simpler proof of those results in the classical Banach space C2,α(Ω).

Three non-zero solutions for a nonlinear eigenvalue problem

In the present paper we prove a novel multiplicity result for a model quasilinear Dirichlet problem (Pλ) depending on a positive parameter λ. By a variational method, we prove that for every λ>1 problem (Pλ) has at least two non-zero solutions, while there exists λˆ>1 such that problem (Pλˆ) has at least three non-zero solutions.

A result of multiplicity of solutions for a class of quasilinear equations

AbstractWe establish the multiplicity of positive weak solutions for the quasilinear Dirichlet problem−Lpu+ |u|p−2u=h(u)in Ωλ,u= 0 on ∂Ωλ, where Ωλ= λΩ, Ω is a bounded domain in ℝN, λ is a positive parameter,Lpu≐ Δpu+ Δp(u2)uand the nonlinear termh(u) has subcritical growth. We use minimax methods together with the Lyusternik–Schnirelmann category theory to get multiplicity of positive solutions.

MULTIPLE SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS IN UNBOUNDED CYLINDER DOMAINS

In this paper, we show that if $Q(x)$ satisfies some suitable conditions, then the quasilinear elliptic Dirichlet problem $-\Delta_p u+|u|^{p-2} u=Q(x)|u|^{q-2}u$ in an unbounded cylinder domain $\Omega$ has at least two solutions in which one is a positive ground state solution and the other is a nodal solution.

Transition-layer solutions of quasilinear elliptic boundary blow-up problems and dirichlet problems

We study profiles of positive solutions for quasilinear elliptic boundary blow-up problems and Dirichlet problems with the same equation: $$ - \varepsilon \Delta _p u = f(x,u)in\Omega ,$$ where 1 0 is a small parameter, $$f(x,u) = \left\{ \begin{gathered} \left| u \right|^{p - 2} u\left| {u - a(x)} \right|^{p - 2} (u - a(x))\left| {1 - u} \right|^{\omega - 1} (1 - u),ifp \geqslant 2 \hfill \\ \left| u \right|^{(2 - p)/(p - 1)} u(u - a(x))\left| {1 - u} \right|^{\omega - 1} (1 - u),if1 < p < 2, \hfill \\ \end{gathered} \right. $$ where ω > 0, a(x) is a continuous function satisfying 0 < a(x) < 1 for x ∈ \(\bar \Omega \), Ω is a bounded smooth domain in ℝN. We will see that the profile of a minimal positive boundary blow-up solution of the equation shares some similarities to the profile of a positive minimizer solution of the equation with homogeneous Dirichlet boundary condition.

Quasilinear equations with dependence on the gradient

We discuss the existence of positive solutions of the problem − ( q ( t ) φ ( u ′ ( t ) ) ) ′ = f ( t , u ( t ) , u ′ ( t ) ) for t ∈ ( 0 , 1 ) and u ( 0 ) = u ( 1 ) = 0 , where the nonlinearity f satisfies a superlinearity condition at 0 and a local superlinearity condition at + ∞ . This general quasilinear differential operator involves a weight q and a main differentiable part φ which is not necessarily a power. Due to the superlinearity of f and its dependence on the derivative, a condition of the Bernstein–Nagumo type is assumed, also involving the differential operator. Our main result is the proof of a priori bounds for the eventual solutions. The presence of the derivative in the right-hand side of the equation requires a priori bounds not only on the solutions themselves, but also on their derivatives, which brings additional difficulties. As an application, we consider a quasilinear Dirichlet problem in an annulus { − div ( A ( | ∇ u | ) ∇ u ) = f ( | x | , u , | ∇ u | ) in r 1 < | x | < r 2 , u ( x ) = 0 on | x | = R 1 and | x | = R 2 .

Quasilinear Dirichlet problem in a periodically perforated domain

Consider the following quasilinear Dirichlet problem � −�pu = in = n T u = 0 on @ (P) where 1 < p ≤ 2, (f) a sequence of functions in L p ' () such that → f in L p ' () where pis the conjugate of p, is a bounded domain in R N (N ≥ 2) and T the union of inclusions contained in which are - periodically distributed. In this paper we give the limit problem and errors estimates for the problem (P).

Structure of nontrivial nonnegative solutions of singularly perturbed quasilinear Dirichlet problems

AbstractWe study structure of nontrivial nonnegative solutions for a class of singularly perturbed quasilinear Dirichlet problems. It is shown that there are infinitely many least‐energy solutions and they are spike‐layer solutions. Moreover, the measure of each spike‐layer is estimated as the parameter ε tends to 0. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Divergent Type Quasilinear Dirichlet Problem with Singularities

A quasilinear equation of divergent type with singular data and singular coefficients is approximated by a net of equations of the same type with enough regular coefficients and data. Solutions of the net of equations are obtained by the classical methods. Known a priory estimates are improved so that a net of solutions can be considered as a solution in an appropriate algebra of generalized functions.

Stability of Unique Solvability in an Ill-Posed Dirichlet Problem

Suppose that Ω ⊂ ∝n is a compact domain with Lipschitz boundary ∂Ω which is the closure of its interior Ω0. Consider functions ϕi, τi: Ω→ ∝ belonging to the space Lq(Ω) for q ∈ (1, +∞] and a locally Holder mapping F: Ω × ∝ → ∝ such that F(·, 0) ≡ 0 on Ω. Consider two quasilinear inhomogeneous Dirichlet problems $$\left\{ {_{u = \tau _i \quad \quad \quad \quad \quad \quad \quad \;\;\,\operatorname{on} \;\partial \Omega ,}^{\Delta u_i = F(x,\;u_i ) + \varphi _i (x)\quad \operatorname{on} \;\Omega _0 ,} } \right.\quad \quad i = 1,\;2.$$ In this paper, we study the following problem: Under certain conditions on the function F generally not ensuring either the uniqueness or the existence of solutions in these problems, estimate the deviation of the solutions ui (assuming that they exist) from each other in the uniform metric, using additional information about the solutions ui. Here we assume that the solutions are continuous, although their continuity is a consequence of the constraints imposed on F, τi, ϕi. For the additional information on the solutions ui, i = 1, 2 we take their values on the grid; in particular, we show that if their values are identical on some finite grid, then these functions coincide on Ω.

Solutions with boundary‐layers and spike‐layers to singularly perturbed quasilinear Dirichlet problems

AbstractThe structure of nontrivial nonnegative solutions to singularly perturbed quasilinear Dirichlet problems of the form –ϵΔpu = f(u) in Ω, u = 0 on ∂Ω, Ω ⊂ RN a bounded smooth domain, is studied as ϵ → 0+, for a class of nonlinearities f(u) satisfying f(0) = f(z1) = f(z2) = 0 with 0 &lt; z1 &lt; z2, f &lt; 0 in (0, z1), f &gt; 0 in (z1, z2) and $ \underline {\lim} _{u \to 0_{+}} $f(u)/up–1 = –∞. It is shown that there are many nontrivial nonnegative solutions with spike‐layers. Moreover, the measure of each spike‐layer is estimated as ϵ → 0+. These results are applied to the study of the structure of positive solutions of the same problems with f changing sign many times in (0,∞). Uniqueness of a solution with a boundary‐layer and many positive intermediate solutions with spike‐layers are obtained for ϵ sufficiently small. (© 2004 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)