Singular Quasilinear Equations Research Articles

Solutions for a class of singular quasilinear equations involving critical growth in R2$\mathbb {R}^2$

AbstractUsing a variational approach, we study the existence of solutions for the following class of quasilinear Schrödinger equations: where , , is a positive potential bounded away from zero and can be “large” at infinity, the nonlinearity is allowed to satisfy the exponential critical growth with respect to the Trudinger–Moser inequality. Precisely, behaves like as for some . This model of equation has been proposed in the theory of superfluid films in plasma physics. As for as we know, this the first work involving this class of operators and singular nonlinearities with exponential critical growth. Moreover, we are able to deal with exponents .

Sobolev and Hölder regularity results for some singular nonhomogeneous quasilinear problems

This article deals with the study of the following singular quasilinear equation: $$\begin{aligned} (P) \left\{ \ -\Delta _{p}u -\Delta _{q}u = f(x) u^{-\delta },\; u>0 \text { in }\; \Omega ; \; u=0 \text { on } \partial \Omega , \right. \end{aligned}$$where \(\Omega \) is a bounded domain in \({\mathbb {R}}^N\) with \(C^2\) boundary \(\partial \Omega \), \(1< q< p<\infty \), \(\delta >0\) and \(f\in L^\infty _{loc}(\Omega )\) is a non-negative function which behaves like \(\text {dist}(x,\partial \Omega )^{-\beta },\) \(\beta \ge 0\) near the boundary of \(\Omega \). We prove the existence of a weak solution in \(W^{1,p}_{loc}(\Omega )\) and its behaviour near the boundary for \(\beta <p\). Consequently, we obtain optimal Sobolev regularity of weak solutions. By establishing the comparison principle, we prove the uniqueness of weak solution for the case \(\beta <2-\frac{1}{p}\). Subsequently, for the case \(\beta \ge p\), we prove the non-existence result. Moreover, we prove Hölder regularity of the gradient of weak solution to a more general class of quasilinear equations involving singular nonlinearity as well as lower order terms (see (1.6)). This result is completely new and of independent interest. In addition to this, we prove Hölder regularity of minimal weak solutions of (P) for the case \(\beta +\delta \ge 1\) that has not been fully answered in former contributions even for p-Laplace operators.

Singular elliptic equations with quadratic gradient term

We establish the existence of a non-trivial weak solution to the following singular quasilinear equation with Hardy potential and singular quadratic gradient term:{−Δu−μu|x|2=|∇u|2u+f(x,u)inΩ,u>0inΩ,u=0on∂Ω, where Ω⊂RN is a smooth bounded domain, μ>0,0∈Ω,N≥3. We show that there exists a solution u∈H01(Ω) to the above problem. The notable characteristic of this problem is that it includes quadratic gradient nonlinearity and strong singularity.

Pointwise gradient bounds for a class of very singular quasilinear elliptic equations

A pointwise gradient bound for weak solutions to Dirichlet problem for quasilinear elliptic equations \begin{document}$ -\mathrm{div}(\mathbb{A}(x,\nabla u)) = \mu $\end{document} is established via Wolff type potentials. It is worthwhile to note that the model case of \begin{document}$ \mathbb{A} $\end{document} here is the non-degenerate \begin{document}$ p $\end{document} -Laplacian operator. The central objective is to extend the pointwise regularity results in [Q.-H. Nguyen, N. C. Phuc, Pointwise gradient estimates for a class of singular quasilinear equations with measure data, J. Funct. Anal. 278(5) (2020), 108391] to the very singular case \begin{document}$ 1 , where the data \begin{document}$ \mu $\end{document} on right-hand side is assumed belonging to some classes that close to \begin{document}$ L^1 $\end{document} . Moreover, a global pointwise estimate for gradient of weak solutions to such problem is also obtained under the additional assumption that \begin{document}$ \Omega $\end{document} is sufficiently flat in the Reifenberg sense.

Pointwise gradient estimates for a class of singular quasilinear equations with measure data

Local and global pointwise gradient estimates are obtained for solutions to the quasilinear elliptic equation with measure data −div(A(x,∇u))=μ in a bounded and possibly nonsmooth domain Ω in Rn. Here div(A(x,∇u)) is modeled after the p-Laplacian. Our results extend earlier known results to the singular case in which 3n−22n−1<p≤2−1n.

On the Solvability of a Boundary-Value Problem for a Second-Order Singular Quasilinear Equation

We obtain solvability conditions for a two-point boundary-value problem for a second-order quasilinear equation. The equation is singular with respect to the independent variable. The result is based on the properties of the Green operator of the corresponding linear problem. In particular, we prove its boundedness and obtain an upper estimate of its norm. Conditions of existence of a solution of the original problem are obtained from the solvability condition of an auxiliary operator equation.

On a nonhomogeneous and singular quasilinear equation involving critical growth in [formula omitted

This paper establishes sufficient conditions for the existence and multiplicity of solutions for nonhomogeneous and singular quasilinear equations of the form −Δu+V(x)u−Δ(u2)u=g(x,u)|x|a+h(x)inR2,where a∈[0,2), V(x) is a continuous positive potential bounded away from zero and which can be “large” at infinity, the nonlinearity g(x,s) is allowed to enjoy the critical exponential growth with respect to the Trudinger–Moser inequality and the nonhomogeneous term h belongs to Lq(R2) for some q∈(1,2]. By combining variational arguments in a nonstandard Orlicz space context with a singular version of the Trudinger–Moser inequality, we obtain the existence of two distinct solutions when ‖h‖q is sufficiently small. Schrödinger equations of this type have been studied as models of several physical phenomena.

Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation

We investigate the following quasilinear and singular problem, { − pu = λ uδ + uq in ; u|∂ = 0, u > 0 in , (P) where is an open bounded domain with smooth boundary, 1 0, and 0 N . We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in W 1,p 0 ( ). While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle and a regularity result for solutions to problem (P) in C1,β ( ) with some β ∈ (0, 1). Furthermore, we show that δ < 1 is a reasonable sufficient (and likely optimal) condition to obtain solutions of problem (P) in C1( ). Mathematics Subject Classification (2000): 35J65 (primary); 35J20, 35J70 (secondary).

Singular quasilinear equations with quadratic growth in the gradient without sign condition

Given a bounded domain Ω in R N , and a function a ∈ L q ( Ω ) with q > N / 2 , we study the existence of a positive solution for the quasilinear problem − Δ w + g ( x , w ) | ∇ w | 2 = a ( x ) , x ∈ Ω , w ∈ H 0 1 ( Ω ) , where g ( x , s ) is a Carathéodory function on Ω × ( 0 , + ∞ ) which may have a singularity at s = 0 and may change of sign.

Derivation of the Zakharov Equations

This article continues the study, initiated in [27, 7], of the validity of the Zakharov model which describes Langmuir turbulence. We give an existence theorem for a class of singular quasilinear equations. This theorem is valid for prepared initial data. We apply this result to the Euler–Maxwell equations which describes laser-plasma interactions, to obtain, in a high-frequency limit, an asymptotic estimate that describes solutions of the Euler–Maxwell equations in terms of WKB approximate solutions, the leading terms of which are solutions of the Zakharov equations. Due to the transparency properties of the Euler–Maxwell equations evidenced in [27], this study is carried out in a supercritical (highly nonlinear) regime. In such a regime, resonances between plasma waves, electromagnetric waves and acoustic waves could create instabilities in small time. The key of this work is the control of these resonances. The proof involves the techniques of geometric optics of JOLY, MÉTIVIER and RAUCH [12, 13]; recent results by LANNES on norms of pseudodifferential operators [14]; and a semiclassical paradifferential calculus.

Derivation of the Zakharov equations

This article continues the study, initiated in [27, 7], of the validity of the Zakharov model which describes Langmuir turbulence. We give an existence theorem for a class of singular quasilinear equations. This theorem is valid for prepared initial data. We apply this result to the Euler–Maxwell equations which describes laser-plasma interactions, to obtain, in a high-frequency limit, an asymptotic estimate that describes solutions of the Euler–Maxwell equations in terms of WKB approximate solutions, the leading terms of which are solutions of the Zakharov equations. Due to the transparency properties of the Euler–Maxwell equations evidenced in [27], this study is carried out in a supercritical (highly nonlinear) regime. In such a regime, resonances between plasma waves, electromagnetric waves and acoustic waves could create instabilities in small time. The key of this work is the control of these resonances. The proof involves the techniques of geometric optics of JOLY, METIVIER and RAUCH [12, 13]; recent results by LANNES on norms of pseudodifferential operators [14]; and a semiclassical paradifferential calculus.

Quasilinear singular equations: a variational approach for nondifferentiable functionals

Our main concern is on existence and nonexistence of solutions of the following class of quasilinear problems: −(r α|u′| βu′)′=r γ(u β+1−g(u)) in (0,R), u>0 in [0,R), u(R)=u′(0)=0, where α, β, γ, R are given real numbers with R>0 and g : (0,∞)→(0,∞) is a C 1 function. This represents the radial form of the Dirichlet problem on the ball of radius R in R N for the operators p-Laplacian (1< p<∞) and k-Hessian (1⩽ k⩽ N), according to the values of α, β, γ. A main feature is that g is possibly singular at u=0. A solution will be shown to exist if R belongs to a suitable range. Our approach exploits variational arguments for an associated nondifferentiable functional.

On multiple solutions of a singular quasilinear equation on unbounded domain

Variational methods are used to prove existence and multiplicity of solutions for the following quasilinear partial differential equation −Δ pu=α|x| −s|u| p ∗(s)−2 u+βk(x)|u| r−2u in Ω, where 0⩽ s⩽ p, Ω⊂ R N may be unbounded. In particular, Ω= R N is permitted.