We are concerned with a class of second order quasilinear elliptic equations driven by a nonhomogeneous differential operator introduced by C.A. Stuart and whose study is motivated by models in Nonlinear Optics. We establish sufficient conditions for the existence of at least one or two non-negative solutions. Our analysis considers the cases when the reaction has either a sublinear or a linear growth. In the sublinear case, we also prove a nonexistence property. The proofs combine energy estimates and variational methods. In particular, the monotonicity trick is applied in order to overcome the lack of a priori bounds on the Palais-Smale sequences.
In this paper, we prove existence of \emph{very weak solutions} to nonhomogeneous quasilinear parabolic equations beyond the duality pairing. The main ingredients are a priori esitmates in suitable weighted spaces combined with the compactness argument developed in \cite{bulicek2018well}. In order to obtain the a priori estimates, we make use of the full Calderón-Zygmund machinery developed in the past few years and combine it with some sharp bounds for the subclass of Muckenhoupt weights considered in this paper.
We consider the nonlinear eigenvalue problem $-{\rm div}(|\nabla u|^{p(x)-2}\nabla u)=λ|u|^{q(x)-2}u$ in $Ω$, $u=0$ on $\partialΩ$, where $Ω$ is a bounded open set in $\RR^N$ with smooth boundary and $p$, $q$ are continuous functions on $\barΩ$ such that $1<\inf\_Ωq< \inf\_Ωp<\sup\_Ωq$, $\sup\_Ωp<N$, and $q(x)<Np(x)/(N-p(x))$ for all $x\in\barΩ$. The main result of this paper establishes that any $λ>0$ sufficiently small is an eigenvalue of the above nonhomogeneous quasilinear problem. The proof relies on simple variational arguments based on Ekeland's variational principle.
It is established some existence and multiplicity of solution results for a quasilinear elliptic problem driven by $Φ$-Laplacian operator. One of these solutions is built as a ground state solution. In order to prove our main results we apply the Nehari method combined with the concentration compactness theorem in an Orlicz-Sobolev framework. One of the difficulties in dealing with this kind of operator is the lost of homogeneity properties.
The aim of this paper is analyzing existence, multiplicity, and regularity issues for the positive solutions of a Neumann boundary value problem of superlinear indefinite type related to the mean curvature operator with a sublinear potential at infinity.
We study the boundary value problem $-{\rm div}(\log(1+ |\nabla u|^q)|\nabla u|^{p-2}\nabla u)=f(u)$ in $Ω$, $u=0$ on $\partialΩ$, where $Ω$ is a bounded domain in $\RR^N$ with smooth boundary. We distinguish the cases where either $f(u)=-λ|u|^{p-2}u+|u|^{r-2}u$ or $f(u)=λ|u|^{p-2}u-|u|^{r-2}u$, with $p$, $q>1$, $p+q<\min\{N,r\}$, and $r<(Np-N+p)/(N-p)$. In the first case we show the existence of infinitely many weak solutions for any $λ>0$. In the second case we prove the existence of a nontrivial weak solution if $λ$ is sufficiently large. Our approach relies on adequate variational methods …
In this paper, we mainly investigate the critical points associated to solutions $u$ of a quasilinear elliptic equation with nonhomogeneous Dirichlet boundary conditions in a connected domain $Ω$ in $\mathbb{R}^2$. Based on the fine analysis about the distribution of connected components of a super-level set $\{x\in Ω: u(x)>t\}$ for any $\mathop {\min}_{\partialΩ}u(x)<t<\mathop {\max}_{\partialΩ}u(x)$, we obtain the geometric structure of interior critical points of $u$. Precisely, when $Ω$ is simply connected, we develop a new method to prove $Σ_{i = 1}^k {m_i}+1=N$, where $m_1,\cdots,m_k$ ar…
This article deals with the study of the following singular quasilinear equation: \begin{equation*} (P) \left\{ \ -Δ_{p}u -Δ_{q}u = f(x) u^{-δ},\; u>0 \text{ in }\; \Om; \; u=0 \text{ on } \pa\Om, \right. \end{equation*} where $\Om$ is a bounded domain in $\mathbb{R}^N$ with $C^2$ boundary $\pa\Om$, $1< q< p<\infty$, $\de>0$ and $f\in L^\infty_{loc}(\Om)$ is a non-negative function which behaves like $\textnormal{dist}(x,\pa\Om)^{-\ba},$ $\ba\ge 0$ near the boundary of $\Om$. We prove the existence of a weak solution in $W^{1,p}_{loc}(\Om)$ and its behaviour near the boundary for $\ba<p$. Cons…
We study a nonlinear parametric Neumann problem driven by a nonhomogeneous quasilinear elliptic differential operator $\operatorname{div}(a(x,\nabla u))$, a special case of which is the $p$-Laplacian. The reaction term is a nonlinearity function $f$ which exhibits $(p-1)$-subcritical growth. By using variational methods, we prove a multiplicity result on the existence of weak solutions for such problems. An explicit example of an application is also presented.
We study the nonlinear eigenvalue problem $-{\rm div}(a(|\nabla u|)\nabla u)=λ|u|^{q(x)-2}u$ in $Ω$, $u=0$ on $\partialΩ$, where $Ω$ is a bounded open set in $\RR^N$ with smooth boundary, $q$ is a continuous function, and $a$ is a nonhomogeneous potential. We establish sufficient conditions on $a$ and $q$ such that the above nonhomogeneous quasilinear problem has continuous families of eigenvalues. The proofs rely on elementary variational arguments. The abstract results of this paper are illustrated by the cases $a(t)=t^{p-2}\log (1+t^r)$ and $a(t)= t^{p-2} [\log (1+t)]^{-1}$.
We consider a nonlinear Robin problem driven by a nonhomogeneous differential operator, with reaction which exhibits the competition of two Carathéodory terms. One is parametric, $(p-1)$-sublinear with a partially concave nonlinearity near zero. The other is $(p-1)$-superlinear and has almost critical growth. Exploiting the special geometry of the problem, we prove a bifurcation-type result, describing the changes in the set of positive solutions as the parameter $λ>0$ varies.
We give a survey of nonlinear potential estimates and their applications obtained recently for positive solutions to sublinear problems of the type \[ u = \mathbf{G}(σu^q) + f \quad \textrm{in} \,\, Ω, \] where $0 < q < 1$, $σ\ge 0$ is a Radon measure in $Ω$, $ f \ge 0$ is a measurable function, and $\mathbf{G}$ is a linear integral operator with positive kernel $G$ on $Ω\timesΩ$. For quasi-metric (or quasi-metrically modifiable) kernels $G$, these bilateral pointwise estimates yield existence criteria and uniqueness of solutions $u \in L^q_{\rm loc} (Ω, σ)$. Applications are considered to sem…
We study global regularity for solutions of quasilinear elliptic equations of the form $÷\A(x,u,\nabla u) = ÷\F $ in rough domains $Ω$ in $\R^n$ with nonhomogeneous Dirichlet boundary condition. The vector field $\A$ is assumed to be continuous in $u$, and its growth in $\nabla u$ is like that of the $p$-Laplace operator. We establish global gradient estimates in weighted Morrey spaces for weak solutions $u$ to the equation under the Reifenberg flat condition for $Ω$, a small BMO condition in $x$ for $\A$, and an optimal condition for the Dirichlet boundary data.
We study the following boundary value problem (P)\ \ \ \ \ {-\mathrm{div}(a(|\nabla u|)\nabla u)=f(x,u),\ & in $Ω$, u=0, & on $\partialΩ$} with nonhomogeneous principal part. By assuming the nonlinearity $f(x, t)$ being subcritical growth, some abstract results of problem (P) are obtained: (1) Regularity; (2) Orlicz-Sobolev versus Hölder local minimizer; (3) Strong comparison principle. Applying these abstract results and critical point theory, we prove the existence of multiple solutions of problem (P) in an Orlicz-Sobolev space.
We consider a nonlinear boundary value problem driven by a nonhomogeneous differential operator. The problem exhibits competing nonlinearities with a superlinear (convex) contribution coming from the reaction term and a sublinear (concave) contribution coming from the parametric boundary (source) term. We show that for all small parameter values $λ>0$, the problem has at least five nontrivial smooth solutions, four of constant sign and one nodal. We also produce extremal constant sign solutions and determine their monotonicity and continuity properties as the parameter $λ>0$ varies. In the sem…
We establish uniqueness results for quasilinear elliptic problems through the criterion recently provided in \cite{DFMST}. We apply it to generalized $p$-Laplacian subhomogeneous problems that may admit multiple nontrivial nonnegative solutions. Based on a generalized hidden convexity result, we show that uniqueness holds among strongly positive solutions and nonnegative global minimizers. Problems involving nonhomogeneous operators as the so-called $(p,r)$-Laplacian are also treated.
The paper concerns boundary value problems for general nonautonomous first order quasilinear hyperbolic systems in a strip. We construct small global classical solutions, assuming that the right hand sides are small. In the case that all data of the quasilinear problem are almost periodic, we prove that the bounded solution is also almost periodic. For the nonhomogeneous version of a linearized problem, we provide stable dissipativity conditions ensuring a unique bounded continuous solution for any smooth right-hand sides. In the autonomous case, this solution is two times continuously differe…
We study existence of solutions for a boundary degenerate (or singular) quasilinear equation in a smooth bounded domain under Dirichlet boundary conditions. We consider a weighted $p-${L}aplacian operator with a coefficient that is {locally bounded inside the domain and satisfying certain additional integrability assumptions}. Our main result applies for boundary value problems involving continuous non-linearities having no growth restriction, but provided the existence of a sub and a supersolution is guaranteed. As an application, we present an existence result for a boundary value pro\-blem …
This paper studies a new gradient regularity in Lorentz spaces for solutions to a class of quasilinear divergence form elliptic equations with nonhomogeneous Dirichlet boundary conditions: \begin{align*} \begin{cases} div(A(x,\nabla u)) &= \ div(|F|^{p-2}F) \quad \text{in} \ \ Ω, \\ \hspace{1.2cm} u &=\ σ\qquad \qquad \qquad \text{on} \ \ \partial Ω. \end{cases} \end{align*} where $Ω\subset \mathbb{R}^n$ ($n \ge 2$), the nonlinearity $A$ is a monotone Carathéodory vector valued function defined on $W^{1,p}_0(Ω)$ for $p>1$ and the $p$-capacity uniform thickness condition is imposed on the compl…
We study quasilinear elliptic equations of the type $$-Δ_pu=σ\, u^q \quad \text{in} \, \, \, \mathbb{R}^n,$$ where $Δ_p u=\nabla \cdot(\nabla u |\nabla u|^{p-2})$ is the $p$-Laplacian (or a more general $\mathcal{A}$-Laplace operator $\text{div} \, \mathcal{A}(x, \nabla u)$), $0<q < p-1$, and $σ\ge 0$ is an arbitrary locally integrable function or measure on $\mathbb{R}^n$. We obtain necessary and sufficient conditions for the existence of positive solutions (not necessarily bounded) which satisfy global pointwise estimates of Brezis-Kamin type given in terms of Wolff potentials. Similar probl…
The existence of positive solutions is considered for the Dirichlet problem \[ \left\{ \begin{array} [c]{rcll}% -Δ_{p}u & = & λω_{1}(x)\left\vert u\right\vert ^{q-2}% u+βω_{2}(x)\left\vert u\right\vert ^{a-1}u|\nabla u|^{b} & \text{in }Ω\\ u & = & 0 & \text{on }\partialΩ, \end{array} \right. \] where $λ$ and $β$ are positive parameters, $a$ and $b$ are positive constants satisfying $a+b\leq p-1$, $ω_{1}(x)$ and $ω_{2}(x)$ are nonnegative weights and $1<q\leq p$. The homogeneous case $q=p$ is handled by making $q\rightarrow p^{-}$ in the sublinear case $1<q<p,$ which is based on the sub- and su…
In this paper we prove the boundedness and Hölder continuity of quasilinear elliptic problems involving variable exponents for a homogeneous Dirichlet and a nonhomogeneous Neumann boundary condition, respectively. The novelty of our work is the fact that we allow critical growth even on the boundary and so we close the gap in the papers of Fan-Zhao [Nonlinear Anal. 36 (1999), no. 3, 295--318.] and Winkert-Zacher [Discrete Contin. Dyn. Syst. Ser. S 5 (2012), no. 4, 865--878.] in which the critical cases are excluded. Our approach is based on a modified version of De Giorgi's iteration technique…
We study generalized solutions of the nonlinear wave equation $$u_{tt}-u_{ss}=au^+-bu^-+p(s,t,u),$$ with periodic conditions in $t$ and homogeneous Dirichlet conditions in $s$, under the assumption that the ratio of the period to the length of the interval is two. When $p\equiv 0$ and $λ$ is a nonzero eigenvalue of the wave operator, we give a proof of the existence of two families of curves (which may coincide) in the Fučik spectrum intersecting at $(λ,λ)$. This result is known for some classes of self-adjoint operators (which does not cover the situation we consider here), but in a smaller r…
We obtain necessary and sufficient conditions for the existence of a positive finite energy solution to the inhomogeneous quasilinear elliptic equation \[ -Δ_{p} u = σu^{q} + μ\quad \text{on} \;\; \mathbb{R}^n \] in the sub-natural growth case $0<q<p-1$, where $Δ_{p}$ ($1<p<\infty$) is the $p$-Laplacian, and $σ$, $μ$ are positive Borel measures on $\mathbb{R}^n$. Uniqueness of such a solution is established as well. Similar inhomogeneous problems in the sublinear case $0<q<1$ are treated for the fractional Laplace operator $(-Δ)^α$ in place of $-Δ_{p}$, on $\mathbb{R}^n$ for $0<α<\frac{n}{2}$,…
In this paper, we prove the existence of infinitely many solutions for a class of quasilinear elliptic $m(x)$-polyharmonic Kirchhoff equations where the nonlinear function has a quasicritical growth at infinity and without assuming the Ambrosetti and Rabinowitz type condition. The new aspect consists in employing the notion of a Schauder basis to verify the geometry of the symmetric mountain pass theorem. Furthermore, for the case $m(x)\equiv Const$, we introduce a positive quantity $λ_M$ similar to the first eigenvalue of the $m$-polyharmonic operator to find a mountain pass solution, and als…
We give necessary and sufficient conditions for the existence of weak solutions to the model equation $$-Δ_p u=σ\, u^q \quad \text{on} \, \, \, \R^n,$$ in the case $0<q<p-1$, where $σ\ge 0$ is an arbitrary locally integrable function, or measure, and $Δ_pu={\rm div}(\nabla u|\nabla u|^{p-2})$ is the $p$-Laplacian. Sharp global pointwise estimates and regularity properties of solutions are obtained as well. As a consequence, we characterize the solvability of the equation $$-Δ_p v \, = {b} \, \frac {|\nabla v|^{p}}{v} + σ\quad \text{on} \, \, \, \R^n,$$ where ${b}>0$. These results are new even…
