Existence of nodal solutions for quasilinear elliptic problems in RN\mathbb{R}^N

We prove the existence of one positive, one negative, and one sign-changing solution of a $p$-Laplacian equation on $\mathbb{R}^N$, with a $p$-superlinear subcritical term. Sign-changing solutions of quasilinear elliptic equations set on the whole of $\mathbb{R}^N$ have only been scarcely investigated in the literature. Our assumptions here are similar to those previously used by some authors in bounded domains, and our proof uses fairly elementary critical point theory, based on constraint minimization on the nodal Nehari set. The lack of compactness due to the unbounded domain is overcome by working in a suitable weighted Sobolev space.Comment: growth assumptions relaxed; main proof more detailed in this versio

Monotonicity and symmetry of singular solutions to quasilinear problems

We consider singular solutions to quasilinear elliptic equations under zero Dirichlet boundary condition. Under suitable assumptions on the nonlinearity we deduce symmetry and monotonicity properties of positive solutions via an improved moving plane procedure

Monotonicity in half-spaces of positive solutions to −Δpu=f(u)-\Delta_p u=f(u) in the case p>2p>2

We consider weak distributional solutions to the equation $-\Delta_pu=f(u)$ in half-spaces under zero Dirichlet boundary condition. We assume that the nonlinearity is positive and superlinear at zero. For $p>2$ (the case $1<p\leq2$ is already known) we prove that any positive solution is strictly monotone increasing in the direction orthogonal to the boundary of the half-space. As a consequence we deduce some Liouville type theorems for the Lane-Emden type equation. Furthermore any nonnegative solution turns out to be $C^{2,\alpha}$ smooth

A regularity result for the p-laplacian near uniform ellipticity

We consider weak solutions to a class of Dirichlet boundary value problems invloving the $p$-Laplace operator, and prove that the second weak derivatives are in $L^{q}$ with $q$ as large as it is desirable, provided $p$ is sufficiently close to $p_0=2$. We show that this phenomenon is driven by the classical Calder\'on-Zygmund constant. As a byproduct of our analysis we show that $C^{1,\alpha}$ regularity improves up to $C^{1,1^-}$, when p is close enough to 2. This result we believe it is particularly interesting in higher dimensions $n>2,$ when optimal $C^{1,\alpha}$ regularity is related to the optimal regularity of $p$-harmonic mappings, which is still open

Multiple solutions to a Caffarelli-Kohn-Nirenberg type equation with asymptotically linear term

In this paper, we study the existence of multiple solutions to a Caffarelli-Kohn-Nirenberg type equation with asymptotically linear term at infinity. In this case, the well-known Ambrosetti-Rabinowtz type condition doesn't hold, hence it is difficult to verify the classical (PS)$_c$ condition. To overcome this difficulty, we use an equivalent version of Cerami's condition, which allows the more general existence result.Comment: 17 pages, no figur

Radial symmetry for a quasilinear elliptic equation with a critical Sobolev growth and Hardy potential

We consider weak positive solutions to the critical $p$-Laplace equation with Hardy potential in $\mathbb R^N$ $$-\Delta_p u -\frac{\gamma}{|x|^p} u^{p-1}=u^{p^*-1}$$ where $1<p<N$, $0\le \gamma <\left(\frac{N-p}{p}\right)^p$ and $p^*=\frac{Np}{N-p}$. The main result is to show that all the solutions in $\mathcal D^{1, p}(\mathbb R^N)$ are radial and radially decreasing about the origin

Ground state alternative for p-Laplacian with potential term

Let $\Omega$ be a domain in $\mathbb{R}^d$, $d\geq 2$, and $1<p<\infty$. Fix $V\in L_{\mathrm{loc}}^\infty(\Omega)$. Consider the functional $Q$ and its G\^{a}teaux derivative $Q^\prime$ given by Q(u):=\int_\Omega (|\nabla u|^p+V|u|^p)\dx, \frac{1}{p}Q^\prime (u):=-\nabla\cdot(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2}u. If $Q\ge 0$ on $C_0^{\infty}(\Omega)$, then either there is a positive continuous function $W$ such that $\int W|u|^p \mathrm{d}x\leq Q(u)$ for all $u\in C_0^{\infty}(\Omega)$, or there is a sequence $u_k\in C_0^{\infty}(\Omega)$ and a function $v>0$ satisfying $Q^\prime (v)=0$, such that $Q(u_k)\to 0$, and $u_k\to v$ in $L^p_\mathrm{loc}(\Omega$). In the latter case, $v$ is (up to a multiplicative constant) the unique positive supersolution of the equation $Q^\prime (u)=0$ in $\Omega$, and one has for $Q$ an inequality of Poincar\'e type: there exists a positive continuous function $W$ such that for every $\psi\in C_0^\infty(\Omega)$ satisfying $\int \psi v \mathrm{d}x \neq 0$ there exists a constant $C>0$ such that $C^{-1}\int W|u|^p \mathrm{d}x\le Q(u)+C|\int u \psi \mathrm{d}x|^p$ for all $u\in C_0^\infty(\Omega)$. As a consequence, we prove positivity properties for the quasilinear operator $Q^\prime$ that are known to hold for general subcritical resp. critical second-order linear elliptic operators.Comment: Corrected error in Appendi

On the Harnack inequality for quasilinear elliptic equations with a first order term

We consider weak solutions to $$-\Delta_pu+a(x,u)|\nabla u|^q=f(x,u),$$ with $p>1$, $q\geq\max\,\{p-1,1\}$. We exploit the Moser iteration technique to prove a Harnack comparison inequality for $C^1$ weak solutions. As a consequence we deduce a strong comparison principle

A characterization of fast decaying solutions for quasilinear and Wolff type systems with singular coefficients

This paper examines the decay properties of positive solutions for a family of fully nonlinear systems of integral equations containing Wolf potentials and Hardy weights. This class of systems includes examples which are closely related to the Euler-Lagrange equations for several classical inequalities such as the Hardy-Sobolev and Hardy-Littlewood-Sobolev inequalities. In particular, a complete characterization of the fast decaying ground states in terms of their integrability is provided in that bounded and fast decaying solutions are shown to be equivalent to the integrable solutions. In generating this characterization, additional properties for the integrable solutions, such as their boundedness and optimal integrability, are also established. Furthermore, analogous decay properties for systems of quasilinear equations of the weighted Lane-Emden type are also obtained.Comment: 26 page

Some recent results on the Dirichlet problem for (p,q)-Laplace equations

A short account of recent existence and multiplicity theorems on the Dirichlet problem for an elliptic equation with $(p,q)$-Laplacian in a bounded domain is performed. Both eigenvalue problems and different types of perturbation terms are briefly discussed. Special attention is paid to possibly coercive, resonant, subcritical, critical, or asymmetric right-hand sides.Comment: Comments welcom