In the present paper we deal with a quasi-linear elliptic equation depending on a sublinear nonlinearity involving the gradient. We prove the existence of a nontrivial nodal solution employing the theory of invariant sets of descending flow together with sub-supersolution techniques, gradient regularity arguments, strong comparison principle for the $p$-Laplace operator. The same conclusion is obtained for an eigenvalue problem under a different set of assumptions.

In this paper, we study weak solutions, possibly unbounded and sign-changing, to the double phase problem

\begin{equation*}-\text{div} (|\nabla u|^{p-2} \nabla u + w(x)|\nabla u|^{q-2} \nabla u) = \left(\frac{1}{|x|^{N-\mu}}*f|u|^r\right) f(x)|u|^{r-2}u \quad\text{in}\ \mathbb{R}^N,\end{equation*}

$q\ge p\ge2$

$0 \lt \mu \lt N$

$w,f \in L^1_{\rm loc}(\mathbb{R}^N)$

$w(x) \le C_1|x|^a$

$f(x) \ge C_2|x|^b$

$|x| \gt R_0$

$R_0,C_1,C_2 \gt 0$

$a,b\in\mathbb{R}$

$\mathbb{R}^N$

The main purpose of this paper is to capture the asymptotic behaviour for solutions to a class of nonlinear elliptic and parabolic equations with the anisotropic weights consisting of two power-type weights of different dimensions near the degenerate or singular point, especially covering the weighted p-Laplace equations and weighted fast diffusion equations. As a consequence, we also establish the local Hölder estimates for their solutions in the presence of single power-type weights.

We study the asymptotic behaviour, as $p\to 1^+$, of the solutions of the following inhomogeneous Robin boundary value problem:P

\begin{equation*} \begin{cases} \displaystyle -\Delta_p u_p = f & \text{ in }\Omega,\\ \displaystyle |\nabla u_p|^{p-2}\nabla u_p\cdot \nu +\lambda |u_p|^{p-2}u_p = g & \text{ on } \partial\Omega, \end{cases} \end{equation*}

$\Omega$

$\mathbb {R}^{N}$

$\nu$

$\Delta _p v$

$p$

$p>1$

$f\in L^{N,\infty }(\Omega )$

$\lambda,\,g$

$\partial \Omega$

$\lambda \ge 0$

$p$

$1$

$p\to 1^+$

We establish some existence results for a class of critical $N$-Laplacian problems in a bounded domain in $\mathbb {R}^{N}$. In the absence of a suitable direct sum decomposition of the underlying Sobolev space to which the classical linking theorem can be applied, we use an abstract linking theorem based on the $\mathbb {Z}_2$-cohomological index to obtain a non-trivial critical point.

We show that the spectrum of the relativistic mean curvature operator on a bounded domain Ω ⊂ ℝN (N ⩾ 1) having smooth boundary, subject to the homogeneous Dirichlet boundary condition, is exactly the interval (λ1(2), ∞), where λ1(2) stands for the principal frequency of the Laplace operator in Ω.

We study the existence of positive ground state solutions for the following class of $(p,q)$-Laplacian coupled systems

$$\begin{eqnarray}\left\{\begin{array}{@{}lr@{}}-\unicode[STIX]{x1D6E5}_{p}u+a(x)|u|^{p-2}u=f(u)+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}(x)|u|^{\unicode[STIX]{x1D6FC}-2}u|v|^{\unicode[STIX]{x1D6FD}}, & x\in \mathbb{R}^{N},\\ -\unicode[STIX]{x1D6E5}_{q}v+b(x)|v|^{q-2}v=g(v)+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D706}(x)|v|^{\unicode[STIX]{x1D6FD}-2}v|u|^{\unicode[STIX]{x1D6FC}}, & x\in \mathbb{R}^{N},\end{array}\right.\end{eqnarray}$$

$1<p\leq q<N$

$\unicode[STIX]{x1D706}(x)$

$|\unicode[STIX]{x1D706}(x)|\leq \unicode[STIX]{x1D6FF}a(x)^{\unicode[STIX]{x1D6FC}/p}b(x)^{\unicode[STIX]{x1D6FD}/q}$

$\unicode[STIX]{x1D6FF}\in (0,1)$

$\unicode[STIX]{x1D6FC}/p+\unicode[STIX]{x1D6FD}/q=1$

Let n, $N \in {\open N}$ with $\Omega \subseteq {\open R}^n$ open. Given ${\rm H} \in C^2(\Omega \times {\open R}^N \times {\open R}^{Nn})$, we consider the functional 1

$${\rm E}_\infty (u,{\rm {\cal O}})\, : = \, \mathop {{\rm ess}\,\sup}\limits_{\rm {\cal O}} {\rm H}(\cdot, u,{\rm D}u),\quad u\in W_{{\rm loc}}^{1,\infty} (\Omega, {\open R}^N),\quad {\rm {\cal O}}{\Subset}\Omega.$$

$L^\infty $

2

$$\left\{\matrix{{\rm H}_{P}(\cdot, u, {\rm D}u)\, {\rm D}\left({\rm H}(\cdot, u, {\rm D} u)\right) = \, 0, \hfill \cr {\rm H}(\cdot, u, {\rm D} u) \, [\![{\rm H}_{P}(\cdot, u, {\rm D} u)]\!]^\bot \left({\rm Div}\left({\rm H}_{P}(\cdot, u, {\rm D} u)\right)- {\rm H}_{\eta}(\cdot, u, {\rm D} u)\right) = 0,\hfill}\right.$$

$[\![A]\!]^\bot := {\rm Proj}_{R(A)^\bot }$

${\cal D}$

The main result of this work is a new extension of the well-known inequality by Díaz and Saa which, in our case, involves an anisotropic operator, such as the p(x)-Laplacian, $\Delta _{p(x)}u\equiv {\rm div}( \vert \nabla u \vert ^{p(x)-2}\nabla u)$. Our present extension of this inequality enables us to establish several new results on the uniqueness of solutions and comparison principles for some anisotropic quasilinear elliptic equations. Our proofs take advantage of certain convexity properties of the energy functional associated with the p(x)-Laplacian.

Let 1 < p < +∞ and let Ω ⊂ ℝN be either a ball or an annulus. We continue the analysis started in [Boscaggin, Colasuonno, Noris, ESAIM Control Optim. Calc. Var. (2017)], concerning quasilinear Neumann problems of the type

$-\Delta _pu = f(u),\quad u > 0\,{\rm in }\,\Omega ,\quad \partial _\nu u = 0\,{\rm on }\,\partial \Omega .$

We study the asymptotic behaviour, as p → ∞, of the least energy solutions of the problem

$$\left\{ {\matrix{ {-(\Delta _p + \Delta _{q(p)})u = \lambda _p \vert u(x_u) \vert ^{p-2}u(x_u)\delta _{x_u}} & {{\rm in}} & \Omega \cr {u = 0} \hfill \hfill \hfill & {{\rm on}} & {\partial \Omega ,} \cr } } \right.$$

$\delta _{x_{u}}$

$$\mathop {\lim }\limits_{p\to \infty } \displaystyle{{q(p)} \over p} = Q\in \left\{ {\matrix{ {(0,1)} & {{\rm if}} & {N < q(p) < p} \cr {(1,\infty )} & {{\rm if}} & {N < p < q(p)} \cr } } \right.$$

$$\min \left\{ {\displaystyle{{{\rm \Vert }\nabla u{\rm \Vert }_\infty } \over {{\rm \Vert }u{\rm \Vert }_\infty }}:0 \ne u\in W^{1,\infty }(\Omega )\cap C_0(\bar{\Omega })} \right\} \les \mathop {\lim }\limits_{p\to \infty } (\lambda _p)^{1/p} < \infty .$$

Let Ω⊂ℝN (N≥2) be a bounded domain with smooth boundary and {pn} be a sequence of real numbers converging to+∞ as n→∞. For each integer n>1, we define the function $\varphi_{n}(t)=p_{n} \vert t \vert^{p_{n}-2}te^{ \vert t \vert^{p_{n}}}$, for all t∈ℝ, and we prove the existence of a unique nonnegative variational solution for the problem−div(((φn(|∇ u(x)|))/(|∇ u(x)|))∇ u(x))=φn(1), when x∈Ω, subject to the homogeneous Dirichlet boundary condition. Next, we establish the uniform convergence in Ω of the sequence of solutions for the above family of equations to the distance function to the boundary of Ω. Our result complements the earlier developments on the topic obtained by Payne and Philippin [26], Kawohl [21], Bhattacharya, DiBenedetto and Manfredi [2], Perez-Llanos and Rossi [27] and Bocea and Mihăilescu [4].

We consider weak solutions to

with p > 1, q ≥ max{p − 1, 1}. We exploit the Moser iteration technique to prove a Harnack comparison inequality for C1 weak solutions. As a consequence we deduce a strong comparison principle.

We investigate the following nonlinear Neumann boundary-value problem with associated p(x)-Laplace-type operator

where the function φ(x, v) is of type |v|p(x)−2v with continuous function p: → (1,∞) and both f : Ω × ℝ → ℝ and g : ∂Ω × ℝ → ℝ satisfy a Carathéodory condition. We first show the existence of infinitely many weak solutions for the Neumann problems using the Fountain theorem with the Cerami condition but without the Ambrosetti and Rabinowitz condition. Next, we give a result on the existence of a sequence of weak solutions for problem (P) converging to 0 in L∞-norm by employing De Giorgi's iteration and the localization method under suitable conditions.

The eigenvalue problem $-{{\Delta }_{p}}u-{{\Delta }_{q}}u=\lambda {{\left| u \right|}^{q-2}}u$ with $p\,\in \,\left( 1,\,\infty \right),\,q\,\in \,\left( 2,\,\infty \right),\,p\ne \,q$ subject to the corresponding homogeneous Neumann boundary condition is investigated on a bounded open set with smooth boundary from ${{\mathbb{R}}^{N}}$ with $N\,\ge \,2$. A careful analysis of this problem leads us to a complete description of the set of eigenvalues as being a precise interval $\left( {{\lambda }_{1,}}+\infty \right)$ plus an isolated point $\lambda \,=\,0$. This comprehensive result is strongly related to our framework, which is complementary to the well-known case $p\,=\,q\,\ne \,2$ for which a full description of the set of eigenvalues is still unavailable.

We consider a nonlinear parametric elliptic equation driven by a nonhomogeneous differential operator with a logistic reaction of the superdiffusive type. Using variational methods coupled with suitable truncation and comparison techniques, we prove a bifurcation type result describing the set of positive solutions as the parameter varies.

In this paper we study a quasi-linear elliptic problem coupled with Dirichlet boundary conditions. We propose a new set of assumptions ensuring the existence of infinitely many solutions.

We prove uniqueness of positive solutions for the boundary value problem

\begin{equation*} \left\{ \begin{array}{l} -\Delta u=\lambda f(u)\text{ in }\Omega , \\ \ \ \ \ \ \ \ u=0\text{ on }\partial \Omega , \end{array} \right. \end{equation*}

$\mathbb{R}$

We establish interior and trace embedding results for Sobolev functions on a class of bounded non-smooth domains. Also, we define the corresponding generalized Maz'ya spaces of variable exponent, and obtain embedding results similar as in the constant case. Some relations between the variable exponent Maz'ya spaces and the variable exponent Sobolev spaces are also achieved. At the end, we give an application of the previous results for the well-posedness of a class of quasi-linear equations with variable exponent.

In the present paper we deal with the existence of multiple solutions for a quasilinear elliptic problem involving an arbitrary perturbation. Our approach, based on an abstract result of Ricceri, combines truncation arguments with Moser-type iteration technique.