Asymmetric (p, 2)-equations with double resonance

We consider a nonlinear Dirichlet elliptic problem driven by the sum of a p-Laplacian and a Laplacian [a (p, 2)-equation] and with a reaction term, which is superlinear in the positive direction (without satisfying the Ambrosetti-Rabinowitz condition) and sublinear resonant in the negative direction. Resonance can also occur asymptotically at zero. So, we have a double resonance situation. Using variational methods based on the critical point theory and Morse theory (critical groups), we establish the existence of at least three nontrivial smooth solutions

Existence and multiplicity of solutions for noncoercive neumann problems with p-Laplacian

We consider a nonlinear Neumann elliptic equation driven by the $p$-Laplacian and a Carathéodory perturbation. The energy functional of the problem need not be coercive. Using variational methods we prove an existence theorem and a multiplicity theorem, producing two nontrivial smooth solutions. Our formulation incorporates strongly resonant equations.

Nonlinear, nonhomogeneous periodic problems with no growth control on the reaction

We consider a nonlinear periodic problem driven by a nonhomogeneous differential operator, which includes as a particular case the scalar p-Laplacian. We assume that the reaction is a Carathéodory function which admits time-dependent zeros of constant sign. No growth control near ±∞ is imposed on the reaction. Using variational methods coupled with suitable truncation and comparison techniques, we prove two multiplicity theorems providing sign information for all the solutions

Positive solutions for nonlinear singular superlinear elliptic equations

We consider a nonlinear nonparametric elliptic Dirichlet problem driven by the p-Laplacian and reaction containing a singular term and a (p−1)-superlinear perturbation. Using variational tools together with suitable truncation and comparison techniques we produce two positive, smooth, ordered solutions

Nonlinear Dirichlet problems with the combined effects of singular and convection terms

We consider a nonlinear Dirichlet elliptic problem driven by the p-Laplacian. In the reaction term of the equation we have the combined effects of a singular term and a convection term. Using a topological approach based on the fixed point theory (the Leray-Schauder alternative principle), we prove the existence of a positive smooth solution

Least energy sign-changing solution for degenerate Kirchhoff double phase problems

In this paper we study the following nonlocal Dirichlet equation of double phase type \begin{align*} -\psi \left [ \int_\Omega \left ( \frac{|\nabla u |^p}{p} + \mu(x) \frac{|\nabla u|^q}{q}\right)\,\mathrm{d} x\right] \mathcal{G}(u) = f(x,u)\quad \text{in } \Omega, \quad u = 0\quad \text{on } \partial\Omega, \end{align*} where $\mathcal{G}$ is the double phase operator given by \begin{align*} \mathcal{G}(u)=\operatorname{div} \left(|\nabla u|^{p-2}\nabla u + \mu(x) |\nabla u|^{q-2}\nabla u \right)\quad u\in W^{1,\mathcal{H}}_0(\Omega), \end{align*} $\Omega\subseteq \mathbb{R}^N$, $N\geq 2$, is a bounded domain with Lipschitz boundary $\partial\Omega$, $1<p<N$, $p<q<p^*=\frac{Np}{N-p}$, $0 \leq \mu(\cdot)\in L^\infty(\Omega)$, $\psi(s) = a_0 + b_0 s^{\vartheta-1}$ for $s\in\mathbb{R}$, with $a_0 \geq 0$, $b_0>0$ and $\vartheta \geq 1$, and $f\colon\Omega\times\mathbb{R}\to\mathbb{R}$ is a Carath\'{e}odory function that grows superlinearly and subcritically. We prove the existence of two constant sign solutions (one is positive, the other one negative) and of a sign-changing solution which has exactly two nodal domains and which turns out to be a least energy sign-changing solution of the problem above. Our proofs are based on variational tools in combination with the quantitative deformation lemma and the Poincar\'{e}-Miranda existence theorem