American Institute of Mathematical Sciences

We consider a nonlinear, nonhomogeneous Dirichlet problem driven by the sum of a \begin{document}$p$\end{document}-Laplacian (\begin{document}$2<p$\end{document}) and a Laplacian. The reaction term is a Carathéodory function \begin{document}$f(z,x)$\end{document} which is resonant with respect to the principal eigenvalue of (\begin{document}$-\Delta_p,\, W^{1,p}_0(\Omega)$\end{document}). Using variational methods combined with truncation and comparison techniques and Morse theory (critical groups) we prove the existence of three nontrivial smooth solutions all with sign information and under three different conditions concerning the behavior of \begin{document}$f(z,\cdot)$\end{document} near zero. By strengthening the regularity of \begin{document}$f(z,\cdot)$\end{document}, we are able to generate a second nodal solution for a total of four nontrivial smooth solutions.

In this paper, a class of nonlinear differential boundary value problems with variable exponent is investigated. The existence of at least one non-zero solution is established, without assuming on the nonlinear term any condition either at zero or at infinity. The approach is developed within the framework of the Orlicz-Sobolev spaces with variable exponent and it is based on a local minimum theorem for differentiable functions.

In this paper, we consider the existence and further qualitative properties of solutions of the Dirichlet problem to quasilinear multi-valued elliptic equations with measures of the form

\begin{document}$Au + G(\cdot,u) \ni f,$\end{document}

where \begin{document}$A$\end{document} is a second order elliptic operator of Leray-Lions type and \begin{document}$f\in \mathcal M_b(\Omega)$\end{document} is a given Radon measure on a bounded domain \begin{document}$\Omega\subset \mathbb R^N$\end{document}. The lower order term \begin{document}$s\mapsto G(\cdot,s)$\end{document} is assumed to be a multi-valued upper semicontinuous function, which includes Clarke's gradient \begin{document}$s\mapsto \partial j(\cdot,s)$\end{document} of some locally Lipschitz function \begin{document}$s\mapsto j(\cdot,s)$\end{document} as a special case. Our main goals and the novelties of this paper are as follows: First, we develop an existence theory for the above multi-valued elliptic problem with measure right-hand side. Second, we propose concepts of sub-supersolutions for this problem and establish an existence and comparison principle. Third, we topologically characterize the solution set enclosed by sub-supersolutions.

In this paper we survey, complete and refine some recent results concerning the Dirichlet problem for the prescribed anisotropic mean curvature equation

\begin{document}$\begin{equation*}{\rm{ -div}}\left({\nabla u}/{\sqrt{1 + |\nabla u|^2}}\right) = -au + {b}/{\sqrt{1 + |\nabla u|^2}},\end{equation*}$ \end{document}

in a bounded Lipschitz domain \begin{document} $Ω \subset \mathbb{R}^N$ \end{document}, with \begin{document} $a,b>0$ \end{document} parameters. This equation appears in the description of the geometry of the human cornea, as well as in the modeling theory of capillarity phenomena for compressible fluids. Here we show how various techniques of nonlinear functional analysis can successfully be applied to derive a complete picture of the solvability patterns of the problem.

We study the positive subharmonic solutions to the second order nonlinear ordinary differential equation

\begin{document}$\begin{equation*}u'' + q(t) g(u) = 0,\end{equation*}$ \end{document}

where \begin{document} $g(u)$ \end{document} has superlinear growth both at zero and at infinity, and \begin{document} $q(t)$ \end{document} is a \begin{document} $T$ \end{document}-periodic sign-changing weight. Under the sharp mean value condition \begin{document} $\int_{0}^{T}{q\left( t \right)dt<0}$ \end{document}, combining Mawhin's coincidence degree theory with the Poincaré-Birkhoff fixed point theorem, we prove that there exist positive subharmonic solutions of order \begin{document} $k$ \end{document} for any large integer \begin{document} $k$ \end{document}. Moreover, when the negative part of \begin{document} $q(t)$ \end{document} is sufficiently large, using a topological approach still based on coincidence degree theory, we obtain the existence of positive subharmonics of order \begin{document} $k$ \end{document} for any integer \begin{document} $k≥2$ \end{document}.

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.

Existence and regularity results for quasilinear elliptic equations driven by \begin{document}$(p, q)$\end{document}-Laplacian and with gradient dependence are presented. A location principle for nodal (i.e., sign-changing) solutions is obtained by means of constant-sign solutions whose existence is also derived. Criteria for the existence of extremal solutions are finally established.

For the homogeneous Dirichlet problem involving a system of equations driven by \begin{document}$(p,q)$\end{document}-Laplacian operators and general gradient dependence we prove the existence of solutions in the ordered rectangle determined by a subsolution-supersolution. This extends the preceding results based on the method of subsolution-supersolution for systems of elliptic equations. Positive and negative solutions are obtained.

We study a semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential term. The nonlinearity \begin{document}$f(x, s)$\end{document} is a Carathéodory function which is asymptotically linear as \begin{document}$ s\to ± ∞$\end{document} and resonant. In fact we assume double resonance with respect to any nonprincipal, nonnegative spectral interval \begin{document}$ \left[ \hat{λ}_k, \hat{λ}_{k+1}\right]$\end{document}. Applying variational tools along with suitable truncation and perturbation techniques as well as Morse theory, we show that the problem has at least three nontrivial smooth solutions, two of constant sign.

We prove an Ambrosetti-Prodi type result for a Neumann problem associated to the equation \begin{document}$u''+f(x, u(x))=μ$\end{document} when the nonlinearity has the following form:\begin{document}$f(x, u):=a(x)g(u)-p(x)$\end{document}. The assumptions considered generalize the classical one, \begin{document}$f(x, u)\to+∞$\end{document} as \begin{document}$|u|\to+∞$\end{document}, without requiring any uniformity condition in \begin{document}$x$\end{document}. The multiplicity result which characterizes these kind of problems will be proved by means of the shooting method.