In this paper we study nonlinear elliptic boundary value problems with monotone and nonmonotone multivalued nonlinearities. First we consider the case of monotone nonlinearities. In the first result we assume that the multivalued nonlinearity is defined on all $ℝ$. Assuming the existence of an upper and of a lower solution, we prove the existence of a solution between them. Also for a special version of the problem, we prove the existence of extremal solutions in the order interval formed by the upper...

In this paper, we study a nonlinear elliptic equation with critical exponent, invariant under the action of a subgroup G of the isometry group of a compact Riemannian manifold. We obtain some existence results of positive solutions of this equation, and under some assumptions on G, we show that we can solve this equation for supercritical exponents.

The paper is devoted to the solvability of a nonlinear elliptic problem in a plane multiply connected domain. On the inner components of its boundary Dirichlet conditions are known up to additive constants which have to be determined together with the sought solution so that the so-called trailing stagnation conditions are satisfied. The results have applications in the stream function solution of subsonic flows past groups of profiles or cascades of profiles.

In this paper existence and multiplicity of solutions of the elliptic problem $ℒu+{\lambda }_{1}u+\mu u^{+}v{u}^{-}+g(x,u)=f$ in $\Omega$$Bu=0$ on $\partial \Omega$, are discussed provided the parameters $\mu$ and $v$ are close to the first eigenvalue ${}_1$. The sufficient conditions...

We investigate some nonlinear elliptic problems of the form $${\Delta }^{2}v+\sigma \left(x\right)v=h(x,v)\text{in}\Omega ,v=\Delta v=0\text{on}\partial \Omega ,\left(\mathrm{P}\right)$$ where $\Omega$ is a regular bounded domain in ${ℝ}^N$, $N\ge 2$, $\sigma \left(x\right)$ a positive function in $L^{\infty }\left(\Omega \right)$, and the nonlinearity $h(x,t)$ is indefinite. We prove the existence of solutions to the problem (P) when the function $h(x,t)$ is asymptotically linear at infinity by using variational method but without the Ambrosetti-Rabinowitz condition. Also, we consider the case when the nonlinearities are superlinear and subcritical.

The nonlinear eigenvalue problem for p-Laplacian $$\left\{\begin{array}{c}-{div\left(a\left(x\right)\right|\nabla u|}^{p-2}{\nabla u)=\lambda g\left(x\right)|u|}^{p-2}u\text{in}{ℝ}^N,u>0\text{in}{ℝ}^N,\underset{\left|x\right|\to \infty }{lim}u\left(x\right)=0,\hfill \end{array}\right.$$ is considered. We assume that $1<p<N$ and that $g$ is indefinite weight function. The existence and $C^{1,\alpha }$-regularity of the weak solution is proved.