Existence of multiple nontrivial solutions for semilinear elliptic problems

AbstractThe aim of this paper is to prove the existence of multiple nontrivial solutions to a semilinear elliptic problem at resonance. The proofs used here are based on combining the Morse theory and the minimax methods

On a uniform estimate for positive solutions of semilinear elliptic equations

We consider semilinear elliptic equations with Dirichlet boundary conditions in a Lipschitz, possibly unbounded, domain. Under suitable assumptions on the nonlinearity, we deduce a condition on the size of the domain that implies the existence of a positive solution satisfying a uniform pointwise estimate. Here, uniform means that the estimate is independent of the domain. Besides of its simplicity, the main advantage of our approach is that we can remove a restrictive assumption on the nonlinearity that was imposed in a recent paper. Moreover, we can remove a non-degeneracy condition that was assumed in the latter paper

Nonzero radial solutions for a class of elliptic systems with nonlocal BCs on annular domains

We provide new results on the existence, non-existence, localization and multiplicity of nontrivial solutions for systems of Hammerstein integral equations. Some of the criteria involve a comparison with the spectral radii of some associated linear operators. We apply our results to prove the existence of multiple nonzero radial solutions for some systems of elliptic boundary value problems subject to nonlocal boundary conditions. Our approach is topological and relies on the classical fixed point index. We present an example to illustrate our theory.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1404.139

Multiplicity of nontrivial solutions for elliptic equations with nonsmooth potential and resonance at higher eigenvalues

We consider a semilinear elliptic equation with a nonsmooth, locally \hbox{Lipschitz} potential function (hemivariational inequality). Our hypotheses permit double resonance at infinity and at zero (double-double resonance situation). Our approach is based on the nonsmooth critical point theory for locally Lipschitz functionals and uses an abstract multiplicity result under local linking and an extension of the Castro--Lazer--Thews reduction method to a nonsmooth setting, which we develop here using tools from nonsmooth analysis.Comment: 23 page

On least Energy Solutions to A Semilinear Elliptic Equation in A Strip

We consider the following semilinear elliptic equation on a strip: $$\left\{{array}{l} \Delta u-u + u^p=0 \ {in} \ \R^{N-1} \times (0, L), u>0, \frac{\partial u}{\partial \nu}=0 \ {on} \ \partial (\R^{N-1} \times (0, L)) {array} \right.$$ where $1< p\leq \frac{N+2}{N-2}$. When $1<p 0$ such that for $L \leq L_{*}$, the least energy solution is trivial, i.e., doesn't depend on $x_N$, and for $L >L_{*}$, the least energy solution is nontrivial. When $N \geq 4, p=\frac{N+2}{N-2}$, it is shown that there are two numbers $L_{*}<L_{**}$ such that the least energy solution is trivial when $L \leq L_{*}$, the least energy solution is nontrivial when $L \in (L_{*}, L_{**}]$, and the least energy solution does not exist when $L >L_{**}$. A connection with Delaunay surfaces in CMC theory is also made.Comment: typos corrected and uniqueness adde

Robin problems with a general potential and a superlinear reaction

We consider semilinear Robin problems driven by the negative Laplacian plus an indefinite potential and with a superlinear reaction term which need not satisfy the Ambrosetti-Rabinowitz condition. We prove existence and multiplicity theorems (producing also an infinity of smooth solutions) using variational tools, truncation and perturbation techniques and Morse theory (critical groups)