A concave–convex problem with a variable operator

Abstract

We study the following elliptic problem $-A(u) = \lambda u^q$ with Dirichlet boundary conditions, where $A(u) (x) = \Delta u (x) \chi _{D_1} (x)+ \Delta _p u(x) \chi _{D_2}(x)$ is the Laplacian in one part of the domain, $D_1$, and the p-Laplacian (with $p>2$) in the rest of the domain, $D_2$. We show that this problem exhibits a concave–convex nature for $1<q<p-1$. In fact, we prove that there exists a positive value $\lambda ^*$ such that the problem has no positive solution for $\lambda > \lambda ^*$ and a minimal positive solution for $0<\lambda < \lambda ^*$. If in addition we assume that p is subcritical, that is, $p<2N/(N-2)$ then there are at least two positive solutions for almost every $0<\lambda < \lambda ^*$, the first one (that exists for all $0<\lambda < \lambda ^*$) is obtained minimizing a suitable functional and the second one (that is proven to exist for almost every $0<\lambda < \lambda ^*$) comes from an appropriate (and delicate) mountain pass argument.

Appendix

We include here a proof of the fact that Palais–Smale sequences are bounded when we assume an Ambrosetti–Rabinowitz type condition with $\kappa >p$. We remark again that this condition does not hold here, but we include this simple computation for the sake of completeness.

Lemma 4.8

Consider the functional $F:\mathcal {W}(\Omega )\rightarrow {\mathbb {R}}$ defined as follows:

$$\begin{aligned} F(u)= \int _{D_1} \frac{|\nabla u|^2}{2} \, dx + \int _{D_2} \frac{|\nabla u|^p}{p} \, dx - \lambda \int _{\Omega } H(x,u(x)) \, dx, \end{aligned}$$

with H such that there exists $\kappa >p$ satisfying

$$\begin{aligned} 0\le \kappa H(x,s)\le s h(x,s), \qquad s \ge 0, \, x \in \Omega , \end{aligned}$$

(40)

where $H(x,s)=\int _0^sh(x,t)dt$.

Then, Palais–Smale sequences for F are bounded.

Proof

Let $\{u_n\}\subset \mathcal {W}(\Omega )$ be a Palais–Smale sequence. That is, $|F(u_n)|\le C$ and $F'(u_n)\rightarrow 0$ in $\mathcal {W}(\Omega )'$. Then

$$\begin{aligned} C&\ge \int _{D_1} \frac{|\nabla u_n|^2}{2} + \int _{D_2} \frac{|\nabla u_n|^p}{p} - \lambda \int _{\Omega } H(x,u_n) \, dx, \\&\ge \int _{D_1} \frac{|\nabla u_n|^2}{2} + \int _{D_2} \frac{|\nabla u_n|^p}{p} -\frac{\lambda }{\kappa }\int _{\Omega }u_n h(x,u_n)dx \\&=\left( \frac{1}{2}-\frac{1}{\kappa } \right) \int _{D_1} |\nabla u_n|^2 + \left( \frac{1}{p}-\frac{1}{\kappa } \right) \int _{D_2} |\nabla u_n|^p+\frac{1}{\kappa } F'(u_n)(u_n) \\&\ge \left( \frac{1}{p}-\frac{1}{\kappa } \right) \left( \int _{D_1} |\nabla u_n|^2+\int _{D_2} |\nabla u_n|^p \right) -\frac{\varepsilon _n}{\kappa }[u_n]_{\mathcal {W}(\Omega )}, \end{aligned}$$

where $\varepsilon _n \rightarrow 0$. This leads to the boundedness of $\{u_n\}$ in $\mathcal {W}(\Omega )$. $\square$

We remark that the condition (40) can be relaxed imposing the inequality for $|s|\ge R>0$.