Existence and multiplicity for elliptic p-Laplacian problems with critical growth in the gradient

Abstract

We consider the boundary value problem

where $\Omega \subset \mathbb {R}^{N}$, $N \ge 2$, is a bounded domain with smooth boundary. We assume $c,\, h \in L^q(\Omega )$ for some $q > \max \{N/p,1\}$ with $c \gneqq 0$ and $\mu \in L^{\infty }(\Omega )$. We prove existence and uniqueness results in the coercive case $\lambda \le 0$ and existence and multiplicity results in the non-coercive case $\lambda >0$. Also, considering stronger assumptions on the coefficients, we clarify the structure of the set of solutions in the non-coercive case.

Appendix A: Sufficient condition

Lemma A.1

Given $f \in L^r(\Omega )$, $r > \max \{ N/p,1\}$ if $p\not =N$ and $1<r<\infty$ if $p=N$, let us consider

$$\begin{aligned} E_f(u) = \left( \int _{\Omega } \left( |\nabla u|^p - f(x)|u|^p\right) \, dx \right) ^{\frac{1}{p}} \end{aligned}$$

for an arbitrary $u \in W_0^{1,p}(\Omega )$. It follows that:

• (i) If $1< p < N$ and $\Vert f^{+}\Vert _{N/p} < S_N$, $E_f(u)$ is an equivalent norm in $W_0^{1,p}(\Omega )$.

• (ii) If $p = N$ and $\Vert f^{+}\Vert _{r} < S_{N,r}$, $E_f(u)$ is an equivalent norm in $W_0^{1,p}(\Omega )$.

• (iii) If $p > N$ and $\Vert f^{+}\Vert _1 < S_N$, $E_f(u)$ is an equivalent norm in $W_0^{1,p}(\Omega )$.

where, for $p \ne N$, $S_N$ denotes the optimal constant in the Sobolev inequality, i.e.

$$\begin{aligned} S_N = \inf \big \{ \Vert \nabla u\Vert _p^p: u \in W_0^{1,p}(\Omega ), \Vert u\Vert _{p^{*}} = 1 \big \}, \end{aligned}$$

and, for $p = N$,

$$\begin{aligned} S_{N,r} = \inf \left\{ \Vert \nabla u\Vert _p^p: u \in W_0^{1,p}(\Omega ), \Vert u\Vert _{\frac{Nr}{r-1}} = 1 \right\} . \end{aligned}$$

Proof

We give the proof for $1< p < N$. The other cases can be done in the same way. First of all, by applying Hölder and Sobolev’s inequalities, observe that, for any $h \in L^{\frac{N}{p}}(\Omega )$, it follows that

$$\begin{aligned} \int _{\Omega } h(x) |u|^p dx \le \Vert h\Vert _{\frac{N}{p}}\Vert u\Vert _{p^{*}}^p \le \frac{1}{S_N}\Vert h\Vert _{\frac{N}{p}}\Vert \nabla u\Vert _p^p. \end{aligned}$$

On the one hand, by using this inequality, observe that

$$\begin{aligned} \int _{\Omega } \left( |\nabla u|^p - f(x)|u|^p \right) \, dx \le \Vert u\Vert ^p \left( 1 + \frac{\Vert f\Vert _{\frac{N}{p}}}{S_N} \right) . \end{aligned}$$

On the other hand, following the same argument, we obtain that

$$\begin{aligned} \int _{\Omega } \left( |\nabla u|^p - f(x)|u|^p \right) \,dx\ge & {} \int _{\Omega } \left( |\nabla u|^p - f^{+}(x)|u|^p\right) \,dx \\\ge & {} \Vert u\Vert ^p\left( 1 - \frac{\Vert f^{+}\Vert _{\frac{N}{p}}}{S_N} \right) = A \Vert u\Vert ^p \end{aligned}$$

with $A > 0$ since $\Vert f^{+}\Vert _{\frac{N}{p}} < S_N$. The result follows. $\square$

As an immediate Corollary, we have a sufficient condition to ensure that $m_p > 0$.

Corollary A.2

Recall that $m_p$ is defined by (1.1). Under the assumptions ($A_{1}$), it follows that:

• (i) If $1< p < N$, then $\Vert h^{+}\Vert _{N/p} < \left( \frac{p-1}{\mu } \right) ^{p-1} S_N$ implies $m_p > 0$.

• (ii) If $p = N$, then $\Vert h^{+}\Vert _q < \left( \frac{p-1}{\mu } \right) ^{p-1} S_{N,q}$ implies $m_p > 0$.

• (iii) If $p > N$, then $\Vert h^{+}\Vert _1 < \left( \frac{p-1}{\mu } \right) ^{p-1} S_N$ implies $m_p > 0$.